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SEMI-EQUATIONAL THEORIES

Published online by Cambridge University Press:  10 May 2023

ARTEM CHERNIKOV*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, LOS ANGELES LOS ANGELES, CA 90095, USA E-mail: alexmennen@math.ucla.edu
ALEX MENNEN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, LOS ANGELES LOS ANGELES, CA 90095, USA E-mail: alexmennen@math.ucla.edu
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Abstract

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

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© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

1 Introduction

Equations and equational theories were introduced by Srour [Reference Srour38Reference Srour40] in order to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well-behaved class of “closed” sets among the definable sets, by analogy to the algebraic sets among the constructible ones in algebraically closed fields. We recall the definition:

Definition 1.1.

  1. (1) A partitioned formula $\varphi (x,y)$ , with $x,y$ tuples of variables, is an equation (with respect to a first-order theory T) if there do not exist $\mathcal {M} \models T$ and tuples $\left ( a_i, b_i : i \in \omega \right )$ in $\mathcal {M}$ such that $\mathcal {M} \models \varphi \left (a_{i},b_{j}\right )$ for all $j<i$ and $\mathcal {M} \models \neg \varphi \left (a_{i},b_{i}\right )$ for all i.

  2. (2) A theory T is equational if every formula $\varphi (x,y)$ , with $x,y$ arbitrary finite tuples of variables, is equivalent in T to a Boolean combination of finitely many equations $\varphi _1(x,y), \ldots , \varphi _n(x,y)$ .

It is immediate from the definition that every equational theory is stable. Structural properties of equational theories in relation to forking and stability theory are studied in [Reference Hrushovski and Srour19Reference Junker and Lascar22, Reference Pillay and Srour33]. Many natural stable theories are equational; [Reference Hrushovski and Srour19] provided the first example of a stable non-equational theory. More recently it was demonstrated that the stable theory of non-abelian free groups is not equational [Reference Müller and Sklinos28, Reference Sela34], and further examples are constructed in [Reference Martin-Pizarro and Ziegler25]. It is demonstrated in [Reference Martin-Pizarro and Ziegler24] that all theories of separably closed fields are equational (generalizing earlier work of Srour [Reference Srour37]). See also [Reference O’Hara29] for an accessible introduction to equationality.

We propose a generalization of equations and equational theories to the larger class of NIP theories (see Section 1.2 for a more detailed discussion):

Definition 1.2. Let T be a first-order theory and $\mathbb {M} \models T$ a monster model of T.

  1. (1) A partitioned formula $\varphi (x,y)$ is a semi-equation (in T) if there is no sequence $\left (a_{i},b_{i} : i \in \omega \right )$ with $a_i \in \mathbb {M}^{x}, b_i \in \mathbb {M}^{y}$ such that for all $i,j \in \omega $ , $\models \varphi \left (a_{i},b_{j}\right )\iff i\neq j$ .

  2. (2) A (partitioned) formula $\varphi \left (x,y\right )$ is a weak semi-equation if there are no $b \in \mathbb {M}^y$ and an ( $\emptyset $ -)indiscernible sequence $\left (a_{i} : i \in \mathbb {Z} \right )$ with $a_i \in \mathbb {M}^{x}$ such that the subsequence $\left (a_{i} : i \neq 0 \right )$ is indiscernible over b, $\models \varphi \left (a_{i},b\right )$ for all $i \neq 0$ , but $\models \neg \varphi \left (a_{0},b\right )$ .

  3. (3) A theory T is (weakly) semi-equational if every formula $\varphi (x,y) \in \mathcal {L}$ , with $x,y$ arbitrary finite tuples of variables, is a Boolean combination of finitely many (weak) semi-equations $\psi _1(x,y), \ldots , \psi _n(x,y) \in \mathcal {L}$ .

Semi-equations are in particular weak semi-equations, every weakly semi-equational theory is NIP, and in a stable theory all three notions coincide (see Proposition 2.10). Some parts of the basic theory of equations naturally generalize to (weak) semi-equations, but there are also some new phenomena and complications appearing outside of stability. In particular, weak semi-equationality provides a simultaneous generalization of equationality and distality, bringing out some curious parallels between those two notions (see Section 4). In this paper we develop the basic theory of (weak) semi-equations, and investigate (weak) semi-equationality in some examples. We view this as a first step, and a large number of questions remain open and can be found throughout the paper.

In Section 1.2 we provide some equivalent characterizations of (weak) semi-equationality in terms of indiscernibles. We discuss closure of (weak) semi-equations under Boolean combinations (Proposition 2.3), reducts and expansions (Proposition 2.6). In Section 2.2 we discuss how (weak) semi-equationality relates to the more familiar notions: all weakly semi-equational theories are NIP, distal theories are weakly semi-equational, and in a stable theory a formula is an equation if and only if it is a (weak) semi-equation (Proposition 2.10). In Section 2.3 we introduce some quantitive parameters associated with semi-equations. This parameter is related to breadth (Definition 2.15) of the family defined by the instances of a formula, and we observe that a formula is a semi-equation if and only if the family of its instances has finite breadth (Proposition 2.16). The case when this parameter is minimal, i.e., $1$ -semi-equations, provides a generalization of weakly normal formulas characterizing $1$ -based stable theories (Proposition 2.19). Hence $1$ -semi-equationality can be viewed as a form of “linearity,” or “ $1$ -basedness” for NIP theories. We discuss its connections to a different form of “linearity” considered in [Reference Basit, Chernikov, Starchenko, Tao and Tran5], namely basic relations and almost linear Zarankiewicz bounds (see Proposition 2.23 and Remark 2.24), observing that $(2,1)$ -semi-equational theories do not define infinite fields.

In Section 3 we consider some examples of semi-equational theories. In Section 3.1 we show that an o-minimal expansion of a group is linear if and only if it is $(2,1)$ -semi-equational. It remains open if the field of reals is semi-equational (Problem 3.4). We demonstrate that arbitrary unary expansions of linear orders (Section 3.2) and many ordered abelian groups (Section 3.4) are $1$ -semi-equational. In Section 3.5 we demonstrate that the theory of infinitely branching dense trees is semi-equational (Theorem 3.12), but not $1$ -semi-equational (even after naming parameters, see Theorem 3.13 and Corollary 3.14). Semi-equationality of arbitrary trees remains open (Problem 3.17). In Section 3.3 we observe that dense circular orders are not semi-equational, but become $1$ -semi-equational after naming a single constant (in contrast to equationality being preserved under naming and forgetting constants).

In Section 4 we consider the relation of weak semi-equationality and distality in more detail. We show that in an NIP theory, weak semi-equationality of a formula is equivalent to the existence of a one-sided strong honest definition for it (Theorem 4.8). This is a simultaneous generalization of the existence of strong honest definitions in distal theories from [Reference Chernikov and Simon10] and the isolation property for the positive part of $\varphi $ -types for equations (replacing a conjunction of finitely many instances of $\varphi $ by some formula $\theta $ ; see Fact 4.2).

In Section 5.2 we show that many theories of NIP valued fields with an infinite stable residue field, e.g., $\operatorname {ACVF}$ , are not weakly semi-equational (see Theorem 5.1 and Remark 5.10). In Section 5.1 we provide a sufficient criterion for when a formula is not a Boolean combination of weak semi-equations (generalizing the criterion for equations from [Reference Müller and Sklinos28]). We then apply it to show that the partitioned formula $\psi (x_1,x_2; y_1,y_2) := \nu \left (x_{1}-y_{1}\right )<\nu \left (x_{2}-y_{2}\right )$ is not a Boolean combination of weak semi-equations via a detailed analysis of the behavior of indiscernible sequences. It remains open if the field $\mathbb {Q}_p$ is semi-equational (Problem 5.12).

In Section 6 we consider preservation of weak semi-equationality in expansions by naming a new predicate, partially adapting a result for NIP from [Reference Chernikov and Simon9]. Namely, we demonstrate in Theorem 6.7 that if $\mathcal {M} \models T$ is distal, A is a subset of $\mathcal {M}$ with a distal induced structure and the pair $\left (M,A \right )$ is almost model complete (i.e., every formula in the pair is equivalent to a Boolean combination of formulas which only quantify existentially over the predicate; see Definition 6.6), then the pair $\left ( \mathcal {M}, A \right )$ is weakly semi-equational. This implies in particular that dense pairs of o-minimal structures are weakly semi-equational (but not distal by [Reference Hieronymi and Nell18]).

2 Semi-equations and their basic properties

Let T be a complete theory in a language $\mathcal {L}$ , and we work inside a sufficiently saturated and homogeneous monster model $\mathbb {M} \models T$ . All sequences of elements are assumed to be small relative to the saturation of $\mathbb {M}$ , and we write $x,y,\ldots $ to denote finite tuples of variables. Given two linear orders $I,J$ , $I+J$ denotes the linear order given by their sum (i.e., $I < J$ ), and $(0)$ denotes a linear order with a single element. We write $\mathbb {N} = \{0, 1, \ldots \}$ and for $k \in \mathbb {N}$ , $[k] = \{1, \ldots , k\}$ . Given a partitioned formula $\varphi (x,y)$ , we let $\varphi ^*(y,x) := \varphi (x,y)$ .

2.1 Some basic properties of (weak) semi-equations

Remark 2.1. By Ramsey and compactness we may equivalently replace $\omega $ by an arbitrary infinite linear order in Definition 1.2(1), and $\mathbb {Z}$ by $I_{L}+\left (0\right )+I_{R} $ with $I_{L}, I_{R}$ arbitrary infinite linear orders in Definition 1.2(2).

By Ramsey, compactness, and taking automorphisms we also have:

Proposition 2.2. A formula $\varphi (x,y)$ is a semi-equation if and only if there are no b, infinite linear orders $I_L, I_R$ , and an indiscernible sequence $\left (a_{i}\right )_{i\in I_{L}+\left (0\right )+I_{R}}$ such that $\models \varphi \left (a_{i},b\right )$ for $i\in I_{L}+I_{R}$ , but $\not \models \varphi \left (a_{0},b\right )$ .

Proposition 2.3.

  1. (1) If $\varphi (x,y)$ is a semi-equation, then $\varphi (x,y)$ is a weak semi-equation. Hence every semi-equational theory is weakly semi-equational.

  2. (2) Semi-equations are closed under conjunctions and exchanging the roles of the variables.

  3. (3) Weak semi-equations are closed under conjunctions and disjunctions.

Proof (1) Clear from definitions using Proposition 2.2.

(2) Suppose $\varphi \left (x,y\right )\land \psi \left (x,y\right )$ is not a semi-equation. By Proposition 2.2, there are b and an indiscernible sequence $\left (a_{i}\right )_{i\in \mathbb {Z}}$ such that $\models \varphi \left (a_{i},b\right )\land \psi \left (a_{i},b\right )\iff i\neq 0$ . Either $\not \models \varphi \left (a_{0},b\right )$ , in which case $\varphi \left (x,y\right )$ is not a semi-equation, or $\not \models \psi \left (a_{0},b\right )$ , in which case $\psi \left (x,y\right )$ is not a semi-equation. And $\varphi (x,y)$ is a semi-equation if and only if $\varphi ^*(y,x) := \varphi (x,y)$ is a semi-equation by the symmetry of the definition.

(3) For conjunctions, the same as the proof of (2), but with the stipulation that $\left (a_{i}\right )_{i\neq 0}$ is b-indiscernible added. Now suppose $\varphi \left (x,y\right )\lor \psi \left (x,y\right )$ is not a weak semi-equation. Then there are b and an indiscernible sequence $\left (a_{i}\right )_{i\in \mathbb {Z}}$ such that $\left (a_{i}\right )_{i\neq 0}$ is b-indiscernible, and $\models \varphi \left (a_{i},b\right )\lor \psi \left (a_{i},b\right )\iff i\neq 0$ . Either $\models \varphi \left (a_{1},b\right )$ or $\models \psi \left (a_{1},b\right )$ , and then, by b-indiscernibility, either $\models \varphi \left (a_{i},b\right )$ for all $i\neq 0$ or $\models \psi \left (a_{i},b\right )$ for all $i\neq 0$ . In the first case, $\varphi \left (x,y\right )$ is not a weak semi-equation, and in the second case, $\psi \left (x,y\right )$ is not a weak semi-equation.

Remark 2.4.

  1. (1) To see that neither property is closed under negation, note that $x=y$ is a semi-equation (hence also a weak semi-equation), but $x\neq y$ is not a weak semi-equation in the theory of infinite sets.

  2. (2) To see that semi-equations need not be closed under disjunction, note that in a linear order, $x<y$ and $y<x$ are both semi-equations, but their disjunction is equivalent to $x\neq y$ , which is not.

Problem 2.5. Are weak semi-equations closed under exchanging the roles of the variables, at least in NIP theories? Fact 6.4 can be viewed as establishing this for the definition of distality; however, the proof is not sufficiently local with respect to a formula witnessing failure of distality.

Proposition 2.6. Assume we are given languages $\mathcal { L} \subseteq \mathcal { L}'$ , a complete $\mathcal {L}$ -theory T and an $\mathcal {L}'$ -theory $T'$ with $T \subseteq T'$ , and a formula $\varphi (x,y) \in \mathcal {L}$ .

  1. (1) The formula $\varphi (x,y)$ is a semi-equation in T if and only if it is in $T'$ .

  2. (2) If $\varphi \left (x,y\right )$ is a weak semi-equation in T, then it is a weak semi-equation in $T'$ .

Proof (1) Left to right is immediate from the definition (Proposition 2.2). For the converse, assume that in some model of T we can find an infinite sequence $\left (a_{i},b_{i}\right )_{i\in I}$ such that for all $i,j \in I$ , $\models \varphi \left (a_{i},b_{j}\right )\iff i\neq j$ . By completeness of T, we can find arbitrarily long finite sequences with the same property in every model of T, in particular in some model of $T'$ . By compactness we can thus find an infinite sequence with the same property in a model of $T'$ , demonstrating that $\varphi (x,y)$ is not a semi-equation in $T'$ .

(2) If $\varphi \left (x,y\right ) \in \mathcal {L}$ is not a weak semi-equation in $T'$ , then (in a monster model of $T'$ , and hence of T) there are b and an $\mathcal { L}'$ -indiscernible $\left (a_{i}\right )_{i\in I_{L}+\left (0\right )+I_{R}}$ such that $\left (a_{i}\right )_{i\in I_{L}+I_{R}}$ is $\mathcal { L}'$ -indiscernible over b and $\models \varphi \left (a_{i},b\right )$ for $i\in I_{L}+I_{R}$ , but $\not \models \varphi \left (a_{0},b\right )$ , for infinite linear orders $I_{L},I_{R}$ . Then, in particular, $\left (a_{i}\right )_{i\in I_{L}+(0) + I_{R}}$ is $\mathcal { L}$ -indiscernible, and $\left (a_{i}\right )_{i\in I_{L}+I_{R}}$ is $\mathcal { L}$ -indiscernible over b, so $\varphi \left (x,y\right )$ is a not a weak semi-equation in T.

Remark 2.7. The converse to Proposition 2.6(2) does not hold. Let $T' := \operatorname {DLO}$ be the theory of dense linear orders, and T its reduct to $\mathcal {L} := \{ = \}$ . Then the $\mathcal {L}$ -formula $x \neq y$ is not a weak semi-equation in T by inspection, but it is a weak semi-equation in $T'$ since it is equivalent to a disjunction of weak semi-equations $(x < y) \lor (x>y)$ (Proposition 2.3).

Problem 2.8. Is weak semi-equationality of a theory preserved under reducts? This appear to be open already for equationality (see [Reference Junker20, Question 3.10]), and fails for semi-equationality (see Section 3.3).

Problem 2.9. Is (weak) semi-equationality of theories invariant under bi-interpretability without parameters? Equivalently, if T is (weakly) semi-equational, does it follow that so is $T^{\operatorname {eq}}$ ?

2.2 Relationship to equations and NIP

We provide some evidence that semi-equationality can be naturally viewed as a generalization of equationality (in the sense of Srour) in stable theories to the NIP context.

Proposition 2.10.

  1. (1) Weak semi-equations are NIP formulas; hence, weakly semi-equational theories are NIP.

  2. (2) Equations are semi-equations.

  3. (3) A formula is an equation if and only if it is both stable and a semi-equation.

  4. (4) In a stable theory, all weak semi-equations are equations. In particular, a stable theory is equational if and only if it is (weakly) semi-equational.

Proof (1) If $\varphi \left (x,y\right )$ is not NIP, then there are an indiscernible sequence $\left (a_{i}\right )_{i\in \mathbb {N}}$ and b such that $\models \varphi \left (a_{i},b\right ) \iff i$ is even. For any finite set of formulas $\Delta \left (x_{1}, \ldots , x_{n},y\right )$ , by Ramsey’s theorem, there is an infinite $I\subseteq 2\mathbb {N}$ on which the truth value of all formulas in $\Delta \left (a_{i_{1}}, \ldots , a_{i_{n}},b\right )$ is constant for all $i_{1}<\cdots < i_{n}\in I$ . Thus, by letting $a_{0}':=a_{i}$ for some sufficiently large odd i, we can find an indiscernible sequence $\left (a_{i}'\right )_{i\in I_{L}+\left (0\right )+I_{R}}$ (using $I_{L}\sqcup I_{R}=I$ , and $a_{i}'=a_{i}$ for $i\in I$ ) for some infinite $I_{R}$ and arbitrarily large finite $I_{L}$ , such that $\left (a_{i}'\right )_{i\in I_{L}+I_{R}}$ is $\Delta $ -indiscernible over b. By compactness, it follows that $\varphi \left (x,y\right )$ is not a weak semi-equation.

(2) If $\varphi \left (x,y\right )$ is not a semi-equation, then there is a sequence $\left (a_{i},b_{i}\right )_{i\in \mathbb {N}}$ such that $\models \varphi \left (a_{i},b_{j}\right )\iff i\neq j$ . In particular, $\models \varphi \left (a_{i},b_{j}\right )$ for all $j<i$ , and $\not \models \varphi \left (a_{i},b_{i}\right )$ , so this is a counterexample to the descending chain condition, and $\varphi \left (x,y\right )$ is not an equation.

(3) If $\varphi \left (x,y\right )$ is not an equation, then by Ramsey and compactness there is an indiscernible sequence $\left (a_{i},b_{i}\right )_{i\in \mathbb {N}}$ such that $\models \varphi \left (a_{i},b_{j}\right )$ for all $j<i$ , and $\not \models \varphi \left (a_{i},b_{i}\right )$ . If $\varphi \left (a_{i},b_{j}\right )$ holds for $i<j$ then $\varphi \left (x,y\right )$ is not a semi-equation. Otherwise, $\varphi \left (x,y\right )$ is not stable.

(4) If $\varphi \left (x,y\right )$ is not an equation, by Ramsey and compactness we can choose an indiscernible sequence $\left (a_{i},b_{i}\right )_{i\in \mathbb {Z}}$ such that $\models \varphi \left (a_{i},b_{j}\right )$ for all $j<i$ , and $\not \models \varphi \left (a_{i},b_{i}\right )$ . The indiscernible sequence $(a_i,b_i)_{i \in \mathbb {Z}}$ is totally indiscernible by stability of T; hence, we have $\models \varphi (a_i, b_0) \iff i \neq 0$ , and also $(a_i : i \neq 0)$ is indiscernible over $b_0$ . This shows that $\varphi \left (x,y\right )$ is not a weak semi-equation.

2.3 Weakly normal formulas, $\left (k,n\right )$ -semi-equations, and breadth

Definition 2.11 (see [Reference Pillay32, Chapter 4, Definition 1.1]).

A formula $\varphi \left (x,y\right )$ is k-weakly normal if for every $b_{1}, \ldots , b_{k}\in \mathbb {M}^{y}$ such that $\models \exists x\,\varphi \left (x,b_{1}\right )\land \cdots \land \varphi \left (x,b_{k}\right )$ , there are some $i\neq j\in \left [k\right ]$ such that $\models \forall x\,\varphi \left (x,b_{i}\right )\leftrightarrow \varphi \left (x,b_{j}\right )$ . It is weakly normal if it is k-weakly normal for some k (by compactness this is equivalent to: an infinite collection of pairwise distinct instances of $\varphi (x,y)$ must have empty intersection).

A formula $\varphi (x,y)$ is normal in the sense of [Reference Pillay31] if and only if it is $2$ -weakly normal. Weakly normal formulas are special kinds of equations characterizing “linearity” of forking in stable theories (see [Reference Pillay32, Chapter 4, Proposition 1.5, Remark 1.8.4, and Lemma 1.9]):

Fact 2.12. A stable theory T is 1-based if and only if in T, every formula $\varphi (x,y) \in \mathcal {L}$ , with $x,y$ arbitrary finite tuples of variables, is equivalent to a Boolean combination of finitely many weakly normal formulas $\psi _1(x,y), \ldots , \psi _n(x,y) \in \mathcal {L}$ .

We introduce some numeric parameters characterizing semi-equations, minimal values of which give rise to a generalization of weak normality.

Definition 2.13. For $k,n \in \mathbb {N}$ , a formula $\varphi \left (x,y\right )$ is a $\left (k,n\right )$ -semi-equation if, for every $b_{1}, \ldots , b_{k}\in \mathbb {M}^y$ , if $\models \exists x \ \varphi \left (x,b_{1}\right )\land \cdots \land \varphi \left (x,b_{k}\right )$ , then for some pairwise distinct $i_{1}, \ldots , i_{n},j\in \left [k\right ]$ , $\models \forall x\,\left (\varphi \left (x,b_{i_{1}}\right )\land \cdots \land \varphi \left (x,b_{i_{n}}\right )\right )\rightarrow \varphi \left (x,b_{j}\right )$ . And $\varphi \left (x,y\right )$ is an n-semi-equation if it is a $\left (k,n\right )$ -semi-equation for some k. A theory T is n-semi-equational (respectively, $(k,n)$ -semi-equational) if every formula $\varphi (x,y) \in \mathcal {L}$ , with $x,y$ arbitrary finite tuples of variables, is equivalent in T to a Boolean combination of n-semi-equations (respectively, $(k,n)$ -semi-equations) $\psi _1(x,y), \ldots , \psi _n(x,y) \in \mathcal {L}$ .

Proposition 2.14.

  1. (1) If $\varphi \left (x,y\right )$ is a $\left (k,n\right )$ -semi-equation, then $n<k$ , and $\varphi \left (x,y\right )$ is also an $\left (\ell ,m\right )$ -semi-equation for any $\ell \geq k$ and $n\leq m<\ell $ . If $\varphi \left (x,y\right )$ is an n-semi-equation, then it is also an m-semi-equation for every $m\geq n$ .

  2. (2) A formula is a semi-equation if and only if it is an n-semi-equation for some n, if and only if it is an $(n,n-1)$ -semi-equation for some n.

Proof (1) Clear from the definitions.

