In this paper, we describe étale Boolean right restriction monoids in terms of Boolean inverse monoids. We show that the Thompson groups arise naturally in this context.

End-spaces of infinite graphs naturally generalise the Freudenthal boundary and sit at the interface between graph theory, geometric group theory and topology.

Our main result is that every end-space can be topologically represented by a special order tree. Our main proof ingredient is a structure theorem that we introduce, which carves out the order-tree-like structure of any graph in such a way that there is a natural bijection between the ends of the graph and the limit-type down-closed chains of the order-tree.

Let $X_1,\ldots, X_n$ be independent integers distributed uniformly on [M], $M\ge 2$. A partition S of [n] into $\nu$ non-empty subsets $S_1,\ldots, S_{\nu}$ is called perfect if all $\nu$ values $\sum_{j\in S_{\alpha}}X_j$ are equal. For a perfect partition to exist, $\sum_j X_j$ has to be divisible by $\nu$. In 2001, for $\nu=2$, Christian Borgs, Jennifer Chayes, and the author proved that, conditioned on $\sum_j X_j$ being even, with high probability a perfect partition exists if $\kappa\;:\!=\; \lim {{n}/{\log M}}>{{1}/{\log 2}}$, and that with high probability no perfect partition exists if $\kappa<{{1}/{\log 2}}$. Responding to a question by George Varghese, we prove that for $\nu\ge 3$ with high probability no perfect partition exists if $\kappa<{{2}/{\log \nu}}$, which is twice as large as the naive threshold $1/\log 3$ for $\nu=3$. We identify the range of $\kappa$ where the expected number of perfect partitions is exponentially high. We show that for $\kappa> {{2(\nu-1)}/{\log[(1-2\nu^{-2})^{-1}]}}$ the total number of perfect partitions is exponentially high with probability $\gtrsim (1+\nu^2)^{-1}$, i.e. below $1/\nu$, the limiting probability that $\sum_j X_j$ is divisible by $\nu$.

In this article, we construct countably many isolated circular orders on the free products $G = F_{2n} \ast \mathbb {Z}_{m_1} \ast \cdots \ast \mathbb {Z}_{m_k}$ of cyclic groups. Moreover, we prove that these isolated circular orders are not the automorphic images of the others. By using these isolated circular orders, we also construct countably many isolated left orders on a certain central $\mathbb {Z}$-extension of G, which are not the automorphic images of the others.

Gödel algebras are the Heyting algebras satisfying the axiom $(x \to y) \vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe free Gödel algebras and coproducts of Gödel algebras. In particular, we realize the Esakia space dual to a Gödel algebra free over a distributive lattice as the, suitably topologized and ordered, collection of all nonempty closed chains of the Priestley dual of the lattice. This provides a tangible dual description of free Gödel algebras without any restriction on the number of free generators, which generalizes known results for the finitely generated case. A similar approach allows us to characterize the Esakia spaces dual to coproducts of arbitrary families of Gödel algebras. We also establish analogous dual descriptions of free algebras and coproducts in every variety of Gödel algebras. As consequences of these results, we obtain a formula to compute the depth of coproducts of Gödel algebras and show that all free Gödel algebras are bi-Heyting algebras.

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of $2$ and every power of $2$ appears at most twice. We give three applications of the length generating function for such partitions, denoted by $h_q(n)$. Morier-Genoud and Ovsienko defined the q-analogue of a rational number $[r/s]_q$ in various ways, most of which depend directly or indirectly on the continued fraction expansion of $r/s$. As our first application we show that $[r/s]_q=q\,h_q(n-1)/h_q(n)$ where $r/s$ occurs as the nth entry in the Calkin-Wilf enumeration of the non-negative rationals. Next we consider fence posets which are those which can be obtained from a sequence of chains by alternately pasting together maxima and minima. For every n we show there is a fence poset ${\cal F}(n)$ whose lattice of order ideals is isomorphic to the poset of hyperbinary partitions of n ordered by refinement. For our last application, Morier-Genoud and Ovsienko also showed that $[r/s]_q$ can be computed by taking products of certain matrices which are q-analogues of the standard generators for the special linear group $\operatorname {\mathrm {SL}}(2,{\mathbb Z})$. We express the entries of these products in terms of the polynomials $h_q(n)$.

In previous work [4] we introduced and examined the class of betweenness algebras. In the current article we study a larger class of algebras with binary operators of possibility and sufficiency, the weak mixed algebras. Furthermore, we develop a system of logic with two binary modalities, sound and complete with respect to the class of frames closely related to the aforementioned algebras, and we prove an embedding theorem which solves an open problem from [4].

