FREE ADEQUATE SEMIGROUPS

We give an explicit description of the free objects in the quasivariety of adequate semigroups, as sets of labelled directed trees under a natural combinatorial multiplication. The morphisms of the free adequate semigroup onto the free ample semigroup and into the free inverse semigroup are realised by a combinatorial ‘folding’ operation which transforms our trees into Munn trees. We use these results to show that free adequate semigroups and monoids are 𝒥-trivial and never finitely generated as semigroups, and that those which are finitely generated as (2,1,1)-algebras have decidable word problem.

GRÖBNER–SHIRSHOV BASES FOR FREE INVERSE SEMIGROUPS

A new construction for free inverse semigroups was obtained by Poliakova and Schein in 2005. Based on their result, we find Gröbner–Shirshov bases for free inverse semigroups with respect to the deg-lex order of words. In particular, we give the (unique and shortest) normal forms in the classes of equivalent words of a free inverse semigroup together with the Gröbner–Shirshov algorithm to transform any word to its normal form.

GROWTH AND EXISTENCE OF IDENTITIES IN A CLASS OF FINITELY PRESENTED INVERSE SEMIGROUPS WITH ZERO

We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

A congruence-free inverse semigroup associated with a pair of infinite cardinals

Let m, n be infinite cardinals such that m < n, and let X be a set of cardinality m. Within the symmetric inverse semigroup on X the elements whose domain and range have complements of cardinality m form an inverse semigroup T. The closure Eω of the semilattice E of idempotents of T is a fundamental bismple inverse semigroup. Its maximum congruence is described. The quotient of Eο by this maximum congruence is a bisimple, congruence is a bisimple, congruence-free inverse semigroup.

An embedding theorem for free inverse semigroups

In this note it is shown that if S is a free inverse semigroup of rank at least two and if e, f are idempotents of S such that e > f then S can be embedded in the partial semigroup eSe/fSf. The proof makes use of Scheiblich's construction for free inverse semigroups [7, 8] and of Reilly's characterisation of a set of free generators in an inverse semigroup [4, 5].

Inverse semigroups determined by their lattices of inverse subsemigroups

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.

Ideals of free inverse semigroups

It is shown that no proper ideal of a free inverse semigroup is free and that every isomorphism between ideals is induced by a unique automorphism of the whole semigroup. In addition, necessary and sufficient conditions are given for two principal ideals to be isomorhic.

Lattice isomorphisms of inverse semigroups

A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐx is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐx upon an inverse semigroup T is induced by a unique isomorphism of ℱℐx upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzuki's monograph on the subject of subgroup lattices (14)).