(2) If $\varphi \left (x,y\right )$ is not a semi-equation, let $\left (a_{i},b_{i}\right )_{i\in \mathbb {N}}$ be such that $\models \varphi \left (a_{i},b_{j}\right )\iff i\neq j$ . Then for any $\left (k,n\right )$ we have $\models \varphi \left (a_{0},b_{1}\right )\land \cdots \land \varphi \left (a_{0},b_{k}\right )$ , but for any pairwise distinct $i_{1}, \ldots , i_{n},j\in \left [k\right ]$ , $\models \varphi \left (a_{j},b_{i_{1}}\right )\land \cdots \land \varphi \left (a_{j},b_{i_{n}}\right )\land \neg \varphi \left (a_{j},b_{j}\right )$ ; hence, $\varphi (x,y)$ is not a $(k,n)$ -semi-equation. Conversely, for any $k \in \mathbb {N}$ , if $\varphi \left (x,y\right )$ is not a $\left (k,k-1\right )$ -semi-equation, then there exist $b_{1}, \ldots , b_{k}$ such that for each $j\in \left [k\right ]$ , there is $a_{j}$ such that $\models \varphi \left (a_{j},b_{i}\right )$ for $i\neq j$ , but $\not \models \varphi \left (a_{j},b_{j}\right )$ . Hence if $\varphi \left (x,y\right )$ is not a $\left (k,k-1\right )$ -semi-equation for any k, then by compactness $\varphi \left (x,y\right )$ is not a semi-equation. And if $\varphi \left (x,y\right )$ is not an n-semi-equation, then it is not an $\left (n+1,n\right )$ -semi-equation by definition, so a formula that is not an n-semi-equation for any n is also not a $\left (k,k-1\right )$ -semi-equation for any k.

We recall the notion of breadth from lattice theory.

Definition 2.15 [Reference Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko2, Section 2.4].

Given a set X and $d \in \mathbb {N}_{\geq 1}$ , a family of subsets $\mathcal {F} \subseteq \mathcal {P}\left (X\right )$ has breadth d if any nonempty intersection of finitely many sets in $\mathcal {F}$ is the intersection of at most d of them, and d is minimal with this property.

Proposition 2.16. A formula $\varphi \left (x,y\right )$ is a $\left (k+1,k\right )$ -semi-equation if and only if the family of sets $\mathcal {F}_{\varphi } := \left \{ \varphi \left (\mathbb { M},b\right )\mid b\in \mathbb { M}^{y}\right \} $ has breadth at most k. In particular, $\varphi (x,y)$ is a semi-equation if and only if the family of sets $\mathcal {F}_{\varphi }$ has finite breadth.

Proof The family of sets $\left \{ \varphi \left (\mathbb {M},b\right )\mid b\in \mathbb { M}^{y}\right \} $ has breadth at most k if and only if every finite consistent conjunction of instances of $\varphi $ is implied by the conjunction of at most k of those instances. In particular this applies to consistent conjunctions of $(k+1)$ instances of $\varphi $ , showing that if the breadth of $\mathcal {F}_{\varphi }$ is $\leq k$ , then it is a $(k+1,k)$ -semi-equation. Conversely, assume $\varphi (x,y)$ is a $(k+1,k)$ -semi-equation. Given any consistent conjunction of $n>k$ instances of $\varphi $ , any $(k+1)$ of them contain an instance implied by the other k instances. Removing this implied instance, we reduce to the case of a consistent conjunction of $(n-1)$ instances, and after $(n-k)$ steps to a conjunction of k instances of $\varphi $ implying all the other ones. The “in particular” part is Proposition 2.14(2).

Example 2.17. Let T be an NIP theory expanding a group, and let a formula $\varphi (x,y)$ be such that for every $b \in \mathbb {M}^y$ , $\varphi ( \mathbb {M}, b)$ is a subgroup. Then, by Baldwin and Saxl [Reference Baldwin and Saxl4], there exists $n \in \omega $ such that for all finite $B\subseteq \mathbb {M}^y$ , there is $B_0\subseteq B$ with $\left |B_0\right |\leq n$ such that $\bigcap _{b\in B_0}\varphi \left (\mathbb {M},b\right )=\bigcap _{b\in B}\varphi \left (\mathbb {M},b\right )$ . So $\varphi (x,y)$ is a semi-equation by Proposition 2.16.

Remark 2.18. If $\varphi (x,y)$ is stable with infinitely many distinct instances $\varphi (\mathbb {M},b), b \in \mathbb {M}^y$ , then either $\varphi (x,y)$ is not a semi-equation, or $\neg \varphi (x,y)$ is not a semi-equation (combining [Reference Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko2, Proposition 2.20] and Proposition 2.16).

The following suggests that $1$ -semi-equationality can be viewed as a generalization of being $1$ -based from stable to the NIP context.

Proposition 2.19. A formula $\varphi \left (x,y\right )$ is weakly normal if and only if it is stable and a $1$ -semi-equation. Hence a stable theory is $1$ -based if and only if it is $1$ -semi-equational.

Proof Clearly every k-weakly normal formula is a $\left (k,1\right )$ -semi-equation and is also an equation, hence stable. Conversely, suppose that $\varphi \left (x,y\right )$ is a $\left (k,1\right )$ -semi-equation and is stable, or just NSOP: there is some $\ell \in \omega $ such that there is no strictly increasing chain of sets of the form $\varphi \left (\mathbb {M},b_{0}\right ) \subsetneq \cdots \subsetneq \varphi \left (\mathbb {M},b_{\ell }\right )$ . We will show that then $\varphi (x,y)$ is $k^{\ell }$ -weakly normal. Let $\left (b_{\eta }\right )_{\eta \in \left [k\right ]^{\ell }}$ be such that $\models \exists x\,\bigwedge _{\eta \in \left [k\right ]^{\ell }}\varphi \left (x,b_{\eta }\right )$ . For $\sigma \in \left [k\right ]^{\leq \ell }$ , we will show by induction on $m:=\ell -\left |\sigma \right |$ that there are pairwise distinct $\eta _{0}, \ldots , \eta _{m}\in \left [k\right ]^{\ell }$ extending $\sigma $ (as sequences) such that $\varphi \left (\mathbb { M},b_{\eta _{0}}\right )\subseteq \varphi \left (\mathbb { M},b_{\eta _{1}}\right )\subseteq \cdots \subseteq \varphi \left (\mathbb { M},b_{\eta _{m}}\right )$ . With $m=\ell $ , so that $\sigma = \langle \rangle $ is the empty sequence, this implies by the choice of $\ell $ that there are $\eta \neq \eta '\in \left [k\right ]^{\ell }$ such that $\varphi \left (\mathbb { M},b_{\eta }\right )=\varphi \left (\mathbb { M},b_{\eta '}\right )$ , as desired. The base case ( $m=0$ ) is trivial, with $\eta _{0}=\sigma $ .

Assume the claim holds for m, and let $\sigma \in \left [k\right ]^{\ell -\left (m+1\right )}$ . For each $i\in \left [k\right ]$ , there exist pairwise distinct $\eta _{i,0}, \ldots , \eta _{i,m}\in \left [k\right ]^{\ell }$ extending $\sigma ^{\frown } i$ such that $\varphi \left (\mathbb { M},b_{\eta _{i,0}}\right )\subseteq \cdots \subseteq \varphi \left (\mathbb { M},b_{\eta _{i,m}}\right )$ . Among the sets $\left \{ \varphi \left (\mathbb { M},b_{\eta _{i,0}}\right )\mid i\in \left [k\right ]\right \} $ , one must be contained in another by $\left (k,1\right )$ -semi-equationality. Say $\varphi \left (\mathbb { M},b_{\eta _{j,0}}\right )\subseteq \varphi \left (\mathbb { M},b_{\eta _{i,0}}\right )$ for some $i\neq j$ . Then $\varphi \left (\mathbb { M},b_{\eta _{j,0}}\right )\subseteq \varphi \left (\mathbb { M},b_{\eta _{i,0}}\right )\subseteq \varphi \left (\mathbb { M},b_{\eta _{i,1}}\right ) \subseteq \cdots \subseteq \varphi \left (\mathbb { M},b_{\eta _{i,m}}\right )$ , and $\eta _{j,0},\eta _{i,0}, \eta _{i,1}, \ldots ,\eta _{i,m}$ are pairwise distinct and extend $\sigma $ , as desired. The “in particular” part follows by Fact 2.12.

Remark 2.20. It is well known that the family of weakly normal formulas is closed under conjunctions (but we could not find a direct reference). While semi-equations are closed under conjunctions by Proposition 2.3(2), this is not the case for the family of $1$ -semi-equations. Indeed, in a dense linear order, the formulas $x < y_1$ and $x> y_2$ are $1$ -semi-equations, but the formula $\varphi (x; y_1, y_2) := y_2 < x < y_1$ is not a $1$ -semi-equation since we can have any number of intervals with a non-empty intersection, so that none of them is contained in the other.

We observe a connection to another notion of “linearity” for NIP theories considered in [Reference Basit, Chernikov, Starchenko, Tao and Tran5], where various combinatorial results are proved for relations that are Boolean combinations of basic relations. The following is [Reference Basit, Chernikov, Starchenko, Tao and Tran5, Definition 2], in the case of binary relations (using the equivalence in [Reference Basit, Chernikov, Starchenko, Tao and Tran5, Proposition 2.8 and Remark 2.9]).

Definition 2.21. A binary relation $R \subseteq X \times Y$ is basic if there exist a linear order $(S,<)$ and functions $f: X \to S, g: Y \to S $ such that for any $a \in X, b \in Y$ , $(a,b) \in R \iff f(a) < g(b)$ .

Fact 2.22 [Reference Guingona and Laskowski16, Claim 1 in the proof of Proposition 2.5].

Let X be a set and $\mathcal {F} \subseteq \mathcal {P}(X)$ a family of subsets of X such that there are no $A,B \in \mathcal {F}$ satisfying $A \cap B \neq \emptyset , B \setminus A \neq \emptyset $ and $B \setminus A \neq \emptyset $ simultaneously. Then there exists a linear order $<$ on X so that every $A \in \mathcal {F}$ is a $<$ -convex subset of X.

Proposition 2.23.

  1. (1) Given a formula $\varphi (x,y) \in \mathcal {L}$ , if the relation $R_{\varphi } := \{(a,b) \in \mathbb {M}^{x} \times \mathbb {M}^y : \models \varphi (a,b) \}$ is basic, then $\varphi (x,y)$ is a $(2,1)$ -semi-equation.

  2. (2) If $\varphi (x,y)$ is $(2,1)$ -semi-equation, then $R_{\varphi } = R_1 \cap R_2$ for some (not necessarily definable) basic relations $R_1, R_2 \subseteq \mathbb {M}^{x} \times \mathbb {M}^y$ .

Proof (1) Let $(S,<), f,g$ be as in Definition 2.21 for $R_{\varphi }$ . Given any $b_1, b_2 \in \mathbb {M}^{y}$ , the sets $\left \{x \in S : x < g(b_i) \right \}$ for $i \in \{1,2\}$ are initial segments of S. Say $g(b_1) \leq g(b_2)$ . Then for any $a \in \mathbb {M}^x$ , $f(a) < g(b_1) \Rightarrow f(a) < g(b_2)$ , so $\varphi (\mathbb {M},b_1) \subseteq \varphi (\mathbb {M},b_2)$ , and the other case is symmetric.