Central to certain versions of logical atomism are claims to the effect that every proposition is a truth-functional combination of elementary propositions. Assuming that propositions form a Boolean algebra, we consider a number of natural formal regimentations of informal claims in this vicinity, and show that they are equivalent. For a number of reasons, such as the need to accommodate quantifiers, logical atomists might consider only complete Boolean algebras, and take into account infinite truth-functional combinations. We show that in such a variant setting, some of the regimentations come apart, and explore how they relate to each other. We also discuss how they relate to the claim that propositions form a double powerset algebra, which has been proposed by a number of authors as a way of capturing the central logical atomist idea.

This paper explores the mathematical connections between the algebraic and relational semantics of Lewis’s logics for counterfactual conditionals. Specifically, we introduce topological variants of Lewis’s well-known possible-worlds semantics—based on spheres, selection functions, and orders—and establish duality results with respect to varieties of Boolean algebras equipped with a counterfactual operator, which serve as the equivalent algebraic semantics of Lewis’s main systems. These results aim to provide a solid mathematical foundation for the study of Lewis’s logics, and offer a new perspective on the most well-known possible worlds-based models. In particular, we write explicit proofs for several results that are often assumed without proof in the literature. Leveraging these duality results, we also derive alternative proofs of strong completeness for Lewis’s variably strict conditional logics with respect to their intended models, and clarify the role of the limit assumption in sphere semantics.

In this paper, we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb {R}^n$, where $e_1,\dots ,e_n$ is the standard basis of $\mathbb {R}^n$. Such a polytope can be encoded by a quiver Q with vertices $V \subseteq \{{\upsilon }_1,\dots ,{\upsilon }_n\} \cup \{\star \}$, where each edge ${\upsilon }_j\to {\upsilon }_i$ or $\star \to {\upsilon }_i$ or ${\upsilon }_i\to \star $ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname {Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver Q is strongly-connected, then the root polytope $\operatorname {Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname {Root}(Q)$. We also show that if Q is planar, then $\operatorname {Root}(Q)$ is (integrally equivalent to) the polar dual of the flow polytope of the planar dual quiver $Q^{\vee }$. Finally, we consider the case that Q comes from the Hasse diagram of a finite ranked poset P and show in this case that $\operatorname {Root}(Q)$ is polar dual to (a translation of) a marked order polytope. We then go on to study the toric variety $Y(\mathcal {F}_Q)$ associated to the face fan $\mathcal {F}_Q$ of $\operatorname {Root}(Q)$. If Q comes from a ranked poset P, we give a combinatorial description of the Picard group of $Y(\mathcal {F}_Q)$, in terms of a new canonical ranked extension of P, and we show that $Y(\mathcal {F}_Q)$ is a small partial desingularisation of the Hibi projective toric variety $Y_{\mathcal {O}(P)}$ of the order polytope $\mathcal {O}(P)$. We show that $Y(\mathcal {F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat {\mathcal {F}}_Q)$ and, as a consequence that the Hibi toric variety $Y_{\mathcal {O}(P)}$ has a small resolution of singularities for any ranked poset P. These results have applications to mirror symmetry [61].

The Lindenbaum lemma saying that completely meet-irreducible closed sets form a basis of any finitary closure system is an easy-to-prove yet crucial result transcending algebraic logic. While the finitarity restriction is crucial for its usual proof, it is not necessary: there are indeed works proving it (or its variant for a larger class of finitely meet-irreducible closed sets) for non-finitary closure systems arising from particular infinitary logics (i.e., substitution-invariant consequence relations). There is also a general result proving it for a wide class of logics with strong p-disjunction and a countable Hilbert-style axiomatization. Identifying the essential properties of strong p-disjunctions we prove a variant of the Lindenbaum lemma for closure systems which are 1) defined over countable sets, 2) countably axiomatized, and 3) frames (in the order-theoretic sense) but not necessarily substitution-invariant.

We introduced positive cones in an earlier paper as a notion of ordering on central simple algebras with involution that corresponds to signatures of hermitian forms. In the current article, we describe signatures of hermitian forms directly out of positive cones, and also use this approach to rectify a problem that affected some results in the previously mentioned paper.

