Within the usual semidirect product S ∗ T of regular semigroups S and T lies the set Reg (S ∗ T ) of its regular elements. Whenever S or T is completely simple, Reg (S ∗T ) is a (regular) subsemigroup. It is this ‘product’ that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, U and V, the e-variety U...

It is well known that intra-regular semigroups (ordered semigroups) play an essential role in studying the structure of semigroups (ordered semigroups). In this paper we present a structure theorem referring to the decomposition of ordered semigroups into left strongly simple components, that is, into components, which are both simple and left quasi-regular. We prove, among others, that if an o...

Left restriction semigroups have appeared at the convergence of several flows of research, including the theories of abstract semigroups, of partial mappings, of closure operations and even in logic. For instance, they model unary semigroups of partial mappings on a set, where the unary operation takes a map to the identity map on its domain. This perspective leads naturally to dual and two-sid...

For all M,N∈P(U) such that M⊂N, we first introduced the definitions of (M,N)-uni-soft ideals and (M,N)-uni-soft interior ideals of an ordered semigroup and studied them. When M=∅ and N=U, we meet the ordinary soft ones. Then we proved that in regular and in intra-regular ordered semigroups the concept of (M,N)-uni-soft ideals and the (M,N)-uni-soft interior ideals coincide. Finally, we introduc...

We study Schr odinger semigroups in the scale of Sobolev spaces and show that for Kato class potentials the range of such semigroups in Lp has exactly two more derivatives than the potential this proves a conjecture of B Simon We show that eigenfunctions of Schr odinger operators are generically smoother by exactly two derviatives in given Sobolev spaces than their potentials We give applicatio...

We study Schrr odinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in L p has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrr odinger operators are generically smoother by exactly two derviatives (in given Sobolev spaces) than their potentials. We giv...

in this paper we find sufficient conditions on primitive inverse topological semigroup s under which: the inversion inv : (h(s)) (h(s)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed h-classes is topologically isomorphic to an orthogonal sum p i2= bi (gi) of topological brandt extensions bi (gi) of countably compac...

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C∗-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective cor...

In this paper, we introduce generalized adjoint semigroups (GA-semigroups) of a ring R. We construct generalized adjoint semigroups on a ring R by means of bitranslations of R. It is shown that GA-semigroups of a π-regular ring are π-regular. As an application we deduce that in any ring, idempotents can be lifted modulo π-regular ideals. GA-semigroups containing idempotents are described in ter...