(2) If $\varphi (x,y)$ is a $(2,1)$ -semi-equation, then the family $\mathcal {F}_{\varphi }$ of subsets of $\mathbb {M}^{x}$ satisfies the assumption in Fact 2.22. Hence there exists a (not necessarily definable) linear ordering $<'$ of $\mathbb {M}^{x}$ so that for every $b \in \mathbb {M}^y$ , $\varphi (\mathbb {M},b)$ is $<'$ -convex. Let $(S,<)$ be the Dedekind completion of $\left ( \mathbb {M}^x, <' \right )$ . Consider the functions $g_1, g_2: \mathbb {M}^{y} \to S$ so that $g_1(b)$ is the infimum of $\varphi (\mathbb {M},b)$ in S, and $g_2(b)$ is the supremum of $\varphi (\mathbb {M},b)$ in S. Then $R_{\varphi } = \left \{ (a,b) \in \mathbb {M}^x \times \mathbb {M}^y : g_1(b)\leq a \right \} \cap \left \{ (a,b) \in \mathbb {M}^x \times \mathbb {M}^y: a \leq g_2(b) \right \}$ , and both of these relations are basic (see [Reference Basit, Chernikov, Starchenko, Tao and Tran5, Remark 2.7]).

Remark 2.24. (1) In view of Proposition 2.23(2), if $\varphi (x,y)$ is a Boolean combination of $(2,1)$ -semi-equations, then by [Reference Basit, Chernikov, Starchenko, Tao and Tran5, Theorem 2.17 and Remark 2.20] the relation $R_{\varphi }$ satisfies an almost linear Zarankiewicz bound. In particular, no infinite field can be defined in a $(2,1)$ -semi-equational theory (see [Reference Basit, Chernikov, Starchenko, Tao and Tran5, Corollary 5.11] or [Reference Walsberg42, Proposition 6.3] for a detailed explanation).

(2) If $\varphi (x,y)$ is a $(2,1)$ -semi-equation, then $R_{\varphi }$ need not be basic. Indeed, the family of cosets of a subgroup is $(2,1)$ -semi-equational. If it was basic, then its complement is also basic, hence $(2,1)$ -semi-equational by the lemma above. But if the index of the subgroup is $\geq 3$ , the family of complements of cosets is clearly not $(2,1)$ -semi-equational.

Problem 2.25. If $\varphi (x,y)$ is a $(k,1)$ -semi-equation for $k \geq 3$ , is it still a Boolean combination of basic relations?

Problem 2.26. Show that no infinite field is definable in a $1$ -semi-equational theory.

Problem 2.27. Is every $1$ -semi-equational theory rosy? (Note that dense trees are not $1$ -semi-equational by Theorem 3.13.)

3 Examples of semi-equational theories

3.1 O-minimal structures

All o-minimal theories (and more generally, ordered dp-minimal theories) are distal (see [Reference Simon36]); hence, they are weakly semi-equational by Remark 4.4. Semi-equationality appears more subtle. An o-minimal structure $\mathcal {M}$ is linear if it has the CF property in the sense of [Reference Loveys and Peterzil23], i.e., if every interpretable normal family of curves is of dimension at most $1$ . This is a weakening of local modularity of the pregeometry induced by the algebraic closure, and by the o-minimal trichotomy [Reference Peterzil and Starchenko30] it is equivalent to no infinite field being definable in $\mathcal {M}$ . We will only need the following fact about linear o-minimal structures from [Reference Loveys and Peterzil23]:

Fact 3.1. Let $T = {\operatorname {Th}}(\mathcal {M})$ , with $\mathcal {M} = \left (M; <, +, \ldots \right )$ a linear o-minimal expansion of a group. Let $\mathcal {L} = (<,+,\ldots )$ be the language of T. A partial endomorphism of $\mathcal {M}$ is a map $f: (-c,c) \to M$ , for c an element of M or $\infty $ , such that if $a,b,a+b$ are all in the domain, then $f(a+b) = f(a) + f(b)$ . Let $\mathcal {M}'$ be the reduct of $\mathcal {M}$ to the language $\mathcal {L}'$ consisting of $+,<$ , constant symbols naming $\operatorname {acl}_{\mathcal {L}}(\emptyset )$ , and for each $\mathcal {L}(\emptyset )$ -definable partial endomorphism $f: (-c,c) \to M$ with $c \in \operatorname {acl}_{\mathcal {L}}(\emptyset )$ or $c = \infty $ , a unary function symbol interpreted as f on $(-c,c)$ and as $0$ outside of the domain of f. Let $T' := {\operatorname {Th}}_{\mathcal {L}'}(\mathcal {M}')$ .

  1. (1) [Reference Loveys and Peterzil23, Proposition 4.2] A subset of $M^n$ is $\emptyset $ -definable in $\mathcal {M}$ if and only if it is $\emptyset $ -definable in $\mathcal {M}'$ .

  2. (2) [Reference Loveys and Peterzil23, Corollary 6.3] $T'$ admits quantifier elimination in the language $\mathcal {L}'$ .

Proposition 3.2. Let $T = {\operatorname {Th}}(\mathcal {M})$ , with $\mathcal {M} = \left (M; <, \ldots \right )$ an o-minimal structure.

  1. (1) If T is an expansion of an ordered group and linear, then T is $(2,1)$ -semi-equational.

  2. (2) Conversely, if T is $(2,1)$ -semi-equational, then T is linear.

Proof (1) Let $\mathcal {L} = (<,+,\ldots )$ , $\mathcal {M}'$ and $\mathcal {L}'$ be as in Fact 3.1. By Fact 3.1(1) it suffices to show that $T' := {\operatorname {Th}}_{\mathcal {L}'} \left (\mathcal {M}'\right )$ is $(2,1)$ -semi-equational. By Fact 3.1(2), it then suffices to show that every atomic $\mathcal {L}'$ -formula $\varphi (x,y)$ , with $x,y$ arbitrary finite tuples of variables, is equivalent in $T'$ to a Boolean combination of $(2,1)$ -semi-equations. By the proof of Theorem 4.3 in [Reference Anderson1], every atomic $\mathcal {L}'$ -formula $\varphi (x,y)$ is equivalent in $T'$ to a Boolean combination of atomic formulas of the form $f(x) \square g(y) + c$ , where $\square \in \left \{<, = ,> \right \}$ , $f: M^{|x|} \to M, g: M^{|y|} \to M$ are total multivariate $\mathcal {L}'(\emptyset )$ -definable homomorphisms and $c \in \operatorname {dcl}_{\mathcal {L}'}(\emptyset )$ . Every formula of this form clearly defines a basic relation on $M^{|x|} \times M^{|y|}$ , hence is a $(2,1)$ -semi-equation by Proposition 2.23(1).

(2) By the o-minimal trichotomy theorem (see [Reference Peterzil and Starchenko30] and Remark 2 after the statement of Theorem 1.7 there), if $\mathcal {M}$ is not linear, then it defines an infinite field. But then Remark 2.24(1) implies that T is not $(2,1)$ -semi-equational.

Problem 3.3. Is every o-minimal $1$ -semi-equational structure linear? A positive answer would follow from a positive answer to Problem 2.26.

Problem 3.4. Which o-minimal theories are semi-equational? In particular, is ${\operatorname {Th}}(\mathbb {R}, +, \times )$ semi-equational?

3.2 Colored linear orders

Given a linearly ordered set $(S,<)$ , a binary relation $R \subseteq S^2$ is monotone if $(x,y) \in R$ , $x' \leq x$ , and $y\leq y'$ implies $\left (x',y'\right ) \in R$ .

Fact 3.5. Let $\mathcal {M} = \left ( M, <, \left (C_i \right )_{i \in I}, \left (R_j \right )_{j \in J} \right )$ be a linear order expanded by arbitrary unary ( $C_i$ ) and monotone binary ( $R_j$ ) relations. Then ${\operatorname {Th}} \left ( \mathcal {M} \right )$ is $(2,1)$ -semi-equational.

Proof Let $\mathcal {M}'$ be an expansion of $\mathcal {M}$ obtained by naming all $\mathcal {L}_{\mathcal {M}}(\emptyset )$ -definable unary and monotone binary relations, then a subset of $M^n$ is $\emptyset $ -definable in $\mathcal {M}$ if and only if it is $\emptyset $ -definable in $\mathcal {M}'$ , so it suffices to show that $T' := {\operatorname {Th}} \left (\mathcal {M}' \right )$ is $(2,1)$ -semi-equational. By [Reference Simon35, Proposition 4.1], $T'$ eliminates quantifiers. Note that if $R(x,y)$ is monotone, then it is a $(2,1)$ -semi-equation (given any $b_1 \leq b_2 \in M$ , for any $a \in M$ we have $\models R(a,b_1) \rightarrow R(a,b_2)$ by monotonicity; hence, $R(M,b_1) \subseteq R(M,b_2)$ ). And any unary relation $C_i(x)$ is trivially a $(2,1)$ -semi-equation; hence, $T'$ is $(2,1)$ -semi-equational.

3.3 Cyclic orders

A cyclic order ${\circlearrowright }(x,y,z)$ (see, e.g., [Reference Chernikov and Kaplan6, Section 5] or [Reference Tran and Walsberg41]) is dense if its underlying set is infinite and for every distinct $a,b$ , there is c such that ${\circlearrowright }\left (a,b,c\right )$ , and d such that ${\circlearrowright }\left (d,b,a\right )$ . The theory $T_{{\circlearrowright }}$ of dense cyclic orders is complete and has quantifier elimination (see, e.g., [Reference Chernikov and Mennen7, Proposition 3.7]).

Proposition 3.6.

  1. (1) The theory $T_{{\circlearrowright }}$ is not semi-equational.

  2. (2) The theory $T_{{\circlearrowright }}$ expanded with one constant symbol c is $(2,1)$ -semi-equational.

Proof (1) We show that $ \psi (x_1,x_2; y) := {\circlearrowright }\left (x_{1},x_{2};y\right )$ is not a Boolean combination of semi-equations. By quantifier elimination, the formulas ${\circlearrowright }\left (x_{1},x_{2};y\right )$ and ${\circlearrowright }\left (x_{2},x_{1};y\right )$ each isolate a complete $3$ -type (over $\emptyset $ ). Any Boolean combination of formulas that is equivalent to ${\circlearrowright }\left (x_{1},x_{2};y\right )$ must contain some formula $\varphi \left (x_{1},x_{2};y\right )$ that is implied by ${\circlearrowright }\left (x_{1},x_{2};y\right )$ and is inconsistent with ${\circlearrowright }\left (x_{2},x_{1};y\right )$ , or vice versa. Assume the former. Let $\left (c_{i}\right )_{i\in \mathbb {Z}}$ be such that $\models {\circlearrowright }\left (c_{k},c_{i},c_{j}\right )$ for $i<j<k$ . Let $a_{1,i}=c_{2i}$ , $a_{2,i}=c_{2i+2}$ , and $b_{i}=c_{2i+1}$ . Then $\models {\circlearrowright }\left (a_{2,i},a_{1,i};b_{j}\right )\Leftrightarrow i=j$ and $\models {\circlearrowright }\left (a_{1,i},a_{2,i};b_{j}\right ) \Leftrightarrow i\neq j$ , so $\models \varphi \left (a_{1,i},a_{2,i};b_{j}\right ) \Leftrightarrow i\neq j$ , so $\varphi \left (x_{1},x_{2};y\right )$ is not a semi-equation. If instead, $\varphi \left (x_{1},x_{2};y\right )$ is implied by ${\circlearrowright }\left (x_{2},x_{1};y\right )$ and inconsistent with ${\circlearrowright }\left (x_{1},x_{2};y\right )$ , we switch the roles of $x_{1}$ and $x_{2}$ to get the same result.

(2) Let $<$ be defined by $x<y\Leftrightarrow {\circlearrowright }\left (x,y,c\right )$ . Then $<$ is a dense linear order on the complement of $\left \{ c\right \} $ , so $x < y$ is a $\left (2,1\right )$ -semi-equation. We have that ${\circlearrowright }\left (x,y,z\right )$ is equivalent to $x<y<z\lor y<z<x\lor z<x<y\lor \left (z=c\land x<y\right )\lor \left (y=c\land z<x\right )\lor \left (x=c\land y<z\right )$ . Hence ${\circlearrowright }\left (x,y,z\right )$ is a Boolean combination of $\left (2,1\right )$ -semi-equations (with c as a parameter), under any partition of the variables. By quantifier elimination, it follows that every formula is a Boolean combination of $\left (2,1\right )$ -semi-equations (using c as a parameter).

This example shows that a theory being semi-equational, or 1-semi-equational, is not preserved under forgetting constants (naming constants clearly preserves k-semi-equationality). This is in contrast to equationality [Reference Junker20, Proposition 3.5] and distality [Reference Simon36, Corollary 2.9], which are invariant under naming or forgetting constants. This is also an example of a distal, non-semi-equational theory.

Problem 3.7. Is weak semi-equationality of theories preserved by forgetting constants?

3.4 Ordered abelian groups

We consider ordered abelian groups, as structures in the language $\mathcal {L}_{\operatorname {CH}}$ introduced in [Reference Cluckers and Halupczok13]. Given an ordered abelian group $(G,+,<)$ and prime p, for $a \in G \setminus pG$ we let $G_p(a)$ be the largest convex subgroup of G such that $a \notin G_p(a) +pG$ , and for $a \in pG$ let $G_p(a) := \{0\}$ . Let $\mathcal {S}_p := \{G_p(a) : a \in G\}$ . Then the $\mathcal {L}_{\operatorname {CH}}$ -structure $\bar {G}$ corresponding to G consists of the main sort $\mathcal {G}$ for G, an auxiliary sort $\mathcal {S}_p$ for each p, along with countably many further auxiliary sorts and relations between them. A relative quantifier elimination result is obtained for such structures in [Reference Cluckers and Halupczok13], to which we refer for the details (see also [Reference Aschenbrenner, Chernikov, Gehret and Ziegler3, Section 3.2] for a quick summary).

Proposition 3.8. Every ordered abelian group (either as a pure ordered abelian group, or the corresponding structure $\bar {G}$ ) with finite auxiliary sorts $\mathcal {S}_p$ for all prime p is $1$ -semi-equational (this includes Presburger arithmetic, and any ordered abelian group with a strongly dependent theory by [Reference Chernikov, Kaplan and Simon12, Reference Dolich and Goodrick14, Reference Farré15, Reference Halevi and Hasson17]).

Proof Since every auxiliary sort is finite and linearly ordered by a (definable) relation in $\mathcal {L}_{\operatorname {CH}}$ , all auxiliary sorts are contained in $\operatorname {dcl}(\emptyset )$ . Hence we only need to verify that every formula $\varphi (x,y)$ with $x,y$ tuples of the main sort $\mathcal {G}$ is a Boolean combination of $1$ -semi-equations in the expansion with every element of every auxiliary sort named by a new constant symbol (countably many in total). As explained in [Reference Aschenbrenner, Chernikov, Gehret and Ziegler3, Proposition 3.14], it then follows from the relative quantifier elimination that $\varphi (x,y)$ is equivalent to a Boolean combination of atomic formulas of the form $\pi _{\alpha }(f(x)) \diamond _{\alpha } \pi _{\alpha }(g(y)) + k_{\alpha }$ , where $\diamond \in \left \{ =, <, \equiv _{m} \right \}$ , $k \in \mathbb {Z}$ , $\alpha $ is an element of an auxiliary sort, $f,g$ are $\mathbb {Z}$ -linear functions on $\mathcal {G}$ , $G_{\alpha }$ is a corresponding convex subgroup of G, $\pi _{\alpha }: G \to G/G_{\alpha }$ is the quotient map, $1_{\alpha }$ is the minimal positive element of $G/G_{\alpha }$ if it is discrete or $0 \in G/G_{\alpha }$ otherwise, and $k_{\alpha } = k \cdot 1_{\alpha }$ in $G/G_{\alpha }$ , and for $g,h \in G/G_{\alpha }$ we have $g \equiv _m h$ if $g - h \in m \left ( G/G_{\alpha } \right )$ (note that these relations on G are expressible in the pure language of ordered abelian groups).

It is straightforward from Definition 2.13 that if $\varphi (x,y)$ is a $(k,n)$ -semi-equation and $f(x), g(y)$ are $\emptyset $ -definable functions, then the formula $\psi (x,y) := \varphi (f(x), g(y))$ is also a $(k,n)$ -semi-equation. Using this (in an expansion of $\bar {G}$ naming $\pi _{\alpha }$ , and the ordered group structure on $G/G_{\alpha }$ together with the constants for $k_{\alpha }$ ), we only have to show that the relations $x = y, x < y, x \in y + m\left (G/G_{\alpha } \right )$ on $G/G_{\alpha }$ are $(2,1)$ -semi-equations, which is straightforward.

Problem 3.9. Is every ordered abelian group $1$ -semi-equational, or at least (weakly) semi-equational?

3.5 Trees

In this section we use “ $\land $ ” to denote “meet,” and “ $\&$ ” to denote conjunction. By a tree we mean a meet-semilattice $(M, \land )$ with an associated partial order $\leq $ (defined by $x \leq y \iff x \land y = x$ ) so that all of its initial segments are linear orders. An infinitely branching dense tree is a tree whose initial segments are dense linear orders and such that for each element x, there are infinitely many elements any two of which have meet x.

Fact 3.10 (see, e.g., [Reference Chernikov and Mennen7, Lemma 3.14] or [Reference Mennuni27, Section 1]).

The theory of infinitely branching dense trees is complete and eliminates quantifiers in the language $\left \{ \land \right \} $ .

Lemma 3.11. In any tree $\mathcal {M} = (M, \land )$ with no additional structure, if every formula of the form $\varphi \left (x;y_{1},y_{2}\right )$ with $x,y_1,y_2$ singletons is a Boolean combination of semi-equations, then every formula is a Boolean combination of semi-equations.

Proof By [Reference Simon35, Corollary 4.6] (using that $x \leq y \iff x \land y = x$ ), in any tree $\mathcal {M}=(M, \land )$ we have: two tuples $\bar {a} = (a_i : i \in [n]),\bar {b} = \left ( b_j: j \in [n] \right ) \in M^{n}$ have the same type if and only if $\left (a_{i},a_{j},a_{k}\right )$ and $\left (b_{i},b_{j},b_{k}\right )$ have the same type for every $i,j,k\in \left [n\right ]$ . Hence for any $\bar {a},\bar {b}$ , $\operatorname {tp}\left (\bar {a}\bar {b}\right )$ is implied by the set of formulas satisfied by three-element subtuples of $\bar {a}\bar {b}$ . So if every partitioned formula with three total free variables is a Boolean combination of semi-equations, then $\operatorname {tp}\left (\bar {a}\bar {b}\right )$ is implied by a Boolean combination of semi-equations. In view of this, it is sufficient to assume that every formula of the form $\varphi \left (x;y_{1},y_{2}\right )$ is a Boolean combination of semi-equations, because then by symmetry, every formula of the form $\varphi \left (x_{1},x_{2};y\right )$ is as well, and every partitioned formula with one of the parts empty (i.e., $\varphi \left (;y_{1},y_{2},y_{3}\right )$ or $\varphi \left (x_{1},x_{2},x_{3};\right )$ ) is automatically a semi-equation.

Theorem 3.12. The theory of infinitely branching dense trees is semi-equational.

Proof Let $\mathcal { M} = (M, \land )$ be an infinitely branching dense tree. By Lemma 3.11, it is enough to check that every formula $\varphi \left (x;y_{1},y_{2}\right )$ is a Boolean combination of semi-equations, and, by Fact 3.10, it is enough to check this for positive atomic formulas $\varphi \left (x;y_{1},y_{2}\right )$ . Using the fact that $\land $ is associative, commutative, and idempotent, each such formula is equivalent to a formula of the form $\bigwedge A=\bigwedge B$ for non-empty $A,B\subseteq \left \{ x,y_{1},y_{2}\right \} $ . By a direct case analysis (see [Reference Chernikov and Mennen7, Theorem 3.16] for the details) every such formula is either a tautology, or does not mention x, or an equality between two variables, or is equivalent to a Boolean combination of the following formulas (possibly replacing $y_2$ by $y_1$ ):

(1) $x=x\land y_{1}$ , i.e., $x\leq y_{1}$ —a semi-equation: given $\left (a_{i},b_{i}\right )_{i\in \mathbb {Z}}$ such that $\models a_{i}\leq b_{j}\iff i\neq j$ , $a_{i}\leq b_{0}$ for $i\neq 0$ , so $\left (a_{i}\right )_{i\neq 0}$ forms a chain. This is not consistent with $a_{1}\leq b_{2}$ , $a_{2}\leq b_{1}$ , $a_{1}\not \leq b_{1}$ , $a_{2}\not \leq b_{2}$ .

(2) $x \land y_1 \land y_2 = y_1 \land y_2$ , i.e., $x \geq y_1 \land y_2$ —a semi-equation for the same reason.

(3) $x\land y_{1}=x\land y_{2}$ —a negated semi-equation: given $\left (a_{i},b_{i},b_{i}'\right )_{i\in \mathbb {Z}}$ such that $\models a_{i}\land b_{j}=a_{i}\land b_{j}'\iff i=j$ , for every $i\neq 0$ we have: either $a_{i}\land b_{0}>a_{0}\land b_{0}$ or $a_{i}\land b_{0}'>a_{0}\land b_{0}$ . By pigeonhole, there are $i_{1}\neq i_{2}$ such that the same case holds for both. Without loss of generality, $a_{1}\land b_{0}>a_{0}\land b_{0}$ and $a_{2}\land b_{0}>a_{0}\land b_{0}$ . But then $a_{1}\land a_{2}>a_{0}\land b_{0}=a_0\land a_1$ , so $a_{1}$ and $a_{2}$ meet strictly closer to each other than to $a_{0}$ . But, since $a_1\land b_1\leq a_0\land b_1$ and $a_1\land b_1\leq a_2\wedge b_1$ , it must also be true that $a_{0}\land a_{2}\geq a_1\land b_1=a_1\land a_0$ , so $a_{0}$ and $a_{2}$ meet at least as closely to each other as to $a_{1}$ . These are inconsistent.

(4) $x\land y_{1}=x\land y_{1}\land y_{2}$ (i.e., $x\land y_{1}\leq y_{2}$ )—a negated semi-equation: given $\left (a_{i},b_{i},b_{i}'\right )_{i\in \mathbb {Z}}$ such that $\models a_{i}\land b_{j}\leq b_{j}'\iff i=j$ , in particular $a_{0}\land b_{0}\leq b_{0}'$ and $a_i\land b_0\not \leq b_{0}'$ for $i\neq 0$ . Since the initial segment below $b_0$ is totally ordered, it follows that $a_0\land b_0 < a_i \land b_0$ for $i\neq 0$ . $a_1\land a_2\geq \left (a_1\land b_0\right )\land \left (a_2\land b_0\right )>a_0\land b_0=a_0\land a_1$ . That is, $a_1$ and $a_2$ meet strictly closer together with each other than with $a_0$ . But, by switching the roles of the indices $0$ and $2$ in that argument, $a_0$ and $a_1$ must meet strictly closer together with each other than with $a_2$ as well, a contradiction.

