Free idempotent generated semigroups: subsemigroups,retracts and maximal subgroups

Let S be a subsemigroup of a semigroup T and let IG(E) and IG(F) be the free idempotent generated semigroups over the biordered sets of idempotents of E of S and F of T, respectively. We examine the relationship between IG(E) and IG(F), including the case where S is a retract of T. We give su?cient conditions satisfied by T and S such that for any e∈E, the maximal subgroup of IG(E) with identity e is isomorphic to the corresponding maximal subgroup of IG(F). We then apply this result to some special cases and, in particular, to that of the partial endomorphism monoid PEnd A and the endomorphism monoid EndA of an independence algebra A of finite rank. As a corollary, we obtain Dolinka’s reduction result for the case where A is a finite set.     

On finitely generated idempotent semigroups

We give two short proofs of the well-known fact that every finitely generated idempotent semigroup is finite.     

The endomorphism monoids and automorphism groups of Cayley graphs of semigroups

In this note, we introduce the notions of color-permutable automorphisms and color-permutable vertex-transitive Cayley graphs of semigroups. As a main result, for a finite monoid S and a generating set C of S, we explicitly determine the color-permutable automorphism group of \(\mathrm {Cay}(S,C)\) [Theorem 1.1]. Also for a monoid S and a generating set C of S, we explicitly determine the color-preserving automorphism group (endomorphism monoid) of \(\mathrm {Cay}(S,C)\) [Proposition 2.3 and Corollary 2.4]. Then we use these results to characterize when \(\mathrm {Cay}(S,C)\) is color-endomorphism vertex-transitive [Theorem 3.4].     

Subgroups of free idempotent generated regular semigroups

In [6] it is shown that the maximal subgroups of the free idempotent generated regular semigroup which is determined by the biordered set of a completely O-simple semigroup are free. In this note we shall extend this result to a wider class of semigroups.     

On maximal subgroups of free idempotent generated semigroups

We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup. As a technical prerequisite for these results we establish a general presentation for the maximal subgroups based on a Reidemeister-Schreier type rewriting.     

Near varieties of idempotent generated completely simple semigroups

Algebra universalis - Completely simple semigroups form a variety if we consider them as algebras with multiplication and inversion. A near variety of idempotent generated completely simple...     

Freeoids: a semi-abstract view on endomorphism monoids of relatively free algebras

A freeoid over a (normally, infinite) set of variables X is defined to be a pair (W, E), where W is a superset of X, and E is a submonoid of W ^W containing just one extension of every mapping X → W. For instance, if W is a relatively free algebra over a set of free generators X, then the pair F(W) := (W, End(W)) is a freeoid. In the paper, the kernel equivalence and the range of the transformation F are characterized. Freeoids form a category; it is shown that the transformation F gives rise to a functor from the category of relatively free algebras to the category of freeoids which yields a concrete equivalence of the first category to a full subcategory of the second one. Also, the concept of a model of a freeoid is introduced; the variety generated by a free algebra W is shown to be concretely equivalent to the category of models of F(W). The sets X, W, and the algebras W may generally be many-sorted.     

Periodic subsemigroups of endomorphism monoids

If S is a periodic subsemigroup of the endomorphism monoid of a polycyclic group, then Endimioni (Mediterr J Math 8:307–313, 2011) proved that S is locally finite. Here we present an alternative proof that also extends the result to groups with suitable rank restrictions. Further we give an alternative proof of McNaughton and Zalcstein’s (J Algebra 34:292–299, 1975) theorem that periodic multiplicative subsemigroups of a matrix ring over a field are also locally finite. Finally we extend the latter to periodic subsemigroups of the endomorphism ring of a finitely generated module over a commutative ring.     

Subgroups of free idempotent generated semigroups need not be free

We study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods. These subgroups are realized as the fundamental groups of a number of 2-complexes naturally associated to the biorder structure of the set of idempotents. We use this to construct the first example of a free idempotent generated semigroup containing a non-free subgroup.     

Bipartite graphs and their endomorphism monoids

It is shown that any connected bipartite graph is determined by its endomorphism monoid up to isomorphism. © 1996 John Wiley & Sons, Inc.     

Hereditary endomorphism monoids of projective acts

Projective acts whose endomorphism monoids are left or right (semi-) hereditary are characterized. For example, it is shown that for a noncyclic free or projective S-act P, End P is left (semi) hereditary if and only if P ≈ Se₁ Π Se₂ and e₁Se₁, e₂Se₂ are groups.     

A note on maximal subgroups of free idempotent generated semigroups over bands

We prove that all maximal subgroups of the free idempotent generated semigroup over a band B are free for all B belonging to a band variety V if and only if V consists either of left seminormal bands, or of right seminormal bands.     

The Banach algebras generated by representations of abelian semigroups

Let S be a suitable subsemigroup of a locally compact abelian group. We consider the class of Banach algebras A for which there exists a continuous homomorphism ω : L ¹ (S) → A with dense range. We study the behavior of the radical in such algebras. As an application, we present some results concerning the structure of weakly compact homomorphisms of semigroup algebras.     

The Banach algebras generated by representations of abelian semigroups

Let S be a suitable subsemigroup of a locally compact abelian group. We consider the class of Banach algebras A for which there exists a continuous homomorphism ω : L ¹ (S) → A with dense range. We study the behavior of the radical in such algebras. As an application, we present some results concerning the structure of weakly compact homomorphisms of semigroup algebras.     

On square-increasing ordered monoids and idempotent semirings

Let \(\mathcal {V}\) be the variety of square-increasing idempotent semirings. Its members can be viewed as semilattice-ordered monoids satisfying \(x\le x^{2}\). We show that the universal theory of \(\mathcal {V}\) is decidable. In order to prove this result, we investigate the class \(\mathcal {Q}\) whose members are ordered-monoid subreducts of members from \(\mathcal {V}\). In particular, we prove that finitely generated members from \(\mathcal {Q}\) are well-partially-ordered and residually finite.