We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the “magic manifold” is bi-orderable, answering a question of Kin and Rolfsen.

This paper is motivated by two conjectures proposed by Bender et al. [‘Complemented zero-divisor graphs associated with finite commutative semigroups’, Comm. Algebra 52(7) (2024), 2852–2867], which have remained open questions. The first conjecture states that if the complemented zero-divisor graph $ G(S) $ of a commutative semigroup $ S $ with a zero element has clique number three or greater, then the reduced graph $ G_r(S) $ is isomorphic to the graph $ G(\mathcal {P}(n)) $. The second conjecture asserts that if $ G(S) $ is a complemented zero-divisor graph with clique number three or greater, then $ G(S) $ is uniquely complemented. We construct a commutative semigroup $ S $ with a zero element that serves as a counter-example to both conjectures.

Let $\Sigma$ be an alphabet and $\mu$ be a distribution on $\Sigma ^k$ for some $k \geqslant 2$. Let $\alpha \gt 0$ be the minimum probability of a tuple in the support of $\mu$ (denoted $\mathsf{supp}(\mu )$). We treat the parameters $\Sigma , k, \mu , \alpha$ as fixed and constant. We say that the distribution $\mu$ has a linear embedding if there exist an Abelian group $G$ (with the identity element $0_G$) and mappings $\sigma _i : \Sigma \rightarrow G$, $1 \leqslant i \leqslant k$, such that at least one of the mappings is non-constant and for every $(a_1, a_2, \ldots , a_k)\in \mathsf{supp}(\mu )$, $\sum _{i=1}^k \sigma _i(a_i) = 0_G$. In [Bhangale-Khot-Minzer, STOC 2022], the authors asked the following analytical question. Let $f_i: \Sigma ^n\rightarrow [\!-1,1]$ be bounded functions, such that at least one of the functions $f_i$ essentially has degree at least $d$, meaning that the Fourier mass of $f_i$ on terms of degree less than $d$ is at most $\delta$. If $\mu$ has no linear embedding (over any Abelian group), then is it necessarily the case that

\begin{equation*}\left | \mathop {\mathbb{E}}_{({\textbf {x}}_1, {\textbf {x}}_2, \ldots , {\textbf {x}}_k)\sim \mu ^{\otimes n}}[f_1({\textbf {x}}_1)f_2({\textbf {x}}_2)\cdots f_k({\textbf {x}}_k)] \right | = o_{d, \delta }(1),\end{equation*}

$\to 0$

$d \to \infty$

$\delta \to 0$

In this paper, we answer this analytical question fully and in the affirmative for $k=3$. We also show the following two applications of the result.

1. 1. The first application is related to hardness of approximation. Using the reduction from [5], we show that for every $3$-ary predicate $P:\Sigma ^3 \to \{0,1\}$ such that $P$ has no linear embedding, an SDP (semi-definite programming) integrality gap instance of a $P$-Constraint Satisfaction Problem (CSP) instance with gap $(1,s)$ can be translated into a dictatorship test with completeness $1$ and soundness $s+o(1)$, under certain additional conditions on the instance.

2. 2. The second application is related to additive combinatorics. We show that if the distribution $\mu$ on $\Sigma ^3$ has no linear embedding, marginals of $\mu$ are uniform on $\Sigma$, and $(a,a,a)\in \texttt{supp}(\mu )$ for every $a\in \Sigma$, then every large enough subset of $\Sigma ^n$ contains a triple $({\textbf {x}}_1, {\textbf {x}}_2,{\textbf {x}}_3)$ from $\mu ^{\otimes n}$ (and in fact a significant density of such triples).

The Kruskal–Friedman theorem asserts: in any infinite sequence of finite trees with ordinal labels, some tree can be embedded into a later one, by an embedding that respects a certain gap condition. This strengthening of the original Kruskal theorem has been proved by I. Kříž (Ann. Math. 1989), in confirmation of a conjecture due to H. Friedman, who had established the result for finitely many labels. It provides one of the strongest mathematical examples for the independence phenomenon from Gödel’s theorems. The gap condition is particularly relevant due to its connection with the graph minor theorem of N. Robertson and P. Seymour. In the present article, we consider a uniform version of the Kruskal–Friedman theorem, which extends the result from trees to general recursive data types. Our main theorem shows that this uniform version is equivalent both to $\Pi ^1_1$-transfinite recursion and to a minimal bad sequence principle of Kříž, over the base theory $\mathsf {RCA_0}$ from reverse mathematics. This sheds new light on the role of infinity in finite combinatorics.

Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $, where the neutral element for the product is an initial object, we consider the poset of $\sqcup $-complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with $\sqcup $, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness.

In well-studied scenarios, the poset of $\sqcup $-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of nondegenerate subspaces of a vector space with a nondegenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.

Let W be a simply laced Weyl group of finite type and rank n. If W has type $E_7$, $E_8$ or $D_n$ for n even, then the root system of W has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of W spanned by n-roots, which are products of n orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains–Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system.

We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of ${\mathbb N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and $\mathrm {C}^{*}$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.

The distinction of the semantic spaces of elements and types is common practice in practically all type systems. A few type systems, including some early ones, have been proposed whose semantic space has functions only, i.e., depending on the context functions may play element roles as well as type roles. All of these systems are either lacking expressive power, in particular, polymorphism, or they violate uniqueness of types. This work presents for the first time a function-based type system in which typing is a relation between functions and which is using an ordering of functions to introduce bounded polymorphism. The ordering is based on an infinite set of top objects, itself strictly linearly ordered, each of which characterizes a certain function space. These top objects are predicative in the sense that a function using some top object cannot be smaller than this object. The interpretation of proposition as types and elements as proofs remains valid and is extended by viewing the ordering between types as logical implication. The proposed system can be shown to satisfy confluence and subject reduction. Furthermore one can show that the ordering is a partial order, every set of expressions has a maximal element, and there is a (unique) minimal, logically strongest, type among all types of an element. The latter result implies an alternative notion of uniqueness of types. Strong normalisation is the deepest property and its proof is based on a well-founded relation defined over a subsystem of expressions without eliminators. Semantic abstraction of the objects involved in typing, i.e., to use functions in element as well as type roles in a relational setting, is the major contribution of function-based type systems. This work shows that dependent products are not necessary for defining type systems with bounded polymorphism, rather it presents a consistent system with bounded polymorphism and minimal types where typing is a relation between partially ordered functions.

Unital quantales constitute a significant subclass within quantale theory, which play a crucial role in the theoretical framework of quantale research. The main purpose of this article is to investigate the construction of unital quantales from a given quantale. Using Q-algebras, we prove that every quantale is embedded into a unital quantale, which generalizes the work of Paseka and Kruml for the construction of unital quantales. Based on which, we further show that every quantale can be transformed into a unitally non-distributive quantale, which expands the foundational work of Guriérrez García and Höhle for unitally non-distributive quantales. Finally, we provide a variety of methods for constructing unital quantales from some special quantales.

On relational structures and on polymodal logics, we describe operations which preserve local tabularity. This provides new sufficient semantic and axiomatic conditions for local tabularity of a modal logic. The main results are the following.

We show that local tabularity does not depend on reflexivity. Namely, given a class $\mathcal {F}$ of frames, consider the class $\mathcal {F}^{\mathrm {r}}$ of frames, where the reflexive closure operation was applied to each relation in every frame in $\mathcal {F}$. We show that if the logic of $\mathcal {F}^{\mathrm {r}}$ is locally tabular, then the logic of $\mathcal {F}$ is locally tabular as well.

Then we consider the operation of sum on Kripke frames, where a family of frames-summands is indexed by elements of another frame. We show that if both the logic of indices and the logic of summands are locally tabular, then the logic of corresponding sums is also locally tabular.

Finally, using the previous theorem, we describe an operation on logics that preserves local tabularity: we provide a set of formulas such that the extension of the fusion of two canonical locally tabular logics with these formulas is locally tabular.

The logico-algebraic study of Lewis’s hierarchy of variably strict conditional logics has been essentially unexplored, hindering our understanding of their mathematical foundations, and the connections with other logical systems. This work starts filling this gap by providing a logico-algebraic analysis of Lewis’s logics. We begin by introducing novel finite axiomatizations for Lewis’s logics on the syntactic side, distinguishing between global and local consequence relations on Lewisian sphere models on the semantical side, in parallel to the case of modal logic. As first main results, we prove the strong completeness of the calculi with respect to the corresponding semantical consequence on spheres, and a deduction theorem. We then demonstrate that the global calculi are strongly algebraizable in terms of a variety of Boolean algebras with a binary operator representing the counterfactual implication; in contrast, we show that the local ones are generally not algebraizable, although they can be characterized as the degree-preserving logic over the same algebraic models. This yields the strong completeness of all the logics with respect to the algebraic models.

