Combinatorial Inverse Semigroups Research Articles

Semiring identities of finite inverse semigroups

We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial rook monoid admits no finite identity basis, and so do almost all additively idempotent semirings whose multiplicative reducts are combinatorial inverse semigroups.

Characterizing pure, cryptic and Clifford inverse semigroups

An inverse semigroup S is pure if e = e 2, a ∈ S, e < a implies a 2 = a; it is cryptic if Green’s relation H on S is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and Clifford semigroups in a similar way by means of divisors. The paper also contains characterizations of completely semisimple inverse and of combinatorial inverse semigroups in a similar manner. It ends with a description of minimal non-V varieties, for varieties V of inverse semigroups considered.

On Incidence Algebras of Combinatorial Inverse Monoids

We study the incidence algebra of the reduced standard division category of a combinatorial bisimple inverse monoid [with (E(S), ≤) locally finite], and we describe semigroups of poset type (i.e., a combinatorial inverse semigroup for which the corresponding Möbius category is a poset) as being combinatorial strict inverse semigroups. Up to isomorphism, the only Möbius-division categories are the reduced standard division categories of combinatorial inverse monoids.

$$\mathcal{P}\mathcal{A}{\text{-isomorphisms}}$$ of inverse semigroups

A partial automorphism of a semigroup $S$ is any isomorphism between its subsemigroups, and the set all partial automorphisms of $S$ with respect to composition is the inverse monoid called the partial automorphism monoid of $S$. Two semigroups are said to be $\cPA$-isomorphic if their partial automorphism monoids are isomorphic. A class $\K$ of semigroups is called $\cPA$-closed if it contains every semigroup $\cPA$-isomorphic to some semigroup from $\K$. Although the class of all inverse semigroups is not $\cPA$-closed, we prove that the class of inverse semigroups, in which no maximal isolated subgroup is a direct product of an involution-free periodic group and the two-element cyclic group, is $\cPA$-closed. It follows that the class of all combinatorial inverse semigroups (those with no nontrivial subgroups) is $\cPA$-closed. A semigroup is called $\cPA$-determined if it is isomorphic or anti-isomorphic to any semigroup that is $\cPA$-isomorphic to it. We show that combinatorial inverse semigroups which are either shortly connected [5] or quasi-archimedean [10] are $\cPA$-determined.

The M�bius Category of Some Combinatorial Inverse Semigroups

\noindent The purpose of this paper is to point out some aspects of the relationship between combinatorial inverse semigroups and their Mobius categories, and to explore combinatorial results arising from combinatorial Brandt semigroups, fundamental simple inverse ω-semigroups and from the free monogenic inverse semigroup.

A finitely generated variety of combinatorial inverse semigroups having uncountably many subvarieties

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.

A Class of Actions of Inverse Semigroups

We generalise the classical Munn representation of an inverse semigroup with the introduction of what we call ordered representations of inverse semigroups. Both the Wagner–Preston Representation and the effective actions of O'Carroll and McAlister are examples of such representations. We show that every ordered representation of an inverse semigroup Sdetermines and is determined by a special kind of cover of S. As applications, we provide a fully categorical account of the theory of idempotent pure congruences, and we show that every inverse semigroup which is a semilattice with respect to the natural partial ordering is an image of a combinatorial inverse semigroup under an L -bijective, prehomomorphism.

On varieties of combinatorial inverse semigroups II

On varieties of combinatorial inverse semigroups II

On varieties of combinatorial inverse semigroups I

On varieties of combinatorial inverse semigroups I

Inverse semigroups with isomorphic partial automorphism semigroups

AbstractIt is shown that a so-called shortly connected combinatorial inverse semigroup is strongly lattice-determined “modulo semilattices”. One of the consequences of this theorem is the known fact that a simple inverse semigroup with modular lattice of full inverse subsemigroups is strongly lattice-determined [7]. The partial automorphism semigroup of an inverse semigroup S consists of all isomorphisms between inverse subsemigroups of S. It is proved that if S is a shortly connected combinatorial inverse semigroup, T an inverse semigroup and the partial automorphism semigroups of S and T are isomorphic, then either S and T are isomorphic or they are dually isomorphic chains (with respect to the natural partial order); moreover, any isomorphism between the partial automorphism semigroups of S and T is induced either by an isomorphism or, if S and T are dually isomorphic chains, by a dual isomorphism between S and T. Counter-examples are constructed to demonstrate that the assumptions about S being shortly connected and combinatorial are essential.

The Algebra of a Combinatorial Inverse Semigroup

The Algebra of a Combinatorial Inverse Semigroup

Basis properties for inverse semigroups

In a previous paper [9], the author showed that a wide class of inverse semigroups (for example free inverse semigroups, semilattices and finite combinatorial inverse semigroups) possess the strong basis property, and thus the basis property, which we now define. If S is an invcrsc semigroup and C < C; ::z S, a CT-basis for I’ is a subset S of V, minimal such that (Z! u .Y) 7: V, and a hasis for V is an O-basis for V, S is said to have the basis property [strong basis property] if for any inverse subsemigroup V of S [and inverse subsemigroup I,’ of I’] any two bases for C’ [U-bases for I’] have the same cardinality. (One notable consequence of the strong basis property for S is that for any j’-class J of S, any two U-bases for V have the same number of elements in J [9, Theorem 3.91). Two questions were put in [9]: