Nil extensions of simple regular ordered semigroups

In this paper, nil extensions of simple regular ordered semigroups, left simple and right regular ordered semigroups, etc. have been characterized. Also, we describe the ordered semigroups which are complete semilattices of nil extensions of left simple and right regular ordered semigroups, left group like ordered semigroups, etc.

On Radicals of Green’s Relations in Ordered Semigroups

In this paper, we give a new definition of radicals of Green’s relations in an ordered semigroup and characterize left regular (right regular), intra regular ordered semigroups by radicals of Green’s relations. We also characterize the ordered semigroups that are unions and complete semilattices of t-simple ordered semigroups.

Strong quasi-ideals and complete semilattice decompositions of an ordered semigroup

There are quasi-ideals of an ordered semigroup which cannot be expressed as an intersection of a left ideal and a right ideal. A quasi-ideal with this intersection property is called strong quasi-ideal. We show that strong quasi-simplicity is equivalent to t-simplicity of an ordered semigroup; and hence it turns out to be the case that the ordered semigroups which are complete semilattices (chains) of t-simple subsemigroups can be characterized by their strong quasi-ideals. An ordered semigroup S is complete semilattice (chain) of t-simple subsemigroups if and only if every strong quasi-ideal of S is a completely semiprime (prime) ideal. Also we introduce and characterize the minimal strong quasi-ideals.

Semilattices of Archimedean Ordered Semigroups

In this paper we deal with the decomposition of some classes of ordered semigroups into archimedean components. In particular, we prove that in ordered semigroups the concepts of semilattices of archimedean semigroups and of complete semilattices of archimedean semigroups are the same. In addition, we discuss similarities and differences between semigroups (without order) and ordered semigroups.

Band Congruences of t-Archimedean Ordered Semigroups

Some semigroups without order are decomposable into t-archimedean semigroups via bands. In this paper, we deal with the decomposition of some ordered semigroups into t-archimedean components. The semilattice congruences play an important role in studying the decomposition of semigroups without order. When we pass from semigroups without order to ordered semigroups, the same role is played by the complete semilattice congruences. The characterization of complete semilattices of ordered semigroups of a given type has been considered by the same authors. The r- and l-band congruences have been proved to be useful in studying the decomposition of some types of ordered semigroups, especially the decomposition of r- and l-archimedean ordered semigroups. Band congruences have been proved to be useful in studying the decomposition of some types of ordered semigroups into t-archimedean (ordered) semigroups. The characterization of the bands of ordered semigroups of a given type [Formula: see text] has been recently considered by the same authors. In this paper, we characterize the bands of t-archimedean ordered semigroups. As a result we get a decomposition of some ordered semigroups into t-archimedean components. The decomposition we obtain is uniquely defined.

On the order-theoretical foundation of a theory of quasicompactly generated spaces without separation axiom

A Hausdorff space X is said to be compactly generated (a k-space) if and only if the open subsets U of X are precisely those subsets for which K ∩ U is open in K for all compact subsets of K of X. We interpret this property as a duality property of the lattice O(X) of open sets of X. This view point allows the introduction of the concept of being quasicompactly generated for an arbitrary sober space X. The methods involve the duality theory of up-complete semilattices, and certain inverse limit constructions. In the process, we verify that the new concept agrees with the classical one on Hausdorff spaces.