Inverse semigroups generated by a pair of subgroups

GENERALIZED INVERSE SEMIGROUPS AND AMALGAMATION

The algebraic theory of inverse semigroups has received considerable attention in recent years with satisfactory structure theorems being obtained for extensive classes of these semigroups. As is the case with so much of semigroup theory, many of these structure theorems were obtained by, or had their origin in fundamental papers of, A. H. Clifford. For example, the structure of inverse semigroups with central idempotents (semilattices of groups) was obtained by Clifford in [1], that of Brandt semigroups was obtained in [2], while, in [3], Clifford described bisimple inverse semigroups in terms of their right unit subsemigroup. The third of these papers has been the basis for much of the research on structural aspects of inverse semigroups. On the one hand, its direct descendants include Reilly's papers on bi-simple inverse semigroups in terms of RP-systems [10], [11], and the author's description of O-bisimple inverse semigroups in terms of groups and semilattices [5]. On the other hand, it has led to Reilly's description [9] of w-bisimple inverse semigroups and Warne's characterisations of various classes of bisimple inverse semigroups with special semilattices of idempotents. These, in turn, have led to the determination of simple inverse semigroups with special semilattices of idempotents.

In a previous paper ([14]) the author showed that a free inverse semigroup is determined by its lattice of inverse subsemigroups, in the sense that for any inverse semigroup T, implies . (In fact, the lattice isomorphism is induced by an isomorphism of upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and E-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

An inverse semigroup S is called E-unitary if the equations ea = e = e2 together imply a2 = a. In a previous paper the author showed that every inverse semigroup is an idempotent separating homomorphic image of an E-unitary inverse semigroup. The main question considered in this paper is the following. Given an inverse semigroup S give a method for constructing E-unitary inverse semigroups P and idempotent separating homomorphisms φ P → S in such a way that the structure of P as a P-semigroup is evident.

Type B semigroups are generalizations of inverse semigroups, and every inverse semigroup admits an [Formula: see text]-unitary cover (M. Petrich, Inverse Semigroups (Wiley, New York, 1984)). Motivated by studying [Formula: see text]-unitary cover for inverse semigroups, and as a continuation of Petrich’s works in inverse semigroups, in this paper, we first introduce the concept of ∗-prehomomorphism of a type B semigroup. After obtaining some basic properties, we get some structure theorems and give some conditions for a type B semigroup which is constructed by using the ∗-prehomomorphism to be proper. In particular, we introduce the notion of [Formula: see text]-unitary good cover for an abundant semigroup, and prove that every type B semigroup with compatible natural partial order admits an [Formula: see text]-unitary good cover.

The inverse semigroup TE of isomorphisms of principal ideals of E onto principal ideals of E, where E is a semilattice, has been introduced and studied by Munn (1966, 1970). He showed that, for any inverse semigroup S with semilattice E, there is a representation of S by an inverse subsemigroup of TE. The Munn representation, however, is not always faithful. In this paper, the possibility is considered of enlarging the carrier set E of the Munn representation in order to obtain a faithful representation of S as an inverse subsemigroup of a structure resembling TE in many ways. A structure X is obtained by replacing each element of E by a set. Then X = ∪{Xe: e ∈ E}, where Xe, denotes some set, has a natural pre-order relation ≤ (where x ≤ y if and only if x ∈ Xe, y ∈ Xf and e ≦ f ) inherited from E such that if T = {(x, y)∈X × X;x ≤ y and y ≤ x} then X/T is isomorphic to E. Such a set X is referred to as a pre-semilattice with semilattice E. If Tx denotes the set of all isomorphisms of principal ideals of X onto principal ideals of X then Tx is an inverse semigroup. Basic properties of Tx are considered. It is shown that when X is locally uniform, that is, when |Xe| = |Xf|, for all e, f ∈ E, Tx may be described as a wreath product of a permutation group with TE.The set s itself is a presemilattice with semilattice E with respect to the pre-order ≤ defined by a ≤ b if and only if a−1a ≦ b−1b. It is then shown that the Vagner-Preston representation embeds S as a full inverse subsemigroup of Ts. As an application of these concepts the following result is established. Let R and S be inverse semigroups and let θ1(θ2) be an isomorphism of a semilattice E onto the semilattice of R(S). Then there exists a locally uniform presemilattice W and embeddings ϕ1, ϕ2 of R and S, respectively, as full inverse subsemigroups of Tw such that (1) θ1ϕ1 = θ2ϕ2 and (2) (eθ1ϕ1, eθ2ϕ2) ∈ if and only if Ee is isomorphic to Ef.

