POLYHEDRAL COMPLETENESS OF INTERMEDIATE LOGICS: THE NERVE CRITERION

2.1 Posets as Kripke frames

A (Kripke) frame for intuitionistic logic is simply a poset. We thus use the term ‘frame’ in this paper as a synonym of ‘poset’. The validity relation $\vDash $ between frames and formulas is defined in the usual way (see, e.g., [Reference Chagrov and Zakharyaschev14, Chapter 2]). Given a class of frames $\mathbf {C}$ , its logic is

Conversely, given a logic $\mathcal {L}$ , define

A logic $\mathcal {L}$ has the finite model property (fmp) if it is the logic of a class of finite frames. Equivalently, if $\mathcal {L} = \operatorname {{\mathrm {Logic}}}(\operatorname {{\mathrm {Frames}_{\mathrm {fin}}}}(\mathcal {L}))$ .

Fix a poset F. For any $x \in F$ , its upset, downset, strict upset, and strict downset are defined, respectively, as follows:

For any set $S \subseteq F$ , its upset and downset are defined, respectively, as follows:

A subframe is a subposet. A subframe $U \subseteq F$ is upwards-closed or a generated subframe if $U = \operatorname {\mathrm {\uparrow }} U$ , and it is downwards-closed if $\operatorname {\mathrm {\downarrow }} U = U$ . The Alexandrov topology on F is the set $\operatorname {{\mathrm {Up}}} F$ of its upwards-closed subsets. This constitutes a topology on F. In the sequel, we will freely switch between thinking of F as a poset and as a topological space. Note that the closed sets in this topology correspond to downwards-closed sets.

A chain in F is $X \subseteq F$ which as a subposet is linearly ordered. The length of the chain X is $\mathopen |X\mathclose |$ . A chain $X \subseteq F$ is maximal if there is no chain $Y \subseteq F$ such that $X \subset Y$ (i.e., such that X is a proper subset of Y). The height of F is the element of $\mathbb N \cup \{\infty \}$ defined by

For notational uniformity, say that this value is also the depth of F, $\operatorname {\mathrm {\mathsf {depth}}}(F)$ . For any $x \in F$ , define its height and depth as follows:

The height of a logic $\mathcal {L}$ is the element of $\mathbb N \cup \{\infty \}$ given by

A top element of F is $t \in F$ such that $\operatorname {\mathrm {\mathsf {depth}}}(t)=0$ . For any $x,y \in F$ , say that x is an immediate predecessor of y, and that y is an immediate successor of x, if $x < y$ and there is no $z \in F$ such that $x < z < y$ . Write $\operatorname {{\mathrm {Succ}}}(x)$ for the collection of immediate successors of x.

The poset F is rooted if it has a minimum element, which is called the root, and is usually denoted by $\bot $ . Define

An antichain in F is a subset $Z \subseteq F$ in which no two elements are comparable. The width, notation $\operatorname {\mathrm {\mathsf {width}}}(F)$ , of F is the cardinality of the largest antichain in F.

A function $f \colon F \to G$ is a p-morphism if for every $x \in F$ we have

$$ \begin{align*} f({\uparrow}(x) ) = {\Uparrow} (f(x)). \end{align*} $$

Equivalently, f should satisfy the following conditions:

(Forth) $$ \begin{align} \forall x,y \in F\colon(x\leqslant y \Rightarrow f(x) \leqslant f(y)), \end{align} $$

(Back) $$ \begin{align} \forall x \in F \colon \forall z \in G \colon (f(x) \leqslant z \Rightarrow \exists y \in F \colon (x \leqslant y \wedge f(y)=z)). \end{align} $$

An up-reduction from F to G is a surjective p-morphism f from an upwards-closed set $U \subseteq F$ to G. Write $f \colon F {\mathrel {{\circ }\mkern -3mu{\rightarrow }}} G$ .

Corollary 2.2. If $\mathbf {C}$ is any collection of frames and $\mathcal {L} = \operatorname {{\mathrm {Logic}}}(\mathbf {C})$ , then

$$ \begin{align*} \mathcal{L} = \operatorname{{\mathrm{Logic}}}(\operatorname{{\mathrm{Frames}_{\bot}}}(\mathcal{L})). \end{align*} $$

2.2 Heyting algebras, topological semantics

A Heyting algebra is a bounded lattice equipped with a Heyting implication $\rightarrow $ that satisfies

$$ \begin{align*} c \leqslant a \rightarrow b \quad\Leftrightarrow\quad c \wedge a \leqslant b. \end{align*} $$

The validity relation $\vDash $ between Heyting algebras and formulas is defined in the usual way; the notation $\operatorname {{\mathrm {Logic}}}(-)$ is extended appropriately. The logic of a Heyting algebra is exactly the logic of its finitely generated subalgebras. Say that A is locally finite if for every $S \subseteq A$ finite, the algebra ${\langle S \rangle }$ generated by S is finite. If F is any poset, the bounded distributive lattice $\operatorname {{\mathrm {Up}}} F$ is a Heyting algebra, and:

Co-Heyting algebras are the order-duals of Heyting algebras. Specifically, a co-Heyting algebra is a bounded lattice equipped with a co-Heyting implication $\leftarrow $ that satisfies

$$ \begin{align*} a \leftarrow b \leqslant c \quad\Leftrightarrow\quad a \leqslant b \vee c. \end{align*} $$

For more information on co-Heyting algebras, the reader is referred to [Reference McKinsey and Tarski22, Section 1] and [Reference Rauszer26], where they are called ‘Brouwerian algebras’.

Given a topological space X, we regard the collection of open sets $\operatorname {\mathrm {\mathcal {O}}}(X)$ of X as a Heyting algebra in the standard manner (cf. [Reference Chagrov and Zakharyaschev14, Proposition 8.31, p. 247]). (Recall that

$$ \begin{align*} U \rightarrow V = \operatorname{{\mathrm{Int}}}(U^{\mathsf{C}} \cup V), \end{align*} $$

where $\operatorname {{\mathrm {Int}}}$ denotes the topological interior operator, and $-^{\mathsf {C}}$ is set-theoretic complement.) We can thus interpret formulas in topological spaces. Write $X \vDash \phi $ for $\operatorname {\mathrm {\mathcal {O}}}(X) \vDash \phi $ , and extend the remaining notations accordingly.

The topological space X also comes with a co-Heyting algebra, namely its collection of closed sets $\operatorname {\mathrm {\mathcal {C}}}(X)$ . The co-Heyting implication on $\operatorname {\mathrm {\mathcal {C}}}(X)$ satisfies

where $\operatorname {{\mathrm {Cl}}}$ denotes the topological closure operator. If a Heyting algebra A is regarded as a poset category $(A, \leqslant )$ , then its opposite category $A^{\mathrm {op}} = (A, \geqslant )$ is a co-Heyting algebra. In the case of the Heyting algebra $\operatorname {\mathrm {\mathcal {O}}}(X)$ of open sets of X, $\operatorname {\mathrm {\mathcal {O}}}(X)^{\mathrm {op}}$ is isomorphic to the co-Heyting algebra $\operatorname {\mathrm {\mathcal {C}}}(X)$ of closed subsets of X.

2.3 Jankov–Fine formulas as forbidden configurations

To every finite, rooted frame Q, we associate a formula $\chi (Q)$ , the Jankov–Fine formula of Q (also called its Jankov–De Jongh formula). The precise definition of $\chi (Q)$ is somewhat involved, but the exact details of this syntactical form are not relevant for our considerations. What matters to us is its notable semantic property.

Jankov–Fine formulas formalise the intuition of ‘forbidden configurations’. The formula $\chi (Q)$ ‘forbids’ the configuration Q from its frames.

The following consequence of Theorem 2.4 will come in handy later on.

Corollary 2.5. Let $\mathcal {L} = \operatorname {{\mathrm {Logic}}}(\mathbf {C})$ where $\mathbf {C}$ is a class of frames. Then

$$ \begin{align*} \operatorname{{\mathrm{Frames}_{\bot,\mathrm{fin}}}}(\mathcal{L}) = \{F \text{ finite, rooted frame} \mid \exists G \in \mathbf{C} \colon G {\mathrel{{\circ}\mkern-3mu{\rightarrow}}} F\}. \end{align*} $$

2.4 Intermediate logics

The logic $\mathbf {IPC}$ is intuitionistic propositional logic. An intermediate logic is any consistent logic extending $\mathbf {IPC}$ . Classical logic, $\mathbf {CPC}$ , is the largest intermediate logic.

For every $n \in \mathbb N$ , let $\mathbf {BD}_n$ be the logic of all finite frames of height at most n. This has the following axiomatisation in terms of Jankov–Fine formulas.

Scott’s Logic, $\mathbf {SL}$ , is usually axiomatised by the Scott sentence:

$$ \begin{align*} \mathbf{SL} = \mathbf{IPC} + \mathbf{IPC} + ((\neg\neg p \rightarrow p) \rightarrow p \vee \neg p) \rightarrow \neg p \vee \neg\neg p. \end{align*} $$

This logic can also be axiomatised using a forbidden configuration, as follows.

