The free idempotent generated locally inverse semigroup

In this paper we construct a model for the free idempotent generated locally inverse semigroup on a set X. The elements of this model are special vertex-labeled bipartite trees with a pair of distinguished vertices. To describe this model, we need first to introduce a variation of a model for the free pseudosemilattice on a set X presented in Auinger and Oliveira (On the variety of strict pseudosemilattices. Stud Sci Math Hungarica 50:207–241, 2013). A construction of a graph associated with a regular semigroup was presented in Brittenham et al. (Subgroups of free idempotent generated semigroups need not be free. J Algebra 321:3026–3042, 2009) in order to give a first example of a free regular idempotent generated semigroup on a biordered set E with non-free maximal subgroups. If G is the graph associated with the free pseudosemilattice on X, we shall see that the models we present for the free pseudosemilattice on X and for the free idempotent generated locally inverse semigroup on X are closely related with the graph G.     

Idempotent Generation in the Endomorphism Monoid of a Uniform Partition

Denote by 𝒯_n and 𝒮_n the full transformation semigroup and the symmetric group on the set {1,…, n}, and ?_n = {1} ∪ (𝒯_n?𝒮_n). Let 𝒯(X, 𝒫) denote the monoid of all transformations of the finite set X preserving a uniform partition 𝒫 of X into m subsets of size n, where m, n ≥ 2. We enumerate the idempotents of 𝒯(X, 𝒫), and describe the submonoid S = ? E ? generated by the idempotents E = E(𝒯(X, 𝒫)). We show that S = S₁ ∪ S₂, where S₁ is a direct product of m copies of ?_n, and S₂ is a wreath product of 𝒯_n with 𝒯_m?𝒮_m. We calculate the rank and idempotent rank of S, showing that these are equal, and we also classify and enumerate all the idempotent generating sets of minimal size. In doing so, we also obtain new results about arbitrary idempotent generating sets of ?_n.     

Amenability constants for semilattice algebras

For any finite commutative idempotent semigroup S, a semilattice, we show how to compute the amenability constant of its semigroup algebra ℓ ¹(S). This amenability constant is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. We also give example of a commutative Clifford semigroups G _n whose semigroup algebras ℓ ¹(G _n ) admit amenability constants of the form 41+4(n−1)/n. We also show there is no commutative semigroup whose semigroup algebra has an amenability constant between 5 and 9. N. Spronk’s research was supported by NSERC Grant 312515-05.     

Fully invariant transformations and associated groups

It is well known that the symmetric group S _ntogether with one idempotent of rank n- 1 on a finite n-element set Nserves as a set of generators for the semigroup T _nof all the total transformations on N. It is also well known that the singular part Sing _n of T _n can be generated by a set of idempotents of rank n- 1. The purpose of this paper is to begin an investigation of the way in which Sing_nand its subsemigroups can be generated by the conjugates of a subset of elements of T _n by a subgroup of S _n . We look for the smallest subset of elements of T _n that will serve and, correspondingly, for a characterization of those subgroups of S _n that will serve. Using some techniques from graph theory we prove our main result:the conjugates of a single transformation of rank n- 1 under Gsuffice to generate Sing_nif and only if Gis what we define to be a 2-block transitive subgroup of S _n .     

Regularity on Near Permutation Semigroups

Abstract

A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group of permutations on N elements and by a set of transformations of rank N ? 1. The aim of this paper is to determine computationally efficient conditions to test whether or not a near permutation semigroup is regular.     

Bipartite graphs and completely 0-simple semigroups

In a manner similar to the construction of the fundamental group of a connected graph, this article introduces the construction of a fundamental semigroup associated with a bipartite graph. This semigroup is a 0-direct union of idempotent generated completely 0-simple semigroups. The maximal nonzero subgroups are the corresponding fundamental groups of the connected components. Adding labelled edges to the graph leads to a more general completely 0-simple semigroup. The basic properties of such semigroups are examined and they are shown to have certain universal properties as illustrated by the fact that the free completely simple semigroup on n generators and its idempotent generated subsemigroup appear as special cases.     

Every group is a maximal subgroup of a naturally occurring free idempotent generated semigroup

The study of the free idempotent generated semigroup IG(E) over a biordered set E has recently received a deal of attention. Let G be a group, let \(n\in\mathbb{N}\) with n≥3 and let E be the biordered set of idempotents of the wreath product \(G\wr \mathcal{T}_{n}\) . We show, in a transparent way, that for e∈E lying in the minimal ideal of \(G\wr\mathcal{T}_{n}\) , the maximal subgroup of e in IG(E) is isomorphic to G. It is known that \(G\wr\mathcal{T}_{n}\) is the endomorphism monoid End F _n (G) of the rank n free G-act F _n (G). Our work is therefore analogous to that of Brittenham, Margolis and Meakin for rank 1 idempotents in full linear monoids. As a corollary we obtain the result of Gray and Ru?kuc that any group can occur as a maximal subgroup of some free idempotent generated semigroup. Unlike their proof, ours involves a natural biordered set and very little machinery.     