We study inverse semigroup amalgams [S 1,S 2;U], where S 1 and S 2 are finitely presented inverse semigroups with decidable word problem and U is an inverse semigroup with decidable membership problem in S 1 and S 2. We use a modified version of Bennett’s work on the structure of Schützenberger graphs of the ℛ-classes of S 1* U S 2 to state sufficient conditions for the amalgamated free products S 1* U S 2 having decidable word problem.

We consider semigroups such that the universal left congruence omega ^{ell } is finitely generated. Certainly a left noetherian semigroup, that is, one in which all left congruences are finitely generated, satisfies our condition. In the case of a monoid the condition that omega ^{ell } is finitely generated is equivalent to a number of pre-existing notions. In particular, a monoid satisfies the homological finiteness condition of being of type left-{text {FP}}_1 exactly when omega ^{ell } is finitely generated.Our investigations enable us to classify those semigroups such that omega ^{ell } is finitely generated that lie in certain important classes, such as strong semilattices of semigroups, inverse semigroups, Rees matrix semigroups (over semigroups) and completely regular semigroups. We consider closure properties for the class of semigroups such that omega ^{ell } is finitely generated, including under morphic image, direct product, semi-direct product, free product and 0-direct union. Our work was inspired by the stronger condition, stated for monoids in the work of White, of being pseudo-finite. Where appropriate, we specialise our investigations to pseudo-finite semigroups and monoids. In particular, we answer a question of Dales and White concerning the nature of pseudo-finite monoids.

SynopsisA Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

For any amalgam ( S , T ; U ) (S,\,T;\,U) of inverse semigroups, it is shown that the natural partial order on S ∗ U T S{{\ast }_U}T , the (inverse semigroup) free product of S S and T T amalgamating U U , has a simple form on S ∪ T S \cup T . In particular, it follows that the semilattice of S ∗ U T S{{\ast }_U}T is a bundled semilattice of the corresponding semilattice amalgam ( E ( S ) , E ( T ) ; E ( U ) ) (E(S),\,E(T);\,E(U)) ; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

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.

Let G=C ∞×Φ where Φ is a nontrivial finite group. In this chapter, subject to conditions on Φ, we begin the task of parametrizing the first syzygy Ω 1(Z) over Z[G]. We do so via a generalization, introduced by Edwards in his thesis (Ph.D. Thesis, University College London, 2006), of a type of module first studied by Swan. The original Swan modules arise as follows; let Φ be a finite group and let ϵ:Z[Φ]→Z be the augmentation homomorphism. For n∈Z we take n:Z→Z to be the homomorphism x↦nx. The eponymous Swan module \((\mathcal{I}, n)\) is defined as \((\mathcal{I}, n) = \varprojlim(\epsilon, \mathbf{n})\). From the commutative diagram $$\begin{array}{clccclc}0 \rightarrow &\mathcal{I} & \longrightarrow &(\mathcal{I}, n)&\stackrel{\epsilon}{\longrightarrow} & \mathbf{Z}&\rightarrow 0\\&\downarrow\scriptstyle\mathrm{Id}&&\cap&&\downarrow\scriptstyle \mathbf{n}&\\0 \rightarrow &\mathcal{I}& \longrightarrow &\mathbf{Z}[\varPhi]&\stackrel{\epsilon}{\longrightarrow} &\mathbf{Z}&\rightarrow 0\end{array}$$ we see that \((\mathcal{I}, n)\) imbeds in Z[Φ] as a Z-sublattice of index n. The main properties of the \((\mathcal{I}, n)\) were established by Swan (Ann. Math. 72:267–291, 1960). Indeed, this was the original context for the projectivity criterion of Chap. 5; Swan showed that \((\mathcal{I}, n)\) is projective over Z[Φ] if and only if n is a unit mod |Φ|. This and other aspects generalize in a manner which we now proceed to describe.

Let [Formula: see text] be a non-trivial finite group different from a cyclic [Formula: see text]-group. The join graph of [Formula: see text] is a graph whose vertex set is the set of all proper subgroups of [Formula: see text] which are not contained in the Frattini subgroup of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are joined by an edge if and only if [Formula: see text]. In this paper we classify finite groups with regular graphs and determine their graphs.