Analysis of semigroups with soft intersection ideals

We give some properties of intuitionistc fuzzy left, right, and two-sided ideals and bi-ideals of a semigroup. And we characterize a regular semigroup, a semi group that is a lattice of left(right) simple semi groups, a semi group that is a semi lattice of left(right) groups and a semi group that is a semi lattice of groups in terms of intuitionistic fuzzy ideals and intuitionistic fuzzy bi-ideals.

Generalizing the notions of -fuzzy left (right) ideal, -fuzzy quasi-ideal, and -fuzzy bi-ideal, the notions of -fuzzy left (right) ideal, -fuzzy quasi-ideal and -fuzzy bi-ideal of semigroups are defined. Regular, intra regular and semisimple semigroups are characterized by the properties of these fuzzy ideals.

The algebraic theory of inverse semigroups has received considerable attention in recent years with satisfactory structure theorems being obtained for extensive classes of these semigroups. As is the case with so much of semigroup theory, many of these structure theorems were obtained by, or had their origin in fundamental papers of, A. H. Clifford. For example, the structure of inverse semigroups with central idempotents (semilattices of groups) was obtained by Clifford in [1], that of Brandt semigroups was obtained in [2], while, in [3], Clifford described bisimple inverse semigroups in terms of their right unit subsemigroup. The third of these papers has been the basis for much of the research on structural aspects of inverse semigroups. On the one hand, its direct descendants include Reilly's papers on bi-simple inverse semigroups in terms of RP-systems [10], [11], and the author's description of O-bisimple inverse semigroups in terms of groups and semilattices [5]. On the other hand, it has led to Reilly's description [9] of w-bisimple inverse semigroups and Warne's characterisations of various classes of bisimple inverse semigroups with special semilattices of idempotents. These, in turn, have led to the determination of simple inverse semigroups with special semilattices of idempotents.

Generalizing the ideals of an algebraic structure has shown to be both beneficial and interesting for mathematicians. In this context, the idea of the bi-interior ideal was introduced as a generalization of the bi-ideal and interior ideal of a semigroup. By introducing "soft intersection (₷-int) bi-interior (ᙝᏆ) ideals of semigroups", we introduce a framework integrating semigroup theory with soft set theory in this study. Finding the relationships between ₷-int ᙝᏆ-ideals and other specific kinds of ₷-int ideals of a semigroup is the main aim of this study. Our results show that an ₷-int ᙝᏆ-ideal is an ₷-int subsemigroup of a soft simple* semigroup, and that an ₷-int left (right/two-sided) ideal, bi-ideal, interior ideal and quasi-ideal is an ₷-int ᙝᏆ-ideal; in other words, the ₷-int ᙝᏆ-ideal is a generalization of the ₷-int left (right/two-sided) ideal, bi-ideal, interior ideal and quasi-ideal, however, we provide counterexamples demonstrating that the converses do not always hold. We demonstrate that the semigroup should be a soft simple* semigroup in order to satisfy the converses. Our key theorem, which states that if a nonempty subset of a semigroup is a ᙝᏆ-ideal, then its soft characteristic function is an ₷-int ᙝᏆ-ideal, and vice versa, enables us to bridge the gap between semigroup theory and soft set theory. Using this theorem, we show how this idea relates to the existing algebraic structures in classical semigroup theory. Furthermore, we present conceptual characterizations and analysis of the new concept in terms of soft set operations supporting our assertions with illuminating examples.

The combinatorial simple principal ideal semigroups generated by two elements were described by L. Megyesi and G. Pollak. The ‘most general’ among them is called the Redei semigroup. The ‘most special’ combinatorial simple principal ideal semigroup generated by two elements is the bicyclic semigroup. D. B. McAlister determined the compatible semilattice orders on the bicyclic semigroup. Our aim is to study the compatible semilattice orders on the homomorphic images of the Redei semigroup. We prove that there are four compatible total orders on these semigroups. We show that on the Redei semigroup, the total orders are the only compatible semilattice orders. Moreover, on each proper homomorphic image of the Redei semigroup, we give a compatible semilattice order which is not a total order.

Intersections of Maximal Ideals in Semigroups

In this paper, we study the structural properties of a non-associative algebraic structure called an AG-groupoid by using soft set theory. We characterize a right regular class of an AG-groupoid in terms of soft intersection ideals and provide counter examples to discuss the converse part of various problems. We also characterize a weakly regular class of an AG***-groupoid by using generated ideals and soft intersection ideals. We investigate the relationship between SI-left-ideal, SI-right-ideal, SI-two-sided-ideal, and SI-interior-ideal of an AG-groupoid over a universe set by providing some practical examples.