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 $\nu$.

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.

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.

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.

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 $1$.

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups, we can establish that the complexity function asymptotically behaves like $r^{{\mathrm {homdim}}(G) \dim (H)}$. Further, we generalize the concept of acceptance domains to locally compact second countable groups.

We introduce semidistrim lattices, a simultaneous generalization of semidistributive and trim lattices that preserves many of their common properties. We prove that the elements of a semidistrim lattice correspond to the independent sets in an associated graph called the Galois graph, that products and intervals of semidistrim lattices are semidistrim and that the order complex of a semidistrim lattice is either contractible or homotopy equivalent to a sphere.

Semidistrim lattices have a natural rowmotion operator, which simultaneously generalizes Barnard’s $\overline \kappa $ map on semidistributive lattices as well as Thomas and the second author’s rowmotion on trim lattices. Every lattice has an associated pop-stack sorting operator that sends an element x to the meet of the elements covered by x. For semidistrim lattices, we are able to derive several intimate connections between rowmotion and pop-stack sorting, one of which involves independent dominating sets of the Galois graph.

We introduce a technique that is helpful in evaluating the reflexivity index of several classes of topological spaces and lattices. The main results are related to products: we give a sufficient condition for the product of a topological space and a nest of balls to have low reflexivity index and determine the reflexivity index of all compact connected 2-manifolds.

We determine the reflexivity index of some closed set lattices by constructing maps relative to irrational rotations. For example, various nests of closed balls and some topological spaces, such as even-dimensional spheres and a wedge of two circles, have reflexivity index 2. We also show that a connected double of spheres has reflexivity index at most 2.

For every n, we evaluate the smallest k such that the congruence inclusion $\alpha (\beta \circ _n \gamma ) \subseteq \alpha \beta \circ _{k} \alpha \gamma $ holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.

We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.

Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.

We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$ , using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$ . Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$ , indecomposable $\unicode[STIX]{x1D70F}$ -rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$ -rigid) modules and layers of $\unicode[STIX]{x1D6F1}$ . The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$ . We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$ -rigid modules for type $A$ and $D$ .

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

We show that any regular pseudocomplemented Kleene algebra defined on an algebraic lattice is isomorphic to a rough set Kleene algebra determined by a tolerance induced by an irredundant covering.

A function $f:X\rightarrow X$ determines a topology $P(f)$ on $X$ by taking the closed sets to be those sets $A\subseteq X$ with $f(A)\subseteq A$ . The topological space $(X,P(f))$ is called a functionally Alexandroff space. We completely characterise the homogeneous functionally Alexandroff spaces.

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction c k of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {c k }. Notoriously hard to compute, the exact fractions c k have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c 4 and c 5 as well; both are ratios of enormous integers, the denominator of c 5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of c k is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.