Let $X$ be an arbitrary set and let $T(X)$ denote the full transformation monoid on $X$. We prove that an element of $T(X)$ is unit-regular if and only if it is semi-balanced. For infinite $X$, we discuss regularity of the submonoid of $T(X)$ consisting of all injective (resp. surjective) transformations. For a partition $\mathcal{P}$ of $X$, we characterize unit-regular elements in the monoid $T(X, \mathcal{P})$, under composition, defined as \[T(X, \mathcal{P}) = \{f\in T(X)\mid (\forall X_i \in \mathcal{P}) (\exists X_j \in \mathcal{P})\; X_i f \subseteq X_j\}.\] We also characterize (unit-)regular elements in various known submonoids of $T(X, \mathcal{P})$.
We completely classify all neutral or costandard elements in the lattice $\mathbb{MON}$ of all monoid varieties. Further, we prove that an arbitrary upper-modular element of $\mathbb{MON}$ except the variety of all monoids is either a completely regular or a commutative variety. Finally, we verify that all commutative varieties of monoids are codistributive elements of $\mathbb{MON}$. Thus, the problems of describing codistributive or upper-modular elements of $\mathbb{MON}$ are completely reduced to the completely regular case.
Using a construction that builds a monoid from a monoid action, this paper exhibits an example of a direct product of monoids that admits a prefix-closed regular cross-section, but one of whose factors does not admit a regular cross-section; this answers negatively an open question from the theory of Markov monoids. The same construction is then used to show that for any full trios $\mathfrak{C}$ and $\mathfrak{D}$ such that $\mathfrak{C}$ is not a subclass of $\mathfrak{D}$, there is a monoid with a cross-section in $\mathfrak{C}$ but no cross-section in $\mathfrak{D}$.
This paper shows that a finitely presented monoid with linear Dehn function need not have a regular cross-section, strengthening the previously-known result that such a monoid need not be presented by a finite complete string rewriting system, and contrasting the fact that finitely presented groups with linear Dehn function always have regular cross-sections.
We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships between parameters associated to Brauer graphs. We calculate the relative ranks of each monoid modulo any other such monoid it may contain, and then apply these results to determine the Sierpinski rank of each monoid, and ascertain which ones have the semigroup Bergman property. We also make some fundamental observations about idempotents and units in arbitrar…
A monoid is said to be special if it admits a presentation in which all defining relations are of the form $w = 1$. Groups are familiar examples of special monoids. This article studies the geometric and structural properties of the Cayley graphs of finitely presented special monoids, building on work by Zhang and Gray-Steinberg. It is shown that the right Cayley graph $Γ$ of a special monoid $M$ is a context-free graph, in the sense of Muller & Schupp, if and only if the group of units of $M$ is virtually free. This generalises the geometric aspect of the well-known Muller-Schupp Theorem from…
We prove that a monoid is sofic, in the sense recently introduced by Ceccherini-Silberstein and Coornaert, whenever the J-class of the identity is a sofic group, and the quotients of this group by orbit stabilisers in the rest of the monoid are amenable. In particular, this shows that the following are all sofic: cancellative monoids with amenable group of units; monoids with sofic group of units and finitely many non-units; and monoids with amenable Schutzenberger groups and finitely many L-classes in each D-class. This provides a very wide range of sofic monoids, subsuming most known example…
For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We sh…
We introduce an algebraic extension $\boldsymbol{B}_ω^{\mathscr{F}}$ of the bicyclic monoid for an arbitrary $ω$-closed family $\mathscr{F}$ subsets of $ω$ which generalizes the bicyclic monoid, the countable semigroup of matrix units and some other combinatorial inverse semigroups. It is proved that $\boldsymbol{B}_ω^{\mathscr{F}}$ is a combinatorial inverse semigroup and Green's relations, the natural partial order on $\boldsymbol{B}_ω^{\mathscr{F}}$, and its set of idempotents are described. We provide criteria of simplicity, $0$-simplicity, bisimplicity, $0$-bisimplicity of the semigroup $…
We consider blind, deterministic, finite automata equipped with a register which stores an element of a given monoid, and which is modified by right multiplication by monoid elements. We show that, for monoids M drawn from a large class including groups, such an automaton accepts the word problem of a group H if and only if H has a finite index subgroup which embeds in the group of units of M. In the case that M is a group, this answers a question of Elston and Ostheimer.
The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa-Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych, In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are pre…
This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form $\operatorname{Inv}\langle A \mid w=1 \rangle$. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes ${\rm {\small ANY}}, {\rm {\small RED}}, {\rm {\small CRED}},$ and ${\rm {\small POS}}$ of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary…
We give a construction of the projective indecomposable modules and a description of the quiver for a large class of monoid algebras including the algebra of any finite monoid whose principal right ideals have at most one idempotent generator. Our results include essentially all families of finite monoids for which this has been done previously, for example, left regular bands, $\mathscr J$-trivial and $\mathscr R$-trivial monoids and left regular bands of groups.
We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a "nice" Garside element, certain monoids $S$ with quadratic relations, whose monoidal algebra $A= k[S]$ has a Frobenius Koszul dual $A^{!}$ with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. Ther…
A number of properties of spherical Artin groups extend to Garside groups, defined as the groups of fractions of monoids where least common multiples exist, there is no nontrivial unit, and some additional finiteness conditions are satisfied \cite{Dgk}. Here we investigate a wider class of groups of fractions, called {\it thin}, which are those associated with monoids where minimal common multiples exist, but they are not necessarily unique. Also, we allow units in the involved monoids. The main results are that all thin groups of fractions satisfy a quadratic isoperimetric inequality, and tha…
We prove that the exponential growth rate of the regular language of penetration sequences is smaller than the growth rate of the regular language of normal form words, if the acceptor of the regular language of normal form words is strongly connected. Moreover, we show that the latter property is satisfied for all irreducible Artin monoids of spherical type, extending a result by Caruso. Our results establish that the expected value of the penetration distance $pd(x,y)$ in an irreducible Artin monoid of spherical type is bounded independently of the length of $x$, if $x$ is chosen uniformly a…
We prove every regular element of a free profinite monoid generates a prime ideal; in particular the minimal ideal is prime. The latter result was first proved by Almeida and Volkov using techniques from symbolic dynamics; our proof is elementary.
For an element $a$ of a monoid $H$, its set of lengths $\mathsf L (a) \subset \mathbb N$ is the set of all positive integers $k$ for which there is a factorization $a=u_1 \cdot \ldots \cdot u_k$ into $k$ atoms. We study the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ with a focus on the unions $\mathcal U_k (H) \subset \mathbb N$ which are the unions of all sets of lengths containing a given $k \in \mathbb N$. The Structure Theorem for Unions -- stating that for all sufficiently large $k$, the sets $\mathcal U_k (H)$ are almost arithmetical progressions with the same difference a…
For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M]. This strengthens the K_2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stable K_2-group is substituted by the unstable ones. The proof is based on a polyhedral/combinatorial techniques, computations in Steinberg groups, and a substantially corrected version of an old result on elementary matrices by Mushkudiani [Mu]. A similar stronger nilpotence result for K_1 and algorithmic consequences for fa…
We study C*-algebras generated by left regular representations of right LCM one-relator monoids and Artin-Tits monoids of finite type. We obtain structural results concerning nuclearity, ideal structure and pure infiniteness. Moreover, we compute K-theory. Based on our K-theory results, we develop a new way of computing K-theory for certain group C*-algebras and crossed products.
The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.
