We develop the theory of transformation semigroups that have degree 2, is, act by partial functions on a finite set such inverse image points at most two elements. show graph fibers an action gives deep connection between semigroup and theory. It is known Krohn–Rhodes complexity 2 2. monoid continuous maps translational hull appropriate 0-simple semigroup. how group mapping can be considered as...

Let A be a finite dimensional Q-algebra and Γ ⊂ A a Z-order. We classify those A with the property that Z 6 →֒U(Γ). We call this last property the hyperbolic property. We apply this in the case that A = KS a semigroup algebra with K = Q or K = Q( √ −d). In particular, when KS is semi-simple and has no nilpotent elements, we prove that S is an inverse semigroup which is the disjoint union of Higm...

Group theory and semigroup theory have developed in somewhat different directions in the past several decades. While Cayley’s theorem enables us to view groups as groups of permutations of some set, the analogous result in semigroup theory represents semigroups as semigroups of functions from a set to itself. Of course both group theory and semigroup theory have developed significantly beyond t...

We give a general description of the free proonite semigroups over a semidirect product of pseudovarieties. More precisely, A (V W) is described as a closed subsemigroup of a proonite semidirect product of the form A WA V A W. As a particular case, the free proonite semigroup over J 1 V is described in terms of the geometry of the Cayley graph of the free proonite semigroup over V (here J 1 is ...

1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ?) where G is a nonempty set and ? is a binary operation on G satisfying: (G1) a ? (b ? c) = (a ? b) ? c, ∀a, b, c ∈ G. A semigroup G is a monoid if it also satisfies: (G2) G has an element e (sometimes denoted by 1G, called the identity of G), such that e ? a = a ? e = a, ∀a ∈ G. A monoid is a group if (G3) below is satisfied...

In this paper we introduce two symmetric variants of amenability, symmetric module amenability and symmetric Connes amenability. We determine symmetric module amenability and symmetric Connes amenability of some concrete Banach algebras. Indeed, it is shown that $ell^1(S)$ is  a symmetric $ell^1(E)$-module amenable if and only if $S$ is amenable, where $S$ is an inverse semigroup with subsemigr...

A common generalization of the author's embedding theorem concerning the £-unitary regular semigroups with regular band of idempotents, and Billhardt's and Ismaeel's embedding theorem on the inverse semigroups, the closure of whose set of idempotents is a Clifford semigroup, is presented. We prove that each orthodox semigroup with a regular band of idempotents, which is an extension of an ortho...

We examine, in a general setting, a notion of inverse semigroup of left quotients, which we call left I-quotients. This concept has appeared, and has been used, as far back as Clifford’s seminal work describing bisimple inverse monoids in terms of their right unit subsemigroups. As a consequence of our approach, we find a straightforward way of extending Clifford’s work to bisimple inverse semi...

We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism (elementary equivalence) of the subsemigroups yield...

We extend the theory of fast Fourier transforms on finite groups to finite inverse semigroups. We use a general method for constructing the irreducible representations of a finite inverse semigroup to reduce the problem of computing its Fourier transform to the problems of computing Fourier transforms on its maximal subgroups and a fast zeta transform on its poset structure. We then exhibit exp...