Theorem 3.13. In an infinitely branching dense tree $\mathcal {M}=(M, \land )$ , the formula $x<y$ is not a Boolean combination of $1$ -semi-equations (without parameters).

Proof By quantifier elimination, there are four complete $2$ -types over $\emptyset $ axiomatized by $\left \{ x=y,\,x>y,\,x<y,\,x\perp y\right \} $ , where $\perp $ denotes incomparable elements. Thus, up to equivalence, there are only $16$ formulas $\varphi \left (x,y\right )$ with $x,y$ singletons without parameters. By a direct case analysis (see [Reference Chernikov and Mennen7, Theorem 3.17] for the details) the only $1$ -semi-equations among them are $x\neq x$ , $x=x$ , $x=y$ , $x>y$ , $x\geq y$ . None of them separate $x<y$ from $x\perp y$ , so any Boolean combination of $1$ -semi-equations implied by $x<y$ must also be implied by $x\perp y$ , so $x<y$ is not equivalent to a Boolean combination of $1$ -semi-equations.

Corollary 3.14. In any expansion of an infinitely branching dense tree $\mathcal {M}=(M, \land )$ by naming constants, the formula $x<y$ is not a Boolean combination of $1$ -semi-equations.

Proof Suppose $x<y$ is equivalent to a Boolean combination of $1$ -semi-equations with parameters $c=\left (c_1, \ldots , c_n\right )$ . Say $x<y\iff \Phi (\varphi _1 (x,y,c ), \ldots , \varphi _k (x,y,c ))$ , where $\Phi $ is a Boolean formula in k variables, and $\varphi _1\left (x,y,c\right ), \ldots , \varphi _k\left (x,y,c\right )$ are $1$ -semi-equations. Let d be an element such that $d\perp \bigwedge _{i\leq n}c_i$ . For each i, let $\psi _i\left (x,y\right )$ be the formula $\exists z \left ( \operatorname {tp}(z) = \operatorname {tp}\left (c\right ) \& \left (x\wedge y\perp \bigwedge _{i\leq n}z_i\right ) \& \, \varphi _i\left (x,y,z\right ) \right )$ . As $\operatorname {tp}\left (c\right )$ is isolated by quantifier elimination, this is indeed a first-order formula. For $a,b>d$ , if $\models \varphi _i\left (a,b,c\right )$ , then $\models \psi _i\left (a,b\right )$ . By quantifier elimination and [Reference Simon35, Lemma 4.4], the converse also holds. Thus, for $a,b>d$ , $ \models a<b\iff \models \Phi \left (\varphi _1\left (a,b,c\right ), \ldots , \varphi _k\left (a,b,c\right )\right )\iff \models \Phi \left (\psi _1\left (a,b\right ), \ldots ,\psi _k\left (a,b\right )\right )$ . Since all singletons have the same type, it follows that this holds for all $a,b$ . It thus remains to show that each $\psi _i\left (x,y\right )$ is a $1$ -semi-equation, contradicting Theorem 3.13. If this were not the case for some $i\leq k$ , then there would be $\left (b_j\right )_{j\in \mathbb N}$ and a such that $\models \psi _i\left (a,b_j\right )$ for all $j\in \mathbb N$ , but such that for every $j\neq \ell \in \mathbb N$ , there is $a_{j,\ell }$ such that $\models \psi _i\left (a_{j,\ell },b_j\right )$ but $\not \models \psi _i\left (a_{j,\ell },b_{\ell }\right )$ . But, again because all singletons have the same type, and every finite set of elements has a lower bound, it is consistent that furthermore all of these elements are above d. But then this would also provide a counterexample to $\varphi _i\left (x,y\right )$ being a $1$ -semi-equation.

Remark 3.15. Since $x>y$ is a $\left (2,1\right )$ -semi-equation and $x<y$ is not, this shows that being an $\left (n,k\right )$ -semi-equation for fixed $n,k$ (or even being a Boolean combination of them) is not preserved under exchanging the roles of the variables (while being a semi-equation is preserved).

Remark 3.16. Note also that every tree admits an expansion in which $x<y$ is a Boolean combination of $\left (2,1\right )$ -semi-equations. In a tree, let $\leq _{\operatorname {lex}}$ be a linear order refining $\leq $ such that for $a,b,b'$ such that $a\perp b$ and $b\wedge b'>b\wedge a$ , $a\leq _{\operatorname {lex}}b\iff a\leq _{\operatorname {lex}}b'$ . Then let $\leq _{\operatorname {revlex}}$ be given by $x\leq _{\operatorname {revlex}}y \ :\iff \ x\leq y\lor \left (x\perp y\&y\leq _{\operatorname {lex}}x\right )$ . Then $\leq _{\operatorname {revlex}}$ satisfies the same conditions as $\leq _{\operatorname {lex}}$ (so both are $\left (2,1\right )$ -semi-equations as both are linear orders), and $x\leq y\iff x\leq _{\operatorname {lex}}y\&x\leq _{\operatorname {revlex}}y$ .

Problem 3.17. Is every theory of trees semi-equational? Is every theory of trees (expanded by constants) not $1$ -semi-equational?

4 Weak semi-equations and strong honest definitions

In this section we discuss how (weak) semi-equationality naturally generalizes both distality and equationality.

Definition 4.1. Given a formula $\varphi \left (x,y\right )\in \mathcal {L}$ and a type p, we denote by $p_{\varphi }^{+} := \left \{ \varphi \left (x,b\right ):\varphi \left (x,b\right )\in p\right \} $ the positive $\varphi $ -part of the type p.

Given small sets $A,B,C \subseteq \mathbb {M}$ , let denote that $\operatorname {tp}\left (A/BC\right )$ is finitely satisfiable in C. We recall the following characterization of equations from [Reference Martin-Pizarro and Ziegler24, Lemma 2.4], which in turn is a variant of [Reference Srour38, Theorem 2.5]. Note that Fact 4.2(3) is equivalent to [Reference Martin-Pizarro and Ziegler24, Lemma 2.4(3)] since in stable theories non-forking is symmetric and equivalent to finite satisfiability over models. Existence of k in Fact 4.2(2) is not stated explicitly in [Reference Martin-Pizarro and Ziegler24, Lemma 2.4(2)], but is immediate from the proof.

Fact 4.2. Given a formula $\varphi \left (x,y\right )$ in a stable theory T, the following are equivalent:

  1. (1) $\varphi \left (x,y\right )$ is an equation (equivalently, $\varphi ^*(y,x) := \varphi (x,y)$ is an equation).

  2. (2) There is some $k\in \mathbb {N}$ such that for any $a \in \mathbb {M}^x$ and small $B \subseteq \mathbb {M}^y$ , there is a subset $B_{0}$ of B of size at most k such that $\operatorname {tp}_{\varphi }^{+}\left (a/B_{0}\right )\vdash \operatorname {tp}_{\varphi }^{+}\left (a/B\right )$ .

On the other hand, we recall one of the standard characterizations of distality (see, e.g., [Reference Aschenbrenner, Chernikov, Gehret and Ziegler3, Corollary 1.11]), which we use as a definition here:

Definition 4.3. A theory is distal if and only if every formula $\varphi \left (x,y\right )$ is distal, that is, for any $I_{L}$ and $I_{R}$ infinite linear orders, $b \in \mathbb {M}^y$ and indiscernible sequence $\left (a_{i}\right )_{i\in I_{L}+\left (0\right )+I_{R}}$ with $a_i \in \mathbb {M}^x$ such that $\left (a_{i}\right )_{i\in I_{L}+I_{R}}$ is indiscernible over b, $\models \varphi \left (a_{0},b\right )\iff \models \varphi \left (a_{i},b\right )$ for $i\in I_{L}+I_{R}$ .

There is a straightforward relationship between weak semi-equationality and distality:

Remark 4.4. A formula $\varphi \left (x,y\right )$ is distal if and only if both $\varphi \left (x,y\right )$ and $\neg \varphi \left (x,y\right )$ are weak semi-equations. In particular, every distal theory is weakly semi-equational.

Problem 4.5. Is there an NIP theory without a (weakly) semi-equational expansion? We note that while the theory $\operatorname {ACF}_p$ for $p>0$ is known not to have a distal expansion [Reference Chernikov and Starchenko11], it is equational, and hence semi-equational.

An NIP theory is distal if and only if every formula admits a strong honest definition:

Fact 4.6 [Reference Chernikov and Simon10, Theorem 21].

A theory T is distal if and only if for every formula $\varphi (x,y)$ there is a formula $\theta \left (x; y_1, \ldots , y_k\right )$ , called a strong honest definition for $\varphi \left (x,y\right )$ , such that for any finite set $C \subseteq \mathbb {M}^y$ ( $\left |C\right |\geq 2$ ) and $a\in \mathbb { M}^{x}$ , there is $b\in C^{k}$ such that $\models \theta \left (a;b\right )$ and $\theta \left (x;b\right )\vdash \operatorname {tp}_{\varphi }\left (a/C\right )$ .

We now show that in an NIP theory, weak semi-equationality is equivalent to the existence of one-sided strong honest definitions, which is also a generalization of Fact 4.2 (replacing a conjunction of finitely many instances of $\varphi $ by some formula $\theta $ ). We will need the following $(p,k)$ -theorem of Matoušek from combinatorics:

Fact 4.7 [Reference Matousek26].

Let $\mathcal {F}$ be a family of subsets of some set X. Assume that the VC co-dimension of $\mathcal {F}$ is bounded by k. Then for every $p \geq k \in \mathbb {N}$ , there is $N \in \mathbb {N}$ such that: for every finite subfamily $\mathcal {G} \subseteq \mathcal {F}$ , if $\mathcal {G}$ has the $(p,k)$ -property, meaning that among any p subsets of $\mathcal {G}$ some k intersect, then there is a subset of X of size N intersecting all sets in $\mathcal {G}$ .

Theorem 4.8. Let T be NIP, and let $\varphi \left (x,y\right )$ be a formula. The following are equivalent:

  1. (1) The formula $\varphi ^*\left (y,x\right ) := \varphi (x,y)$ is a weak semi-equation.

  2. (2) For every small $B \subseteq \mathbb {M}^{y}$ and $a \in \mathbb {M}^x$ with $\models \varphi (a,b)$ for all $b \in B$ there are $\theta (x;y_1, \ldots , y_k), c \in \left ( \mathbb {M}^y \right )^k$ such that , $\models \theta (a,c)$ and $\theta (x,c) \vdash \operatorname {tp}_{\varphi }^+ \left (a/B \right )$ .

  3. (3) There is some formula $\theta \left (x;y_{1},\ldots ,y_{k}\right )$ and number N such that for any finite set $B \subseteq \mathbb {M}^y$ with $\left |B\right |\geq 2$ and $a \in \mathbb {M}^x$ , there is some $B_0 \subseteq B$ with $|B_0| \leq N$ such that $\operatorname {tp}_{\theta }^+\left (a/B_0\right ) \vdash \operatorname {tp}_{\varphi }^{+}\left (a/B\right )$ .

Proof (1) implies (2). We follow closely the proof of [Reference Chernikov and Simon10, Proposition 19]. Assume that $a, B$ are such that $\models \varphi (a,b)$ for all $b \in B$ . Let $\mathcal {M} \preceq \mathbb {M}$ contain $a,B$ , and let $\left (\mathcal {M}',B' \right ) \succ \left (\mathcal {M},B \right )$ be a $\kappa := \left | M \right |{}^+$ -saturated elementary extension (with B named by a new predicate). We may assume $\mathcal {M}' \prec \mathbb {M}$ is a small submodel. Take $p(x) := \operatorname {tp} \left (a / B' \right )$ .