We explore the Weihrauch degree of the problems “find a bad sequence in a non-well quasi order” ($\mathsf {BS}$) and “find a descending sequence in an ill-founded linear order” ($\mathsf {DS}$). We prove that $\mathsf {DS}$ is strictly Weihrauch reducible to $\mathsf {BS}$, correcting our mistaken claim in [18]. This is done by separating their respective first-order parts. On the other hand, we show that $\mathsf {BS}$ and $\mathsf {DS}$ have the same finitary and deterministic parts, confirming that $\mathsf {BS}$ and $\mathsf {DS}$ have very similar uniform computational strength. We prove that König’s lemma $\mathsf {KL}$ and the problem $\mathsf {wList}_{{2^{\mathbb {N}}},\leq \omega }$ of enumerating a given non-empty countable closed subset of ${2^{\mathbb {N}}}$ are not Weihrauch reducible to $\mathsf {DS}$ or $\mathsf {BS}$, resolving two main open questions raised in [18]. We also answer the question, raised in [12], on the existence of a “parallel quotient” operator, and study the behavior of $\mathsf {BS}$ and $\mathsf {DS}$ under the quotient with some known problems.

Quantales can be regarded as a combination of complete lattices and semigroups. Unital quantales constitute a significant subclass within quantale theory, which play a crucial role in the theoretical framework of quantale research. It is well known that every complete lattice can support a quantale. However, the question of whether every complete lattice can support a unital quantale has not been considered before. In this article, we first give some counter-examples to indicate that the answer to the above question is negative, and then investigate the complete lattices of supporting unital quantales.

Let G be a finite transitive permutation group on $\Omega $. The G-invariant partitions form a sublattice of the lattice of all partitions of $\Omega $, having the further property that all its elements are uniform (that is, have all parts of the same size). If, in addition, all the equivalence relations defining the partitions commute, then the relations form an orthogonal block structure, a concept from statistics; in this case the lattice is modular. If it is distributive, then we have a poset block structure, whose automorphism group is a generalised wreath product. We examine permutation groups with these properties, which we call the OB property and PB property respectively, and in particular investigate when direct and wreath products of groups with these properties also have these properties.

A famous theorem on permutation groups asserts that a transitive imprimitive group G is embeddable in the wreath product of two factors obtained from the group (the group induced on a block by its setwise stabiliser, and the group induced on the set of blocks by G). We extend this theorem to groups with the PB property, embedding them into generalised wreath products. We show that the map from posets to generalised wreath products preserves intersections and inclusions.

We have included background and historical material on these concepts.

Building on the correspondence between finitely axiomatised theories in Łukasiewicz logic and rational polyhedra, we prove that the unification type of the fragment of Łukasiewicz logic with $n\geqslant 2$ variables is nullary. This solves a problem left open by V. Marra and L. Spada [Ann. Pure Appl. Logic 164 (2013), pp. 192–210]. Furthermore, we refine the study of unification with bounds on the number of variables. Our proposal distinguishes the number m of variables allowed in the problem and the number n in the solution. We prove that the unification type of Łukasiewicz logic for all $m,n \geqslant 2$ is nullary.

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.

In this note, some conditions are investigated under which the left amenability of a semigroup S is a consequence of the left amenability of its subsemigroups. It is known that for the Green’s relation $\mathcal {H}^S$ on S, an $\mathcal {H}^S$-class of S is a semigroup if and only if it is a subgroup of S, and hence it contains a unique identity. Let S be a semigroup such that every $\mathcal {H}^S$-class of S is a group and E, the set of idempotents of S, is a subsemigroup of S. As the main result of this note, applying the above fact, a connection between left amenability of S, left amenability of E, and left amenability of its $\mathcal {H}^S$-classes is established.

As an application, I completely determine left amenable Clifford semigroups and left amenable rectangular groups, when they are left amenable with some measure such that the union of every collection of $\mathcal {H}^S$-classes of S with zero measure has zero measure (especially, when E is finite or when E is countable and it is left amenable with a measure which is countably additive). Indeed, I show that under this assumption, (i) a Clifford semigroup S is left amenable if and only if E has a zero element z and $H_z$, the $\mathcal {H}^S$-class of S which contains z, is a left amenable group and (ii) a rectangular group S is left amenable if and only if it is a right group and its $\mathcal {H}^S$-classes are left amenable groups.

The Chan–Robbins–Yuen polytope ($CRY_n$) of order n is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow $(1,0,0, \ldots , 0, -1)$. The volume and lattice points of this polytope have been actively studied; however, its face structure has received less attention. We give generating functions and explicit formulas for computing the f-vector by using Hille’s (2003) result bijecting faces of a flow polytope to certain graphs, as well as Andresen–Kjeldsen’s (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes of the complete graph having arbitrary (non-negative) netflow vectors and recover the f-vector of the Tesler polytope of Mészáros–Morales–Rhoades (2017).