arXiv subjects
Publications and source records attributed to Michael Tsingelis.
We characterize the intra-regular $Γ$-semigroups and the left regular $Γ$-semigroups $M$ in which $xΓM\subseteq MΓx$ for every $x\in M$ in terms of filters and we prove, among others, that every intra-regular $Γ$-semigroup is decomposable into simple components, and every $Γ$-semigroup $M$ for which $xΓM\subseteq MΓx$ is left regular, is decomposable into left simple components.
For an intra-regular or a left regular and left duo ordered $Γ$-semigroup $M$, we describe the principal filter of $M$ which plays an essential role in the structure of this type of $po$-$Γ$-semigroups. We also prove that an ordered $Γ$-semigroup $M$ is intra-regular if and only if the ideals of $M$ are semiprime and it is left (right) regular and left (right) duo if and only if the left (right) ideals of $M$ are semiprime.
In this paper we show the way we pass from semigroups (without order) to hypersemigroups. Moreover we show that, exactly as in semigroups, in the results of hypersemigroups based on right (left) ideals, quasi-ideals and bi-ideals, points do not play any essential role, but the sets, which shows their pointless character. The aim of writing this paper is not just to add a publication on hypersemigroups but, mainly, to publish a paper which serves as an example to show what an hypersemigroup is and give the right information concerning this structure.
A fuzzy subset $f$ of an ordered semigroup (or semigroup) $S$ is called fuzzy semiprime if $f(x)\ge f(x^2)$ for every $x\in S$ (Definition 1). Following the terminology of semiprime subsets of ordered semigroups (semigroups), the terminology of ideal elements of $poe$-semigroups (: ordered semigroups possessing a greatest element), and the terminology of ordered semigroups, in general, a fuzzy subset $f$ of an ordered semigroups (semigroup) should be called fuzzy semiprime if for every fuzzy subset $g$ of $S$ such that $g^2:=g\circ g\preceq f$, we have $g\preceq f$ (Definition 2). And this is because if $S$ is a semigroup or ordered semigroup, then the set of all fuzzy subsets of $S$ is a semigroup (ordered semigroup) as well. What is the relation between these two definitions? that is between the usual definition (Definition 1) we always use and the definition we give in the present paper (Definition 2) saying that that definition should actually be the correct one? The present paper gives the related answer.
Concerning the paper in the title by K. Hila and E. Pisha in Commun. Korean Math. Soc. Volume 26, Issue 3 (2011), 373--384, we give our results and make the main corrections.