Generating Sets of Completely 0-Simple Semigroups

ABSTRACT

A formula for the rank of an arbitrary finite completely 0-simple semigroup, represented as a Rees matrix semigroup ?⁰[G; I, Λ; P], is given. The result generalizes that of Ru?kuc concerning the rank of connected finite completely 0-simple semigroups. The rank is expressed in terms of |I|, |Λ|, the number of connected components k of P, and a number r _min, which we define. We go on to show that the number r _min is expressible in terms of a family of subgroups of G, the members of which are in one-to-one correspondence with, and determined by the nonzero entries of, the components of P. A number of applications are given, including a generalization of a result of Gomes and Howie concerning the rank of an arbitrary Brandt semigroup B(G,{1,…,n}).     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

A note on maximal regular subsemigroups of the finite transformation semigroups \mathcal{T}(n,r)

Let $\mathcal{T}_{n}$ be the semigroup of all full transformations on the finite set X _n ={1,2,…,n}. For 1≤r≤n, set $\mathcal {T}(n, r)=\{ \alpha\in\mathcal{T}_{n} | \operatorname{rank}(\alpha)\leq r\}$ . In this note we show that, for 2≤r≤n?2, any maximal regular subsemigroup of the semigroup $\mathcal{T} (n,r)$ is idempotent generated, but this may not happen in the semigroup $\mathcal{T}(n, n-1)$ .     

D n -normal semigroups of transformations

Given a subgroup G of the symmetric group S _n on n letters, a semigroup S of transformations of X _n is G-normal if G _S =G, where G _S consists of all permutations h∈S _n such that h ⁻¹ fh∈S for all f∈S. A semigroup S is G-normax if it is a maximal semigroup in the set of all G-normal semigroups. In 1996, I. Levi showed that the alternating group A _n can not serve as the group G _S for any semigroup of total transformations of X _n . In 2000 and 2001, I. Levi, D.B. McAlister and R.B. McFadden described all A _n -normal semigroups of partial transformations of X _n . Also, in 1994, I. Levi and R.B. McFadden described all S _n -normal semigroups. In this paper, we show that the dihedral group D _n may serve as the group G _S for semigroups of transformations of X _n . We characterize a large class of D _n -normax semigroups and describe certain D _n -normal semigroups.     

On a Problem of M. Kambites Regarding Abundant Semigroups

A semigroup is regular if it contains at least one idempotent in each ?-class and in each ?-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each ?-class and in each ?-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each ?*-class and in each ?*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each ?* and ?*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each ?* and ?*-class, must the idempotents commute? In this note, we provide a negative answer to this question.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? admits an n-torsion section, its quotient by the group of automorphisms induced by this section admits again an n-torsion section, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group.

Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.     

Shades of orthodoxy in Rees-Sushkevich varieties

It is shown that, within the class of Rees-Sushkevich varieties that are generated by completely (0-) simple semigroups over groups of exponent dividing n, there is a hierarchy of varieties determined by the lengths of the products of idempotents that will, if they fall into a group ℋ-class, be idempotent. Moreover, the lattice of varieties generated by completely (0-) simple semigroups over groups of exponent dividing n, with the property that all products of idempotents that fall into group ℋ-classes are idempotent, is shown to be isomorphic to the direct product of the lattice of varieties of groups with exponent dividing n and the lattice of exact subvarieties of a variety generated by a certain five element completely 0-simple semigroup.     

Krull Dimension and Deviation in Certain Parafree Groups

Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference μ(G) ? n, where μ(G) is the minimum possible number of generators of G.

Let G be fg; then Hom(G, SL ₂?) inherits the structure of an algebraic variety, denoted by R(G). If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2 and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer.     

Finite generation of iterated wreath products

Let (G _n , X _n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product ${\ldots\wr G_2\wr G_1}Let (G _n , X _n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product ?\wr G₂\wr G₁{\ldots\wr G_2\wr G_1} is topologically finitely generated if and only if the profinite abelian group ?_{n 3 1} G_n/G￠_n{\prod_{n\geq 1} G_n/G'_n} is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power G\wr ?\wr G\wr G{G\wr \ldots\wr G\wr G} (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.     

On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices

Let n be a positive integer, and C _n (r) the set of all n × n r-circulant matrices over the Boolean algebra B = {0, 1}, . For any fixed r-circulant matrix C (C ≠ 0) in G _n , we define an operation “*” in G _n as follows: A * B = ACB for any A, B in G _n , where ACB is the usual product of Boolean matrices. Then (G _n , *) is a semigroup. We denote this semigroup by G _n (C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. Let F be an idempotent element in G _n (C) and M(F) the maximal subgroup in G _n (C) containing the idempotent element F. In this paper, the elements in M(F) are characterized and an algorithm to determine all the elements in M(F) is given.