On some medial semigroups with an associate subgroup

Let S be a semigroup and s,t∈S. We say that t is an associate of s if s=sts. If S has a maximal subgroup G such that every element s of S has a unique associate in G, say s ^∗, we say that G is an associate subgroup of S and consider the mapping s→s ^∗ as a unary operation on S. In this way, semigroups with an associate subgroup may be identified with unary semigroups satisfying three simple axioms. Among them, only those satisfying the identity (st)^∗=t ^∗ s ^∗, called medial, have a structure theorem, due to Blyth and Martins.     

On local integrability and transience of convolution semigroups of measures

The analytic proof due to M. Itô of the Kesten-Ornstein transience criterion for continuous convolution semigroups of nonnegative contraction measures on a compactly generated Abelian locally compact group has been reworked and given a self-contained form. The new proof still relies on the existence of the equilibrium measure but dispenses with the complete maximum principle.     

The structure of medial weakly U-abundant semigroups

We consider the class of weakly U-abundant semigroups satisfying the congruence condition (C) containing both the class of regular semigroups and the class of abundant semigroups as its subclasses. The class of weakly U-abundant semigroups with a medial projection satisfying the congruence condition (C) will be particularly studied. This kind of semigroups will be called medial weakly U-abundant semigroups. In this paper, we establish a structure theorem for such semigroups. It is proved that every medial weakly U-abundant semigroup can be expressed by some kind of bands and quasi-Ehresmann semigroups. Our theorem generalizes and enriches the structure theorem given by M. Loganathan in 1987 for regular semigroups with a medial idempotent.     

Approximation of local convoluted semigroups

We extend the Trotter-Kato theorem on C₀-semigroups to local convoluted semigroups on dual spaces and apply these results to the general Banach space setting. Compared to known results we obtain weaker convergence assumptions on the resolvent.     

On Ehresmann semigroups

We formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.

     

On U-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semi-groups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in HU on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup WU and a V-ample semigroup ...     

On exclusive semigroups

The following problem was posed by the second author in ‘Semigroup Forum’, Vol. 1, No. 1, 1970, p. 91: Describe the structure of semigroups S such that for all a, b, c ε S, abc=ab, ac or bc.     

On implemented semigroups

We shall be concerned with implemented semigroups of the form U(t)X := T(t)XS(t) for X ∈ L(F,E) and t ≥ 0 , where (T(t)) _{t ≥ 0} and (S(t)) _{t ≥ 0} are C ₀ -semigroups on the Banach spaces E and F , respectively. The aim of this article is to clarify the structure of this semigroup and its ``generator' with the help of inter- and extrapolation techniques. Moreover, using positivity arguments, we present a short proof of the infinite dimensional Liapunov Stability Theorem on Hilbert spaces as an application of our results. May 15, 2000     

Morita invariants for semigroups with local units

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

Morita invariants for semigroups with local units

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.