Proposition 2.8.

.

2.5 Polytopes, polyhedra, and simplices

Polyhedra are certain subsets of finite-dimensional real affine spaces $\mathbb {R}^n$ . An affine combination of $x_0, \ldots , x_d \in \mathbb {R}^n$ is a point $r_0 x_0 + \cdots + r_d x_d$ , where $r_0, \ldots , r_d \in \mathbb {R}$ are such that $r_0 + \cdots + r_d = 1$ . A convex combination is an affine combination in which additionally each $r_i \geqslant 0$ . Given a set $S \subseteq \mathbb {R}^n$ , its convex hull, notation $\operatorname {{\mathrm {Conv}}} S$ , is the collection of convex combinations of its elements. (We stress that each convex combination involves, by definition, a finite subset of S only.) A subspace $S \subseteq \mathbb {R}^n$ is convex if $\operatorname {{\mathrm {Conv}}} S = S$ . A polytope is the convex hull of a finite set. A polyhedron in $\mathbb {R}^n$ is a set which can be expressed as the finite union of polytopes. A subpolyhedron of a polyhedron P in $\mathbb {R}^n$ is a subset of P which is itself a polyhedron. Note that every polyhedron is closed and bounded, hence compact, in the canonical (Euclidean) topology carried by the real affine space $\mathbb {R}^n$ . All topological notions pertaining to polyhedra in the following refer to this topology.

A set of points $x_0, \ldots , x_d$ is affinely independent if whenever

$$ \begin{align*} r_0 x_0 + \cdots + r_d x_d = \mathbf 0 \quad\text{and}\quad r_0 + \cdots + r_d = 0, \end{align*} $$

we must have that $r_0, \ldots , r_d = 0$ . This is equivalent to saying that the vectors

$$ \begin{align*} x_1 - x_0, \ldots, x_d - x_0 \end{align*} $$

are linearly independent. A d-simplex is the convex hull $\sigma $ of $d+1$ affinely independent points $x_0, \ldots , x_d$ , which we call its vertices. Write $\sigma = x_0\cdots x_d$ ; the dimension of $\sigma $ is d.

A face of $\sigma $ is the convex hull $\tau $ of some non-empty subset of $\{x_0, \ldots , x_d\}$ (note that $\tau $ is then a simplex too). Write $\tau \preccurlyeq \sigma $ , and $\tau \prec \sigma $ if $\tau \neq \sigma $ .

Since $x_0, \ldots , x_d$ are affinely independent, every point $x \in \sigma $ can be expressed uniquely as a convex combination $x = r_0 x_0 + \cdots + r_d x_d$ with $r_0, \ldots , r_d \geqslant 0$ and $r_0 + \cdots + r_d = 1$ . Call the tuple $(r_0, \ldots , r_d)$ the barycentric coordinates of x in $\sigma $ . The barycentre $\widehat \sigma $ of $\sigma $ is the special point whose barycentric coordinates are $(\frac {1}{d+1}, \ldots , \frac {1}{d+1})$ . The relative interior of $\sigma $ is defined:

The relative interior of $\sigma $ is ‘ $\sigma $ without its boundary’ in the following sense. The affine subspace spanned by $\sigma $ is the set of all affine combinations of $x_0, \ldots , x_d$ . Then the relative interior of $\sigma $ coincides with the topological interior of $\sigma $ inside this affine subspace, the latter being equipped with the subspace topology it inherits from $\mathbb {R}^n$ . Note that $\operatorname {{\mathrm {Cl}}}\operatorname {{\mathrm {Relint}}}\sigma = \sigma $ , the closure being taken in the ambient space $\mathbb {R}^n$ .

2.6 Triangulations

A simplicial complex in $\mathbb {R}^n$ is a finite set $\Sigma $ of simplices satisfying the following conditions.

1. (a) $\Sigma $ is $\prec $ -downwards-closed: whenever $\sigma \in \Sigma $ and $\tau \prec \sigma $ we have $\tau \in \Sigma $ .

2. (b) If $\sigma ,\tau \in \Sigma $ , then $\sigma \cap \tau $ is either empty or a common face of $\sigma $ and $\tau $ .

