We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Gamma) does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding and more general results for monoids.
A variety of algebras is called limit if it is non-finitely based but all its proper subvarieties are finitely based. A monoid is aperiodic if all its subgroups are trivial. We classify all limit varieties of aperiodic monoids with commuting idempotents.
We continue our programme of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially-defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Svarc-Milnor Lemma to this setting.
In this paper, a connection between rewriting systems and embedding of monoids in groups is found. We show that if a group with a positive presentation has a complete rewriting system $\Re$ that satisfies the condition that each rule in $\Re$ with positive left-hand side has a positive right-hand side, then the monoid presented by the subset of positive rules from $\Re$ embeds in the group. As an example, we give a simple proof that right angled Artin monoids embed in the corresponding right angled Artin groups. This is a special case of the well-known result of Paris \cite{paris} that Artin m…
The commuting graph of a finite non-commutative semigroup S, denoted by Δ(S), is the simple graph whose vertices are the non-central elements of S and two distinct vertices x; y are adjacent if xy = yx. In the present paper, we study various graph-theoretic properties of the commuting graph Δ(B_n) of Brandt semigroup B_n including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, we obtain the automorphism group Aut(Δ(Bn)) and the endomorphism monoid End(Δ(Bn)) of Δ(Bn). We show that Aut(Δ(Bn)) = S_n \times Z_2, where S_…
Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset.
Let X be a set and Bin(X) the set of all binary operations on X. We say that a subset of Bin(X) is a distributive set of operations if all pairs of elements are right distributive. J.Przytycki posed the question of which groups can be realized as distributive sets. The initial guess that any group may be embedded into Bin(X) for some X was complicated by an observation that if a binary operation is idempotent (a*a=a), then it commutes with every element of S. The first noncommutative subgroup of Bin(X), the symmetric group on three elements, was found in October of 2011 by Y.Berman. We show th…
In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(π,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a local…
For a group $G$ and a finite set $A$, denote by $\text{End}(A^G)$ the monoid of all continuous shift commuting self-maps of $A^G$ and by $\text{Aut}(A^G)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{End}(A^G)$ and $\text{Aut}(A^G)$. In the first part, when $G$ is a finite group, we give upper and lower bounds for the rank of $\text{Aut}(A^G)$ in terms of the number of conjugacy classes of subgroups of $G$. In the second part, we apply our bounds to show that if $G$ has an infinite descending chain of normal subgroups of finite index, t…
Topological embeddings into transformation monoids
P.J. Šafárik University in Košice（科希策帕·约·沙法里克大学）；TU Wien（维也纳工业大学）；Binghamton University（宾汉姆顿大学）；University of St Andrews（圣安德鲁斯大学）；University of Hertfordshire（赫特福德大学）
Let G be a finite group that acts on an abelian monoid A. If f: A -> G is a map so that f(a f(a)(b)) = f(a)f(b), for all a, b in A, then the submonoid S = {(a, f(a)) | a in A} of the associated semidirect product of A and G is said to be a monoid of IG-type. If A is a finitely generated free abelian monoid of rank n and G is a subgroup of the symmetric group Sym_n of degree n, then these monoids first appeared in the work of Gateva-Ivanova and Van den Bergh (they are called monoids of I-type) and later in the work of Jespers and Okninski. It turns out that their associated semigroup algebras s…
Let $G$ be a countable monoid and let $A$ be an Artinian group (resp. an Artinian module). Let $Σ\subset A^G$ be a closed subshift which is also a subgroup (resp. a submodule) of $A^G$. Suppose that $Γ$ is a finitely generated monoid consisting of pairwise commuting cellular automata $Σ\to Σ$ that are also homomorphisms of groups (resp. homomorphisms of modules) with monoid binary operation given by composition of maps. We show that the valuation action of $Γ$ on $Σ$ satisfies a natural intrinsic shadowing property. Generalizations are also established for families of endomorphisms of admissib…
The goal of this article is to clarify the relationship between the topos of triads and the neo-Riemannian PLR-group. To do this, we first develop some theory of generalized interval systems: 1) we prove the well known fact that every pair of dual groups is isomorphic to the left and right regular representations of some group (Cayley's Theorem), 2) given a simply transitive group action, we show how to construct the dual group, and 3) given two dual groups, we show how to easily construct sub dual groups. Examples of this construction of sub dual groups include Cohn's hexatonic systems, as we…
Let $\mc G$ be a reductive group over an algebraically closed field of characteristic $p>0$. We study homogeneous $\mc G$-spaces that are induced from the $G\times G$-space $G$, $G$ a suitable reductive group, along a parabolic subgroup of $\mc G$. We show that, under certain mild assumptions, any (normal) equivariant embedding of such a homogeneous space is canonically Frobenius split compatible with certain subvarieties and has an equivariant rational resolution by a toroidal embedding. In particular, all these embeddings are Cohen-Macaulay. Examples are the $G\times G$-orbits in normal redu…
Let $G$ be a finitely generated group, and let $Σ$ be a finite subset that generates $G$ as a monoid. The \emph{word problem of $G$ with respect to $Σ$} consists of all words in the free monoid $Σ^{\ast}$ that are equal to the identity in $G$. The \emph{co-word problem of $G$ with respect to $Σ$} is the complement in $Σ^{\ast}$ of the word problem. We say that a group $G$ is \emph{co$\mathcal{CF}$} if its co-word problem with respect to some (equivalently, any) finite generating set $Σ$ is a context-free language. We describe a generalized Thompson group $V_{(G, θ)}$ for each finite group G an…
Tsirelson's problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by t…
In this paper we show that polycyclic monoids are universal objects in the class of graph inverse semigroups. In particular, we prove that a graph inverse semigroup $G(E)$ over a directed graph $E$ embeds into the polycyclic monoid $\mathcal{P}_λ$ where $λ=|G(E)|$. We show that each graph inverse semigroup $G(E)$ admits the coarsest inverse semigroup topology $τ$. Moreover, each injective homomorphism from $(G(E),τ)$ to the $(\mathcal{P}_{|G(E)|},τ)$ is a topological embedding.
Let $n$ be any positive integer and $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ be the semigroup of all order isomorphisms between principal filters of the $n$-th power of the set of positive integers $\mathbb{N}$ with the product order. We study algebraic properties of the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$. In particular, we show that $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is a bisimple, $E$-unitary, $F$-inverse semigroup, describe Green's relations on $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ and its maximal subgroups. We show that the semigroup $\mathscr{I\!\!P\!F}(\mathbb{N}^n)$ is isomorphic t…
The Poincaré polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space $G/B$, while the $h$-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the $H$-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to any algebraic variety $X$ where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the $H$-polynomials of certa…
To every group $G$ we associate a linear monoidal category $\mathcal{P}\mathit{ar}(G)$ that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of $\mathcal{P}\mathit{ar}(G)$ into the group Heisenberg category associated to $G$. This embedding intertwines the natural actions of both categories on modules for wreath products of $G$. Finally, we prove that the additive Karoubi envelope of $\mathcal{P}\mathit{ar}(G)$ is equivalent to a wreath product …
