GENERALIZED INVERSE SEMIGROUPS AND AMALGAMATION

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

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.

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.

A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.

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.

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.

E-unitary inverse monoids and the Cayley graph of a group presentation

A class of algebras A is said to have the strong amalgamation property if for any indexed set of algebras {Ai: i ∈ J} from A, each having an algebra U ∈ A as a subalgebra, there exist an algebra B ∈ A and monomorphisms φi: Ai → B (for each i ∈ J) such that (i) φi|U = φi| U for all i,j∈J, (ii) φi(Ai) ∩ φi(Aj) =φi(U) for atl i, j ∈ J with i ≠ j, where φi|U denotes the restriction of φi to U. Omitting condition (ii) gives us the definition of the weak amalgamation property.

We consider inverse semigroup amalgams [S1,S2;U] such that for any u ∈ U and e ∈ E(Si) with u ≥ e in Si, where i ∈ {1,2}, there exists f ∈ E(U) with u ≥ f ≥ e in Si; we say that U is lower bounded in S1 and S2. We construct and describe the Schutzenberger automata of S1∗U S2 and give conditions for decidable word problem. The homomorphisms of the Schutzenberger graphs of S1∗U S2 are studied and conditions are given for S1∗U S2 to be completely semisimple. In the case when S1 and S2 have decidable word problems and U is finite, we show that S1∗U S2 has decidable word problem.

Lattice isomorphisms of free products of inverse semigroups

A graphical representation for the free product of E-unitary inverse semigroups