The support of $\Sigma $ is the set

. Note that by definition this set is automatically a polyhedron. We say that $\Sigma $ is a triangulation of the polyhedron $\mathopen |\Sigma \mathclose |$ . The set $\Sigma $ is a poset under $\prec $ , called the face poset of the triangulation. A subcomplex of $\Sigma $ is subset which is itself a simplicial complex. Note that a subcomplex, as a poset, is precisely a downwards-closed set. Given $\sigma \in \Sigma $ , its open star is defined:

In light of Proposition 2.10, for any $x \in \mathopen |\Sigma \mathclose |$ let us write $\sigma ^x$ for the unique $\sigma \in \Sigma $ such that $x \in \operatorname {{\mathrm {Relint}}}\sigma $ . The simplex $\sigma ^x$ is known as the carrier of x.

The next result is a basic fact of polyhedral geometry, and is of fundamental importance in its connection with logic. For $\Sigma $ a triangulation and S a subset of the ambient space $\mathbb {R}^n$ , define

This, being a downwards-closed subset of $\Sigma $ , is a subcomplex of $\Sigma $ .

2.7 Stellar and barycentric subdivisions

For $\Sigma $ and $\Delta $ simplicial complexes, $\Delta $ is a subdivision or refinement of $\Sigma $ , notation $\Delta \lhd \Sigma $ , if $\mathopen |\Sigma \mathclose |=\mathopen |\Delta \mathclose |$ and every simplex of $\Delta $ is contained in a simplex of $\Sigma $ .

Lemma 2.13. If $\Delta \lhd \Sigma $ then for every $\sigma \in \Sigma $ we have

$$ \begin{align*} \sigma = \bigcup\{\tau \in \Delta \mid \tau \subseteq \sigma\}. \end{align*} $$

We now introduce a special class of subdivisions, for which the original source [Reference Alexander4] remains a fundamental reference. Let $\Sigma $ be a simplicial complex, and let $c\in |\Sigma |$ . The elementary stellar subdivision of $\Sigma $ at c is the set of simplices $\Delta $ obtained from $\Sigma $ via the following transformation: Replace each simplex $\sigma \in \Sigma $ that contains c by the set of all simplices $\operatorname {{\mathrm {Conv}}}{\{\tau \cup \{c\}\}}$ , where $\tau $ ranges over all faces of $\sigma $ that do not contain c. It can then be proved that $\Delta $ is again a triangulation, and a subdivision of $\Sigma $ . The equality $\Sigma =\Delta $ holds precisely when the chosen c is a vertex of $\Sigma $ . If $\Delta $ is a subdivision of $\Sigma $ that is obtained via a finite number of successive elementary subdivisions of $\Sigma $ , then $\Delta $ is a stellar subdivision of $\Sigma $ . See Figure 1.

Figure 1 Examples of elementary stellar subdivisions.

If $\Delta $ is obtained from $\Sigma $ via an elementary stellar subdivision at $c\in |\Sigma |$ , and c is moreover the barycentre of the vertices of its carrier simplex $\sigma ^c\in \Sigma $ (see Proposition 2.10 and the comments following it), then $\Delta $ is an elementary barycentric subdivision of $\Sigma $ (at the barycentre c).

The barycentric subdivision $\operatorname {{\mathrm {Sd}}}\Sigma $ of $\Sigma $ is then defined as the refinement of $\Sigma $ obtained by successively applying elementary barycentric subdivisions at each simplex of $\Sigma $ , proceeding in decreasing order of dimension. It can be proved that $\operatorname {{\mathrm {Sd}}}\Sigma $ does not depend on the chosen ordering of the simplices of $\Sigma $ . See the examples in Figure 2. In the literature, $\operatorname {{\mathrm {Sd}}}\Sigma $ is also often called the first derived subdivision of $\Sigma $ (cf., e.g., [Reference Rourke and Sanderson27]). We inductively define, for each $k\in \mathbb {N}$ , the $k^{\mathrm {th}}$ derived subdivision of $\Sigma $ : ; and $\Sigma ^{(k)} =\operatorname {{\mathrm {Sd}}} \Sigma ^{(k-1)}$ .

Figure 2 Examples of barycentric subdivision. Each simplex in the simplicial complex is divided at its barycentre, proceeding in decreasing order of dimension. The bottom right tetrahedron is drawn without filled-in faces to aid clarity.

3.1 Polyhedral semantics

Given a polyhedron P, let $\mathrm {Sub} P$ denote the set of its subpolyhedra.

