Quotient semigroups and extension semigroups

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of C ^*-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.     

Invariant semigroups of orthodox semigroups

We consider, in a right inverse semigroupS with a multiplicative inverse transversalS ^o, the notion of anS ^o-invariant subsemigroup and use this to describe all the left amenable orders definable onS. The results obtained, together with their duals, are used to prove that ifS is an orthodox semigroup with a multiplicative inverse transversalS ^o, then every amenable order onS ^o can be extended to a unique amenable order onS. NATO Collaborative Research Grant 910765 is gratefully acknowledged. The second-named author also gratefully acknowledges support from the Calouste Gulbenkian Foundation, Lisbon.     

Preaffine semigroups and convex matrix semigroups

Algebraic conditions are given which guarantee that a semigroup on a subset of real topological vector space can be embedded in a convex matrix semigroup. We also study when the minimal ideal of a convex matrix semigroup will be convex.     

Preaffine semigroups and convex matrix semigroups

Algebraic conditions are given which guarantee that a semigroup on a subset of real topological vector space can be embedded in a convex matrix semigroup. We also study when the minimal ideal of a convex matrix semigroup will be convex.     

Embedding inverse semigroups in coset semigroups

The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G). This research was supported by a grant from the National Science Foundation.     

Inverse semigroups and varieties of finite semigroups

In this paper we shall present a new construction of the free inverse monoid on a set X. Contrary to the previous constructions of [9, 11], our construction is symmetric and originates from classical ideas of language theory. The ingredients of this construction are the free group on X and the relation that associates to a word w of the free monoid on X, the set of all pairs (u, v) such that uv = w. It follows at once from our construction that the free inverse monoid on X can be naturally embedded into the Schützenberger product of two free groups of basis X. We shall also give some connections with the theory of expansions as developed by Rhodes and Birget [2, 3].     

Embedding locally inverse semigroups into Rees matrix semigroups

We give a necessary and sufficient condition for a locally inverse semigroup to be embeddable into a Rees matrix semigroup over a generalized inverse semigroup.     

Categorical semigroups

This investigation was stimulated by a question raised by F.R. McMorris and M. Satyanarayana [Proc. Amer. Math. Soc. 33 (1972), 271–277] which asked whether a regular semigroup with a tree of idempotents is categorical. The question is answered in the affirmative. Characterizations of categorical semigroups are found within the following classes of semigroups: regular semigroups, bands, commutative regular semigroups, unions of simple semigroups, semilattices of groups, and commutative semigroups. Some results are related to part of the work of M. Petrich [Trans. Amer. Math. Soc. 170 (1972), 245–268]. For instance, it is shown that the poset of J-classes of any regular categorical semigroup is a tree; however, an example of a regular non-categorical semigroup is given in which the poset of J-classes is a chain. It is also shown that the condition that the subsemigroup of idempotents be categorical is sufficient, but not necessary, for an orthodox semigroup to be categorical.     

Stratified semigroups

Stratified semigroups include free semigroups, finite nilsemigroups, and homogeneous semigroups (=with homogeneous presentations). Basic properties of this class of semigroups are given, including the role that certain combinatorial structures (homogeneous equivalence relations) play in their construction. After this article was typeset the author discovered that stratified semigroups were also considered by Sasaki and Tamura (Proc. Japan. Acad. Ser. A Math. Phys.63 (1987), 315–317), who called themboundly factorizable.     

Quasi-hyperbolic semigroups

We introduce the notion of quasi-hyperbolic operators and C₀-semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C₀-semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property.     

IdeallyF-decomposable semigroups

We consider ideallyy -decomposable semigroups (y — is a system of semigroup identities). It is proved that every ideallyy -decomposable semigroup has ay -decomposable series of ideals.Translated from Matematicheskie Zametki, Vol. 8, No, 6, pp. 681–691, December, 1970.