Some completely semisimple HNN-extensions of inverse semigroups

We give necessary and sufficient conditions in order that lower bounded HNN-extensions of inverse semigroups and HNN-extensions of finite inverse semigroups are completely semisimple semigroups. Since it is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup, we first characterize such HNN-extensions containing a bicyclic subsemigroup making use of the special feature of their Schützenberger automata.

On completely semisimple Lie algebra bundles

We study semisimple, completely semisimple Lie algebra bundles in terms of characteristic ideal bundles following Seligman. Further decomposition theorem for Lie algebra bundles over any field is proved following Dieudonné.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

Trace functions on inverse semigroup algebras

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

A note on shortly connected inverse semigroups

Shortly connected and shortly linked inverse semigroups arise in the study of inverse semigroups determined by the lattices of inverse subsemigroups and by the partial automorphism semigroups. It has been shown that every shortly linked inverse semigroup is shortly connected, but the question of whether the converse is true has not been addressed. Here we construct two examples of combinatorial shortly connected inverse semigroups which are not shortly linked. One of them is completely semisimple while the other is not.

Weakly reductive semigroups with atomistic congruence lattices

The structure of semigroups with atomistic congruence lattices (that is, each congruence is the supremum of the atoms it contains) is studied. For the weakly reductive case the problem of describing the structure of such semigroups is solved up to simple and congruence free semigroups, respectively. As applications, all commutative, finite, completely semisimple semigroups, respectively, with atomistic congruence lattices are described.

Inverse semigroups with certain types of partial automorphism monoids

For an inverse semigroup S, the set of all isomorphisms betweeninverse subsemigroups of S is an inverse monoid under composition which is denoted by (S) and called the partial automorphism monoid of S. Kirkwood [7] and Libih [8] determined which groups have Clifford partial automorphism monoids. Here we investigate the structure of inverse semigroups whose partial automorphism monoids belong to certain other important classes of inverse semigroups. First of all, we describe (modulo so called “exceptional” groups) all inverse semigroups S such that (S) is completely semisimple. Secondly, for an inverse semigroup S, we find a convenient description of the greatest idempotent-separating congruence on (S), using a well-known general expression for this congruence due to Howie, and describe all those inverse semigroups whose partial automorphism monoids are fundamental.

The lattice of full regular subsemigroups of a regular semigroup

SynopsisAlthough the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S.A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.