By an open subpolyhedron of a polyhedron P in this paper we mean the complement (in P) of a subpolyhedron of P. Denote by $\mathrm {Sub}_{\mathrm {o}} P$ the collection of open subpolyhedra in P. Evidently, there is an isomorphism $\mathrm {Sub}_{\mathrm {o}} P\cong (\mathrm {Sub} P)^{\mathrm {op}}$ , and Theorem 3.1 yields the following.

For any formula $\phi $ and polyhedron P, say that $P \vDash \phi $ if and only if $\mathrm {Sub}_{\mathrm {o}} P \vDash \phi $ as a Heyting algebra. Call an intermediate logic polyhedrally complete if it is the logic of some class of polyhedra. In [Reference Bezhanishvili, Marra, McNeill and Pedrini13], it is shown that $\mathbf {IPC}$ is polyhedrally complete, being the logic of all polyhedra, while $\mathbf {BD}_n$ is the logic of all polyhedra of dimension at most n. It is also shown that all polyhedrally complete logics must have the finite model property (cf. Theorem 3.7).

3.2 Triangulation subalgebras

Let $\Sigma $ be a triangulation of the polyhedron P. Then $\Sigma \subseteq \mathrm {Sub} P$ . Let $\mathrm {P}_{\mathrm {c}}(\Sigma )$ be the sublattice of $\mathrm {Sub} P$ generated by $\Sigma $ .

Call any algebra of the form $\mathrm {P}_{\mathrm {c}}(\Sigma )$ a triangulation subalgebra.

Turning now to the dual, every triangulation $\Sigma $ of a polyhedron P gives rise to a Heyting subalgebra $\mathrm {P}_{\mathrm {o}}(\Sigma )$ of $\mathrm {Sub}_{\mathrm {o}} P$ , which we also call a triangulation subalgebra, generated by the complements of the simplices in $\Sigma $ .

The algebra $\mathrm {P}_{\mathrm {o}}(\Sigma )$ , though not necessarily easy to visualise geometrically, is in fact precisely the algebra of upsets of the poset $\Sigma $ .

Lemma 3.6. The map

$$ \begin{align*} \gamma^\uparrow \colon \operatorname{{\mathrm{Up}}} \Sigma &\to \mathrm{P}_{\mathrm{o}}(\Sigma) \\ U &\mapsto \bigcup_{\sigma \in U}\operatorname{{\mathrm{Relint}}}(\sigma) \end{align*} $$

is an isomorphism of Heyting algebras.

As a consequence of the preceding results, we have:

The following additional facts about triangulation algebras will be useful later on.

3.3 PL homeomorphisms

Let $P\subseteq \mathbb {R}^m$ and $Q\subseteq \mathbb {R}^n$ be polyhedra. A continuous function $f\colon P\to Q$ is piecewise-linear, or is a PL map, if the graph of f is a polyhedron in the product space $\mathbb {R}^m\times \mathbb {R}^n$ . A PL homeomorphism is a PL map that is a homeomorphism.

4.1 Rational polyhedra and unimodular triangulations

The geometric intuition behind Theorem 4.3 is that any triangulation can be approximated from any other by taking iterated barycentric subdivisions. One difficulty with spelling out such an intuition is that if we start with a triangulation $\Sigma $ on vertices with irrational coordinates, and try to approximate it using the iterated barycentric subdivisions of a triangulation on vertices with rational coordinates, the approximations can never quite capture (a refinement of) $\Sigma $ . The approach taken here is effectively to show that it suffices to restrict attention to the rational case. In order to make this idea precise, we need tools on rational triangulations that go beyond the standard polyhedral topology handbooks, which typically deal with the real case only. For these tools we mainly use [Reference Mundici23] as a background reference.

A polytope in $\mathbb {R}^n$ is rational if it may be written as the convex hull of finitely many points in $\mathbb {Q}^n \subseteq \mathbb {R}^n$ . A polyhedron in $\mathbb {R}^n$ is rational if it may be written as a union of a finite collection of rational polytopes. A simplicial complex $\Sigma $ is rational if it consists of rational simplices. Note that when this is the case, $\mathopen |\Sigma \mathclose |$ is a rational polyhedron.