Claim 4.9. Assume that $q\left (y\right )\in S_{y}\left (B'\right )$ is a type finitely satisfiable in B. Then $p\left (x\right )\cup q\left (y\right )\vdash \varphi \left (x,y\right )$ .

Proof Let $\hat {q} \in S_y(\mathbb {M})$ be an arbitrary global type extending q and finitely satisfiable in B, and form the Morley product $\hat {q}^{(\omega )}(y_1, y_2, \ldots ) := \bigotimes _{i \in \mathbb {N}} q(y_i) \in S_{(y_1, y_2, \ldots )}(\mathbb {M})$ , also finitely satisfiable in B. For any set $C \subseteq \mathbb {M}$ , we let $q|_{C} := \hat {q} \restriction _{C}$ (respectively, $q^{(\omega )}|_{C} := \hat {q}^{(\omega )} \restriction _{C}$ ) be the restriction of $\hat {q}$ (respectively, of $\hat {q}^{(\omega )}$ ) to formulas with parameters in C. As T is NIP, by [Reference Chernikov and Simon10, Lemma 5] there is some D with $B \subseteq D \subseteq B', |D| < \kappa $ such that for any two realizations $I, I' \subseteq B'$ of $q^{(\omega )}|_{D}$ we have $aI \equiv _{D} aI'$ . Fix some $I \models q^{(\omega )}|_{D}$ in $B'$ (exists by saturation of $(\mathcal {M}',B')$ and finite satisfiability of $q^{(\omega )}|_{\mathbb {M}}$ in B) and $J \models q^{(\omega )}|_{\mathbb {M}}$ (in some larger monster model $\mathbb {M}' \succ \mathbb {M}$ ).

We claim that $I+J$ is indiscernible over $aB$ . Indeed, as $q^{(\omega )}|_{\mathbb {M}}$ is finitely satisfiable in B, by compactness and saturation of $\left (\mathcal {M}', B' \right )$ there is some $J' \models q^{(\omega )}|_{a D I}$ in $B'$ . If $I+J$ is not $aB$ -indiscernible, then $I' + J'$ is not $aB$ -indiscernible for some finite subsequence $I'$ of I. As by construction both $I' + J'$ and $J'$ realize $q^{(\omega )}|_{D}$ in $B'$ , it follows by the choice of D that $J'$ is not indiscernible over $aB$ —contradicting the choice of $J'$ .

Now let $b^* \in \mathbb {M}$ be any realization of q, then the sequence $I + (b^*) + J$ is Morley in $q|_{\mathbb {M}}$ over B, hence indiscernible (even over B). And $I+J$ is indiscernible over a (even over $aB$ ) by the previous paragraph. Note also that $\models \varphi (a,b)$ for every $b \in B'$ (by assumption we had $\models \varphi (a,b)$ for all $b \in B$ , but $a \in \mathcal {M}$ and $\left (\mathcal {M}',B' \right ) \succ \left (\mathcal {M},B \right )$ ). Hence $\models \varphi (a,b)$ for every $b \in I +J$ . And since $\varphi ^*(y,x)$ is a weak semi-equation, this implies $\models \varphi (a, b^*)$ . That is, for any $a \models p$ and $b^* \models q$ , we have $\models \varphi (a,b^*)$ , as wanted.

Now let $S'$ be the set of types over $B'$ finitely satisfiable in B, then $S'$ is a closed subset of $S_y(B')$ . By the claim, for every $q \in S'$ we have $p(x) \cup q(y) \vdash \varphi (x,y)$ ; hence, by compactness $\theta _q(x) \cup \psi _q(y) \vdash \varphi (x,y)$ for some formulas $\theta _q(X) \in p, \psi _q(y) \in q$ . As $\left \{\psi _q(y) : q \in S' \right \}$ is a covering of the closed set $S'$ , it has a finite sub-covering $\left \{\psi _{q_k} : k \in K\right \}$ . Let $\theta (x) := \bigwedge _{k \in K} \theta _{q_k} (x)\in p(x)$ . As in particular $\operatorname {tp}(b/B) \in S'$ for every $b \in B$ , we thus have $\theta (x) \in \mathcal {L}(B')$ (and $B^{\prime }$ ), $\models \theta (a)$ and $\theta (x) \vdash \operatorname {tp}^+_{\varphi }(a/B)$ .

(2) implies (3). Let $a,B$ be given. We either have that $\models \neg \varphi (a,b)$ holds for all $b \in B$ , in which case $\operatorname {tp}_{\theta }^+\left (a/B_0\right ) = \operatorname {tp}_{\varphi }^{+}\left (a/B\right ) = \emptyset $ , and $\emptyset \vdash \emptyset $ trivially. Or we replace B by $\left \{ b \in B : \ \models \varphi (a,b) \right \}$ , and follow the proof of (1) implies (2) in [Reference Chernikov and Simon10, Theorem 21].

We provide the details. By (2), given small $B\subseteq \mathbb {M}^y$ and $a\in \mathbb {M}^x$ such that $\models \varphi \left (a,b\right )$ for all $b\in B$ , there exist $\theta \left (x;y_1,\ldots ,y_{\ell }\right )$ and $c\in \left (\mathbb {M}^y\right )^{\ell }$ such that , $\models \theta \left (a,c\right )$ , and $\theta \left (x,c\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/B\right )$ . Then given any finite $B_0\subseteq B$ , there is $d \in B^{\ell }$ such that $\operatorname {tp}_{\varphi }(d/a B_0) = \operatorname {tp}_{\varphi }(c/a B_0)$ , so in particular $\models \theta \left (a,d\right )$ and $\theta \left (x,d\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/B_0\right )$ .

Now fix an arbitrary function $f: \mathcal {L} \to \mathbb {N}$ and let $n_{\theta } := f \left ( \theta \left (x;y_1,\ldots ,y_{\ell }\right ) \right )$ for every partitioned formula $\theta \in \mathcal {L}$ with x the same as before and $\ell $ arbitrary. Let $T_f$ be a theory in the language $\mathcal {L} \cup \{P(x),a \}$ with P a new unary predicate and a a new constant symbol, so that $T_f$ expands T with the following axioms: $\forall x\in P\,\varphi \left (a,x\right )$ and, for every formula $\theta \left (x;y_1,\ldots ,y_{\ell }\right ) \in \mathcal {L}$ , an axiom $\exists b_1,\ldots ,b_{n_{\theta }}\in P\,\forall c\in P^{\ell }\,\left (\neg \theta \left (a,c\right )\right ) \lor \exists x\,\left (\theta \left (x,c\right ) \land \bigvee _{i\leq n_{\theta }}\neg \varphi \left (a,b_i\right )\right )$ . By the previous paragraph, the theory $T_f$ is inconsistent. By compactness, there is a finite inconsistent subset of $T_f$ only requiring finitely many of these formulas $\theta _1,\ldots ,\theta _k$ .

Thus there are finitely many formulas $\theta _1\left (x;y_1,\ldots ,y_{\ell _1}\right ),\ldots ,\theta _k\left (x;y_1,\ldots ,y_{\ell _k}\right ) \in \mathcal {L}$ such that: given $B\subseteq \mathbb {M}^y$ and $a\in \mathbb {M}^x$ such that $\models \varphi \left (a,b\right )$ for all $b\in B$ , there is $i\leq k$ such that for all $B_0\subseteq B$ with $\left |B_0\right |\leq n_{\theta _i}$ , there is $c\in B^{\ell _i}$ such that $\models \theta _i\left (a;c\right )$ and $\theta _i\left (x;c\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/B_0\right )$ .

For each formula $\theta \left (x;y_1,\ldots ,y_{\ell }\right ) \in \mathcal {L}$ , let $\rho _{\theta }\left (x,y;z\right ):=\theta \left (x;z\right )\land \forall w\,\theta \left (w;z\right )\rightarrow \varphi \left (w,y\right )$ , and $n_{\theta }:=\text {VC}\left (\rho _{\theta }\right )+1$ in the above argument (where $\operatorname {VC}$ is the VC-dimension), and let $\theta _1, \ldots , \theta _k$ be as given by the previous paragraph for this choice of the $n_{\theta }$ ’s. Then for an arbitrary $a\in \mathbb {M}^x$ and finite $B\subseteq \mathbb {M}^y$ such that $\models \varphi \left (a;b\right )$ for $b\in B$ , there is $i_{a,B}\leq k$ such that: for all $B_0\subseteq B$ with $\left |B_0\right |\leq n_{\theta _{i_{a,B}}}$ , there is $c\in B^{\ell _{i_{a,B}}}$ with $\models \theta _{i_{a,B}}\left (a;c\right )$ and $\theta _{i_{a,B}}\left (x;c\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/B_0\right )$ . For $b\in B$ , consider the set

$$ \begin{align*}F_{a,B}^b:=\left\{c\in B^{\ell_{i_{a,B}}}: \,\models\theta_{i_{a,B}}\left(a;c\right),\,\theta_{i_{a,B}}\left(x;c\right)\vdash\varphi\left(x;b\right)\right\}.\end{align*} $$

Note that $F_{a,B}^b=\left \{c\in B^{\ell _{i_{a,B}}} : \,\models \rho _{\theta _{i_{a,B}}}\left (a,b;c\right )\right \}$ , and let $\mathcal {F}_{a,B}:=\left \{F_{a,B}^b : b\in B\right \}$ . By Fact 4.7 applied to $\mathcal {F}_{a,B}$ , with $p=k=n_{\theta _{i_{a,B}}}$ , there is $N_{i_{a,B}}$ (depending on $i_{a,B}$ but not otherwise depending on $a,B$ ) such that if every $n_{\theta _{i_{a,B}}}$ sets from $\mathcal {F}_{a,B}$ intersect, then there is $B_0\subseteq B^{\ell _{i_{a,B}}}$ with $\left |B_0\right |\leq N_{i_{a,B}}$ intersecting all sets from $\mathcal {F}_{a,B}$ . Furthermore, by choice of $i_{a,B}$ , the condition that every $n_{\theta _{i_{a,B}}}$ sets from $\mathcal {F}_{a,B}$ intersect holds. And there are only k many possible values of $i_{a,B}$ , so we let $N:=\max _{1 \leq i \leq k} N_i$ .

We thus found $N\in \mathbb N$ such that: for all $a\in \mathbb {M}^x$ and finite $B\subseteq \mathbb {M}^y$ with $\models \varphi \left (a;b\right )$ for all $b\in B$ , there is $i_{a,B}\leq k$ and $B_1\subseteq B^{\ell _{i_{a,B}}}$ with $\left |B_1\right |\leq N$ intersecting all sets from $\mathcal {F}_{a,B}$ , meaning that for every $b\in B$ there is $c\in B_1$ such that $\models \theta _{i_{a,B}}\left (a;c\right )$ and $\theta _{i_{a,B}}\left (x;c\right )\vdash \varphi \left (x;b\right )$ . That is, $\operatorname {tp}_{\theta _{i_{a,B}}}^+\left (a/B_1\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/B\right )$ .

Finally, let $\theta \left (x;y_1,\ldots ,y_{\ell }\right )$ be a formula that can code for any $\theta _i\left (x;y_1,\ldots ,y_{\ell _i}\right )$ when parameters range over a set with at least two elements. For all $a\in \mathbb {M}^x$ and finite $B\subseteq \mathbb {M}^y$ with $\left |B\right |\geq 2$ , for which $\models \varphi \left (a;b\right )$ for all $b\in B$ , there is $B_0\subseteq B$ with $2\leq \left |B_0\right |\leq \ell N+2$ (consisting of the coordinates of $B_1$ from the previous paragraph, and two points for coding) such that $\operatorname {tp}_{\theta }^+\left (a/B_0\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/B\right )$ , as desired.