It is a classic result of Segerberg and Maksimova that a variety of $\mathsf {S4}$-algebras is locally finite iff it is of finite depth. Since the logic $\mathsf {MS4}$ (monadic $\mathsf {S4}$) axiomatizes the one-variable fragment of $\mathsf {QS4}$ (predicate $\mathsf {S4}$), it is natural to try to generalize the Segerberg–Maksimova theorem to this setting. We obtain several results in this direction. Our positive results include the identification of the largest semisimple variety of $\mathsf {MS4}$-algebras. We prove that the corresponding logic $\mathsf {MS4_S}$ has the finite model property. We show that both $\mathsf {S5}^2$ and $\mathsf {S4}_u$ are proper extensions of $\mathsf {MS4_S}$, and that a direct generalization of the Segerberg–Maksimova theorem holds for a family of varieties containing the variety of $\mathsf {S4}_u$-algebras. Our negative results include a translation of varieties of $\mathsf {S5}_2$-algebras into varieties of $\mathsf {MS4_S}$-algebras of depth 2, which preserves and reflects local finiteness. This, in particular, shows that the problem of characterizing locally finite varieties of $\mathsf {MS4}$-algebras (even of $\mathsf {MS4_S}$-algebras) is at least as hard as that of characterizing locally finite varieties of $\mathsf {S5}_2$-algebras—a problem that remains wide open.

The group of order-preserving automorphisms of a finitely generated Archimedean ordered group of rank $2$ is either infinite cyclic or trivial according as the ratio in $\mathbb {R}$ of the generators of the subgroup is or is not quadratic over $\mathbb {Q}.$ In the case of an Archimedean ordered group of rank $2$ that is not finitely generated, the group of order-preserving automorphisms is free abelian. Criteria determining the rank of this free abelian group are established.

The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of logarithmic derivation modules of hyperplane arrangements. This concept was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards investigated further by Cuntz-Mücksch (2020).

In the particular case of graphic arrangements, the last two authors (2023) recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this paper, we first introduce a poset characterization of a vine. Then we show that, interestingly, there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vines and MAT-labeled complete graphs.

Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mücksch that MAT-freeness can be characterized by a generalization of the root poset in the case of graphic arrangements.

We show that for $\mathrm {C}^*$-algebras with the global Glimm property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a $\mathrm {C}^*$-algebra is determined by the soft part of its Cuntz semigroup.

Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

We study the problem of extending an order-preserving real-valued Lipschitz map defined on a subset of a partially ordered metric space without increasing its Lipschitz constant and preserving its monotonicity. We show that a certain type of relation between the metric and order of the space, which we call radiality, is necessary and sufficient for such an extension to exist. Radiality is automatically satisfied by the equality relation, so the classical McShane–Whitney extension theorem is a special case of our main characterization result. As applications, we obtain a similar generalization of McShane’s uniformly continuous extension theorem, along with some functional representation results for radial partial orders.

We give a short proof of a theorem of Beleznay asserting that the set $L2$ of reals coding linear orders of the form $I + I$ is complete analytic.

Assuming Stanley’s P-partitions conjecture holds, the regular Schur labeled skew shape posets are precisely the finite posets P with underlying set $\{1, 2, \ldots , |P|\}$ such that the P-partition generating function is symmetric and the set of linear extensions of P, denoted $\Sigma _L(P)$, is a left weak Bruhat interval in the symmetric group $\mathfrak {S}_{|P|}$. We describe the permutations in $\Sigma _L(P)$ in terms of reading words of standard Young tableaux when P is a regular Schur labeled skew shape poset, and classify $\Sigma _L(P)$’s up to descent-preserving isomorphism as P ranges over regular Schur labeled skew shape posets. The results obtained are then applied to classify the $0$-Hecke modules $\mathsf {M}_P$ associated with regular Schur labeled skew shape posets P up to isomorphism. Then we characterize regular Schur labeled skew shape posets as the finite posets P whose linear extensions form a dual plactic-closed subset of $\mathfrak {S}_{|P|}$. Using this characterization, we construct distinguished filtrations of $\mathsf {M}_P$ with respect to the Schur basis when P is a regular Schur labeled skew shape poset. Further issues concerned with the classification and decomposition of the $0$-Hecke modules $\mathsf {M}_P$ are also discussed.

Describing the equality conditions of the Alexandrov–Fenchel inequality [Ale37] has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem and is a complexity counterpart of the recent result by Shenfeld and van Handel [SvH23], which gave a geometric characterization of the equality conditions. The proof involves Stanley’s [Sta81] order polytopes and employs poset theoretic technology.

For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$, and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa.

We investigate the initial structure of pairs $(X,\mathcal {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces.

Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $-pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$.