For any $x \in \mathbb {Q}^n \subseteq \mathbb {R}^n$ , there is a unique way to write out x in coordinates as $x = (\frac {p_1}{q_1},\ldots ,\frac {p_n}{q_n})$ such that for each i, we have $p_i,q_i \in \mathbb {Z}$ coprime. The denominator of x is defined:

Thus, $\operatorname {{\mathrm {Den}}}(x)=1$ if and only if x has integer coordinates. Letting $q = \operatorname {{\mathrm {Den}}}(x)$ , the homogeneous correspondent of x is defined to be the integer vector:

A rational d-simplex $\sigma = x_0 \cdots x_d$ is unimodular if there is an $(n+1)\times (n+1)$ matrix with integer entries whose first $d+1$ columns are $\widetilde {x}_0,\ldots ,\widetilde {x}_d$ , and whose determinant is $\pm 1$ . This is equivalent to requiring that the set $\{\widetilde {x}_0,\ldots ,\widetilde {x}_d\}$ can be completed to a $\mathbb {Z}$ -module basis of $\mathbb {Z}^{d+1}$ . A simplicial complex is unimodular if each one of its simplices is unimodular.

4.2 Farey subdivisions

Proof Let $H_{n+1} \subseteq \mathbb {R}^{n+1}$ be the hyperplane specified by

Identify $\mathbb {Q}^n$ with the set of rational points of $H_{n+1}$ via the map $(q_1,\ldots ,q_n)\mapsto (q_1,\ldots ,q_n,1)$ . Under this identification, $\widetilde m$ lies in the rational cone:

$$ \begin{align*} \left\{\sum_{i=0}^d c_i\widetilde x_i\mid c_i\in\mathbb{R}, c_i \geqslant 0\right\}. \end{align*} $$

A routine computation then proves the geometrically evident fact that m is the point of intersection of the line spanned in $\mathbb {R}^{n+1}$ by the vector $\widetilde m$ , with the hyperplane $H_{n+1}$ ; from which the result follows.

The element $m \in \mathbb {Q}^n$ in Proposition 4.4 is called the Farey mediant of (the vertices of) the simplex $\sigma $ . Note that when $d=0$ , i.e., when $\sigma $ is a vertex of $\Sigma $ , then m coincides with the vertex $\sigma $ . Also observe that the Farey mediant and the barycentre of $\sigma $ are in general distinct, though both lie in $\operatorname {{\mathrm {Relint}}}{\sigma }$ .

We can now define a specific type of stellar subdivision based on Farey mediants (cf. [Reference Mundici23, Section 5.1, p. 55]). Let $\Sigma $ be a simplicial complex, let $\sigma \in \Sigma $ , and let m be the Farey mediant of $\sigma $ . The elementary Farey subdivision of $\Sigma $ at m is the elementary stellar subdivision of $\Sigma $ at m. In general, the triangulation $\Delta $ is a Farey subdivision of $\Sigma $ if it is obtained from the latter via finitely many successive elementary Farey subdivisions.

At the combinatorial level, Farey and barycentric subdivisions are indiscernible:

However, going beyond the combinatorial level, the construction of universal approximations of arbitrary rational polyhedra does require Farey subdivisions and cannot be done with barycentric ones. This is made precise in the following fundamental fact of rational polyhedral geometry.

4.3 From $\mathbb {R}$ to $\mathbb {Q}$

To deploy the power of Lemma 4.6, we need to relate general polyhedra to rational polyhedra, and general triangulations to unimodular ones.

Finally, we shall need the non-trivial fact that arbitrary triangulations of a rational polyhedron realise no more combinatorial types than its rational triangulations; this is due to Meurig Beynon:

4.4 End of proof of Theorem 4.3

4.5 Nerves, subdivisions, and geometric realisations

The reason that Theorem 4.3 is relevant to the Nerve Criterion is the following classical connection between nerves of posets and derived subdivisions, which is deeply rooted in the work of Pavel S. Alexandrov.

Proposition 4.10. Let $\Sigma $ be a simplicial complex, regarded as a poset under inclusion of faces. Then the barycentric subdivision of $\Sigma $ is isomorphic as a poset to the nerve of $\Sigma $ :

$$ \begin{align*} \operatorname{{\mathrm{Sd}}} \Sigma \cong \mathcal{N}(\Sigma). \end{align*} $$

Corollary 4.11. For P a polyhedron and $\Sigma $ a triangulation of P we have

$$ \begin{align*} \operatorname{{\mathrm{Logic}}}(P) = \operatorname{{\mathrm{Logic}}}(\mathcal{N}^k(\Sigma) \mid k \in \mathbb N). \end{align*} $$

Proof Indeed

([14, Chapter 7]) $$ \begin{align} \operatorname{{\mathrm{Logic}}}(P) &= \operatorname{{\mathrm{Logic}}}(\mathrm{Sub}_{\mathrm{o}} P) \nonumber \\ &= \operatorname{{\mathrm{Logic}}} (A \mid A\text{ finitely generated subalgebra of } \mathrm{Sub}_{\mathrm{o}} P) \end{align} $$

(Theorem 4.3) $$ \begin{align} &= \operatorname{{\mathrm{Logic}}}(\mathrm{P}_{\mathrm{o}}(\Sigma^{(k)}) \mid k \in \mathbb N)\qquad \end{align} $$

(as above) $$ \begin{align} &\hspace{-21pt}= \operatorname{{\mathrm{Logic}}}(\Sigma^{(k)} \mid k \in \mathbb N)\qquad\qquad\qquad\quad\ \ \ \end{align} $$

(Proposition 4.10) $$ \begin{align} &= \operatorname{{\mathrm{Logic}}}(\mathcal{N}^k(\Sigma) \mid k \in \mathbb N).\ \ \kern1pt \end{align} $$

In order to complete a proof of the Nerve Criterion, we will also need to use geometric realisations of finite posets via nerves, another classical tool. Let $F = \{x_1, \ldots , x_m\}$ be a finite poset, and let $e_1, \ldots , e_m$ be the standard basis vectors of $\mathbb {R}^m$ . The set

can be proved to be a triangulation by elementary arguments; its underlying polyhedron $\mathopen |\nabla F\mathclose |$ is the geometric realisation of F. For us, the key fact about geometric realisations is:

4.6 End of proof of the Nerve Criterion

Proof of Theorem 4.1, the Nerve Criterion.

Assume that $\mathcal {L}$ is the logic of a class $\mathbf {C}$ of polyhedra. For each $P \in \mathbf {C}$ fix a triangulation $\Sigma _P$ , and let

Then

(Corollary 4.11) $$ \begin{align} \operatorname{{\mathrm{Logic}}}(\mathbf{C}^*) &= \bigcap_{P \in \mathbf{C}}\operatorname{{\mathrm{Logic}}}(\mathcal{N}^k(\Sigma_P) \mid k \in \mathbb N)\nonumber \\ &= \bigcap_{p \in \mathbf{C}}\operatorname{{\mathrm{Logic}}}(P) \\ &= \operatorname{{\mathrm{Logic}}}(\mathbf{C}) = \mathcal{L}.\nonumber \end{align} $$

Conversely, assume that $\mathcal {L} = \operatorname {{\mathrm {Logic}}}(\mathbf {D})$ , where $\mathbf {D}$ is a class of finite frames closed under $\mathcal {N}$ . Let

where $\mathopen |\nabla (F)\mathclose |$ is the geometric realisation of F as in Section 4.5. We will show that $\mathcal {L} = \operatorname {{\mathrm {Logic}}}(\mathbf {D}_*)$ . First suppose that $\mathcal {L} \nvdash \phi $ , so that $F \nvDash \phi $ for some $F \in \mathbf {D}$ . Then we have that $\mathopen |\nabla (F)\mathclose | \nvDash \phi $ , so that $\operatorname {{\mathrm {Logic}}}(\mathbf {D}_*) \nvdash \phi $ . Conversely, suppose that $\operatorname {{\mathrm {Logic}}}(\mathbf {D}_*) \nvdash \phi $ , so that $\mathopen |\nabla (F)\mathclose | \nvDash \phi $ for some $F \in \mathbf {D}$ . By definition $\nabla (F)$ is a triangulation of $\mathopen |\nabla (F)\mathclose |$ , and hence by Corollary 4.11 there is $k \in \mathbb N$ such that $\nabla (F)^{(k)} \nvDash \phi $ . But $\nabla (F) \cong \mathcal {N}(F)$ by definition, and so by Proposition 4.10 we get $\mathcal {N}^{k+1}(F) \cong \nabla (F)^{(k)}$ . Thus, as $\mathbf {D}$ is closed under $\mathcal {N}$ , we get that $\mathcal {L} \nvdash \phi $ .

6.1 Starlike trees

A finite poset T is a tree if it has a root $\bot $ , and every other $x \in T \setminus \{\bot \}$ has exactly one immediate predecessor. A branch in T is a maximal chain. Say that T is a starlike tree if every $x \in T \setminus \{\bot \}$ has at most one immediate successor. (The terminology ‘starlike’ comes from graph theory [see [Reference Watanabe and Schwenk33]].) A starlike tree is determined by the multiset of its branch heights, which motivates the following notation.

