arXiv subjects
Publications and source records attributed to Niovi Kehayopulu.
The aim is to correct part of the Remark 3 of my paper "On regular, intra-regular ordered semigroups" in Pure Math. Appl. (PU.M.A.) 4, no. 4 (1993), 447--461. On this occasion, some further results and the similarity between the $po$-semigroups and the $le$-semigroups is discussed.
This is about the paper in the title by Kostaq Hila in Rocky Mt. J. Math. 41, no. 1 (2011), 189-203 for which corrections should be done.
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.
We add here some further characterizations to the characterizations of strongly regular ordered $Γ$-semigroups already considered in Hacettepe J. Math. 42 (2013), 559--567. Our results generalize the characterizations of strongly regular ordered semigroups given in the Theorem in Math. Japon. 48 (1998), 213--215, in case of ordered $Γ$-semigroups. The aim of writing this paper is not just to add a publication in $Γ$-semigroups but, mainly, as a continuation of the paper "On regular duo $po$-$Γ$-semigroups in Math. Slovaca 61 (2011), 871--884, to publish a paper which serves as an example to show what a $Γ$-semigroup is and give the right information about this structure.
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.
This is about the paper in the title published by K. Hila and J. Dine in Acta Math. Sinica, English Series, 29, no. 3, (2013), 609--624.
Our aim is to show the way we pass from the results of ordered semigroups (or semigroups) to ordered $Γ$-semigroups (or $Γ$-semigroups). The results of this note have been transferred from ordered semigroups. The concept of strongly regular $po$-$Γ$-semigroups has been first introduced here and a characterization of strongly regular $po$-$Γ$-semigroups is given.