(3) implies (1). This follows almost verbatim from the proof of (2) implies (1) in [Reference Chernikov and Simon10, Theorem 21]. Let $I+d+J$ be an indiscernible sequence in $\mathbb M^y$ , with I and J infinite, and $I+J$ indiscernible over $a\in \mathbb M^x$ , and suppose $\models \varphi \left (a,b\right )$ for $b\in I+J$ . Let $I_1\subset I$ with $\left |I_1\right |=N+1$ . Then there is some $I_0\subseteq I_1$ such that $\left |I_0\right |\leq N$ and $\operatorname {tp}_{\theta }^+\left (a/I_0\right )\vdash \operatorname {tp}_{\varphi }^+\left (a/I_1\right )$ . Let $b\in I_1\setminus I_0$ . By indiscernibility of $I+d+J$ , there is some $\sigma \in \text {Aut}\left (\mathbb M\right )$ such that $\sigma \left (I_1\right )\subset I+d+J$ and $\sigma \left (b\right )=d$ . We have $\sigma \left (I_0\right )\subseteq I+J$ , so by a-indiscernibility of $I+J$ , $\models \theta \left (a,\sigma \left (c\right )\right )$ for every $c\in I_0^k$ for which $\models \theta \left (a,c\right )$ , and hence $a\models \sigma \left (\operatorname {tp}_{\varphi }^+\left (a/I_1\right )\right )$ . And $\varphi \left (x,b\right )\in \operatorname {tp}_{\varphi }^+\left (a/I_1\right )$ , so $\varphi \left (x,d\right )\in \sigma \left (\operatorname {tp}_{\varphi }^+\left (a/I_1\right )\right )$ , and hence $\models \varphi \left (a,d\right )$ .

Problem 4.10. Can the assumption that T is NIP be omitted? (Note that the proof of (3) implies (1) does not use it.)

Proposition 2.16 immediately implies an analog of Fact 4.2 for semi-equations, telling us that $\varphi \left (x;y\right )$ is a one-sided strong honest definition for itself:

Corollary 4.11. A formula $\varphi \left (x,y\right )$ (equivalently, $\varphi ^*(y,x)$ ) is a semi-equation if and only if there is some $k\in \mathbb {N}$ such that: for every finite $B\subset \mathbb { M}^{y}$ and $a\in \mathbb { M}^{x}$ there is some $B_0 \subseteq B$ with $|B_0| \leq k$ such that $ \operatorname {tp}_{\varphi }^+(a/B_0) \vdash \operatorname {tp}_{\varphi }^{+}\left (a/B\right )$ .

5 Non weakly semi-equational valued fields

In this section we demonstrate that many valued fields are not weakly semi-equational. By an $\operatorname {ac}$ -valued field field we mean a three-sorted structure $\left (K,k,\Gamma , \nu , \operatorname {ac} \right )$ in the Denef-Pas language, where K is a field, $\nu : K \to \Gamma $ is a valuation, with (ordered) value group $\Gamma $ and residue field k, and $\operatorname {ac}: K \to k$ is the angular component map. As usual, $\mathcal {O} = \mathcal {O}_{\nu }$ denotes the valuation ring of $\nu $ , and for $x \in \mathcal {O}$ , $\bar {x}$ denotes the residue of x in k. The following is the main theorem of the section:

Theorem 5.1. Let K be an ac-valued field for which the residue field k contains a non-constant totally indiscernible sequence (for instance, if k is infinite and stable), and which eliminates quantifiers of the main field sort (for example, a Henselian ac-valued field of equicharacteristic $0$ with an algebraically closed residue field). Then K is not weakly semi-equational.

Before presenting its proof, we need to develop some auxiliary results. First we provide a general sufficient criterion for when a formula is not a Boolean combination of weak semi-equations in Section 5.1. Then we discuss valuational independence in Section 5.2. In Section 5.3 we describe a particular configuration of elements in a valued field indented to satisfy this sufficient criterion with respect to the formula $\psi (x_1,x_2; y_1,y_2) := \nu \left (x_{1}-y_{1}\right )<\nu \left (x_{2}-y_{2}\right )$ , and reduce demonstrating that it has all of the required properties to Claims 5.7 and 5.8 which express a certain amount of indiscernibility of our configuration. We also explain how both claims can be proved by induction on the complexity of the formula and reduce to several essential cases that have to be considered; and show in Claim 5.9 valuational independence of some elements of our configuration which will be helpful in the proof of the claims. We then prove Claim 5.7 in Section 5.4 and Claim 5.8 in Section 5.5, concluding the proof of Theorem 5.1. Finally, in Section 5.6 we discuss some further applications of Theorem 5.1 and examples.

5.1 Boolean combinations of weak semi-equations

We provide a sufficient criterion for when a formula is not a Boolean combination of weak semi-equations (analogous to a criterion for equations from [Reference Müller and Sklinos28]).

Lemma 5.2. If $\varphi \left (x,y\right )$ and $\psi \left (x,y\right )$ are weak semi-equations, then there are no $b \in \mathbb {M}^{y}$ and array $\left (a_{i,j}\right )_{i,j\in \mathbb {Z}}$ with $a_{i,j} \in \mathbb {M}^x$ such that:

  • Every row (i.e., $(a_{i,j} : j \in \mathbb {Z})$ for a fixed $i \in \mathbb {Z}$ ) and every column (i.e., $(a_{i,j} : i \in \mathbb {Z})$ for a fixed $j \in \mathbb {Z}$ ) is indiscernible (over $\emptyset $ ).

  • Rows and columns without their $0$ -indexed elements (i.e., $\left (a_{i,j}\right )_{j\neq 0}$ for fixed i, and $\left (a_{i,j}\right )_{i\neq 0}$ for fixed j) are b-indiscernible.

  • $\models \varphi \left (a_{i,j},b\right )\land \neg \psi \left (a_{i,j},b\right )\iff i=0\lor j\neq 0$ .

Proof Assume there exist an array $\left ( a_{i,j} : i,j \in \mathbb {Z} \right )$ and b with these properties. For any fixed $i\neq 0$ , we have $\models \varphi \left (a_{i,j},b\right )$ for all $j \neq 0$ , $\left (a_{i,j}\right )_{j \in \mathbb {Z}}$ is indiscernible, and $\left (a_{i,j}\right )_{j\neq 0}$ is b-indiscernible, so, by weak semi-equationality of $\varphi $ , $\models \varphi \left (a_{i,0},b\right )$ . But $\not \models \varphi \left (a_{i,0},b\right )\land \neg \psi \left (a_{i,0},b\right )$ , so $\models \psi \left (a_{i,0},b\right )$ . Now the sequence $\left (a_{i,0}\right )_{i \in \mathbb {Z}}$ is indiscernible, $\left (a_{i,0}\right )_{i\neq 0}$ is b-indiscernible, and $\models \psi (a_{i,0},b)$ for all $i \neq 0$ , so, by weak semi-equationality of $\psi $ , $\models \psi \left (a_{0,0},b\right )$ —contradicting $\models \varphi \left (a_{0,0},b\right )\land \neg \psi \left (a_{0,0},b\right )$ .

Lemma 5.3. If $\varphi \left (x,y\right )$ is a Boolean combination of weak semi-equations, then there are no $b \in \mathbb {M}^y$ and array $\left (a_{i,j}\right )_{i,j\in \mathbb {Z}}$ with $a_{i,j} \in \mathbb {M}^x$ such that:

  • Rows and columns of $\left (a_{i,j}\right )_{i,j\in \mathbb {Z}}$ are indiscernible.

  • Rows and columns without their $0$ -indexed elements (i.e., $\left (a_{i,j}\right )_{j\neq 0}$ for fixed i, and $\left (a_{i,j}\right )_{i\neq 0}$ for fixed j) are b-indiscernible.

  • $\models \varphi \left (a_{i,j},b\right )\iff i=0\lor j\neq 0$ .

  • All $a_{i,j}$ with $i=0$ or $j\neq 0$ have the same type over b.

Proof Any conjunction of finitely many weak semi-equations and negations of weak semi-equations is of the form $\psi \left (x,y\right )\land \neg \theta \left (x,y\right )$ for some weak semi-equations $\psi \left (x,y\right )$ and $\theta \left (x,y\right )$ , because weak semi-equations are closed under conjunction and under disjunction (Proposition 2.3(3)), so negations of weak semi-equations are also closed under conjunction. Thus any Boolean combination of weak semi-equations is equivalent, via its disjunctive normal form, to $\bigvee _{k\in I} \left ( \psi _{k}\left (x,y\right )\land \neg \theta _{k}\left (x,y\right ) \right )$ for some finite index set I and weak semi-equations $\psi _{k}\left (x,y\right )$ and $\theta _{k}\left (x,y\right )$ for $k\in I$ . Given b and $\left (a_{i,j}\right )_{i.j\in \mathbb {Z}}$ as above, since $i=0\lor j\neq 0\iff \models \varphi \left (a_{i,j},b\right )\iff \models \bigvee _{k\in I} \left ( \psi _{k}\left (a_{i,j},b\right )\land \neg \theta _{k}\left (a_{i,j},b\right ) \right )$ , and all $a_{i,j}$ with $i=0$ or $j\neq 0$ have the same type over b, there is some k such that $\models \psi _{k}\left (a_{i,j},b\right )\land \neg \theta _{k}\left (a_{i,j},b\right )\iff i=0\lor j\neq 0$ , contradicting Lemma 5.2.

5.2 Valuational independence

Definition 5.4. Let K be a field with valuation $\nu $ .

  1. (1) We say that $a_{1}, \ldots , a_{n}\in K$ are valuationally independent if, for every polynomial $f\left (x_{1}, \ldots , x_{n}\right )=\sum _{i}c_{i}x_{1}^{\alpha _{1,i}} \ldots x_{n}^{\alpha _{n,i}}$ (where i runs over some finite index set, $c_{i},\alpha _{1,i}, \ldots , \alpha _{n,i}\in \mathbb {Z}$ , and $\left (\alpha _{1,i}, \ldots , \alpha _{n,i}\right )\neq \left (\alpha _{1,j}, \ldots , \alpha _{n,j}\right )$ for $i\neq j$ ) we have

    $$ \begin{align*}\nu\left(f\left(a_{1}, \ldots, a_{n}\right)\right)=\min_{i}\nu\left(c_{i}a_{1}^{\alpha_{1,i}} \ldots a_{n}^{\alpha_{n,i}}\right).\end{align*} $$
    That is, if the valuation of every polynomial applied to $a_{1}, \ldots , a_{n}$ is the minimum of the valuations of its monomials (including their coefficients).
  2. (2) An infinite set is valuationally independent if every finite subset is.

Example 5.5. (1) A set of elements with valuation $0$ is valuationally independent if and only if their residues are algebraically independent.

(2) In a valued field of pure characteristic, every set of elements whose valuations are $\mathbb {Z}$ -linearly independent is valuationally independent. In mixed characteristic $\left (0,p\right )$ , every set of elements whose valuations, together with $\nu \left (p\right )$ , are $\mathbb {Z}$ -linearly independent, is valuationally independent. In an ac-valued field, this is the only way for a set of elements with angular component $1$ to be valuationally independent.

5.3 Reducing the proof of Theorem 5.1 to two claims

We will show that the partitioned formula $\psi (x_1,x_2; y_1,y_2) := \nu \left (x_{1}-y_{1}\right )<\nu \left (x_{2}-y_{2}\