Let $n_1, \ldots , n_k, m_1, \ldots , m_k \in \mathbb N^{>0}$ , with $n_1, \ldots , n_k$ distinct. Then let us define $T={\langle {n_1^{m_1} \cdots n_k^{m_k}} \rangle }$ as the starlike tree, uniquely determined to within an isomorphism, with the property that if we remove the root $\bot $ we are left with exactly, for each i, $m_i$ chains of length $n_i$ . Let ${\langle {\epsilon } \rangle } = \bullet $ , the singleton poset. Call $\alpha = n_1^{m_1} \cdots n_k^{m_k}$ (or $\epsilon $ ) the signature of T. We will always assume that $n_1> n_2 > \cdots > n_k$ . See Figure 4 for some examples of starlike trees together with their signatures. We will sometimes write $1^0$ for $\epsilon $ .

Figure 4 Some examples of starlike trees.

The length of a signature $\alpha = n_1^{m_1} \cdots n_k^{m_k}$ is defined as . Let . For $j \leqslant \mathopen |\alpha \mathclose |$ , the jth height, $\alpha (j)$ , is $n_i$ , where

$$ \begin{align*} m_1 + \cdots + m_{i-1} \leqslant j < m_1 + \cdots + m_i. \end{align*} $$

Let $\alpha $ and $\beta $ be signatures. Say that $\alpha \leqslant \beta $ if $\mathopen |\alpha \mathclose | \leqslant \mathopen |\beta \mathclose |$ and for every $j \leqslant \mathopen |\alpha \mathclose |$ we have $\alpha (j) \leqslant \beta (j)$ . Considering the examples in Figure 4, we have the following relations:

$$ \begin{align*} 1^3 < 3 \cdot 1^2 < 3^2 \cdot 2 \cdot 1, \quad 2 < 3 \cdot 1^2. \end{align*} $$

Note that if $\alpha = n_1^{m_1} \cdots n_k^{m_k}$ , we have $\alpha \leqslant \beta $ if and only if $\mathopen |\alpha \mathclose | \leqslant \mathopen |\beta \mathclose |$ and for every $i \leqslant k$ , we have

$$ \begin{align*} \beta(m_1 + \cdots + m_i) \geqslant n_i. \end{align*} $$

Note that the starlike tree ${\langle {k} \rangle }$ is the chain on $k+1$ elements; we will use this notation for chains from now on. We will write the signature as $k^1$ , to disambiguate it from k as a number. For $k \in \mathbb N^{>0}$ , the k-fork is the starlike tree ${\langle {1^k} \rangle }$ .

6.2 Starlike logics

We are now in a position to define the principal class of logics that will be investigated in this section. Let . Take $\Lambda \subseteq \mathcal S$ (possibly infinite). The starlike logic $\mathbf {SFL}(\Lambda )$ based on $\Lambda $ is the logic axiomatised by $\mathbf {IPC}$ plus $\chi ({\langle {\alpha } \rangle })$ for each $\alpha \in \Lambda $ . Write $\mathbf {SFL}(\alpha _1, \ldots , \alpha _k)$ for $\mathbf {SFL}(\{\alpha _1, \ldots , \alpha _k\})$ .

Let us examine what $\mathbf {SFL}(\Lambda )$ ‘means’ in terms of its class of frames. The formula $\chi ({\langle {\alpha } \rangle })$ turns out to express a kind of connectedness property. We make this precise using the following definitions.

Let F be a finite poset. A path in F is a sequence $p=x_0\cdots x_k$ of elements of F such that for each i we have $x_i < x_{i+1}$ or $x_i> x_{i+1}$ . Write $p \colon x_0 \rightsquigarrow x_k$ . The path p is closed if $x_0=x_k$ . The poset F is path-connected if between any two points there is a path.

A connected component of F is a subposet $U \subseteq F$ which is connected as a topological subspace and is such that there is no connected V with $U \subset V$ .

Define $\mathrm {ConComps}(F)$ to be the set of connected components of F. The connectedness type $\mathrm {ConType}(F)$ of F is the signature $n_1^{m_1} \cdots n_k^{m_k}$ such that $\mathrm {ConComps}(F)$ contains for each i exactly $m_i$ sets of height $n_i-1$ , and nothing else. Let .

Let $\alpha> \epsilon $ be a signature. An $\alpha $ -partition of F is an open partition in which the number and heights of the connected components are specified by $\alpha $ . In other words, it is a partition:

$$ \begin{align*} F = C_1 \sqcup \cdots \sqcup C_{\mathopen|\alpha\mathclose|} \end{align*} $$

into open sets such that $C_j$ has height at least $\alpha (j)-1$ . For notational uniformity, say that F has an $\epsilon $ -partition if $F=\varnothing $ . The following lemma is a straightforward consequence of the definitions.