Fuzzy right (left) ideals in hypergroupoids and fuzzy bi-ideals in hypersemigroups

We introduce the concepts of fuzzy right and fuzzy left ideals of hypergroupoids and the concept of a fuzzy bi-ideal of an hypersemigroup and we show that a fuzzy subset $f$ of an hypergroupoid $H$ is a fuzzy right (resp. fuzzy left) ideal of $H$ if and only if $f\circ 1\preceq f$ (resp. $1\circ f\preceq f)$ and for an hypersemigroup $H$, a fuzzy subset $f$ of $H$ is a bi-ideal of $H$ if and only if $f\circ 1\circ f\preceq f$. These characterizations are very useful for the investigation. The paper serves as an example to show the way we pass from fuzzy groupoids (semigroups) to fuzzy hypergroupoids (hypersemigroups)

Remark on quasi-ideals of ordered semigroups

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

Comment "On dual ordered semigroups"

This is about the paper by Thawhat Changphas and Nawamin Phaipong in Quasigroups and Related Systems 22 (2014), 193--200

On fuzzy prime and fuzzy semiprime ideals of ≤\le-hypergroupoids

We deal with an hypergroupoid endowed with a relation denoted by "$\le$", we call it $\le$--hypergroupoid. We prove that a nonempty subset $A$ of a $\le$--hypergroupoid $H$ is a prime (resp. semiprime) ideal of $H$ if and only if its characteristic function $f_A$ is a fuzzy prime (resp. fuzzy semiprime) ideal of $H$

An application of Γ\Gamma-semigroups techniques to the Green's Theorem

The concept of a $\Gamma$-semigroup has been introduced by Mridul Kanti Sen in the Int. Symp., New Delhi, 1981. It is well known that the Green's relations play an essential role in studying the structure of semigroups. In the present paper we deal with an application of $\Gamma$-semigroups techniques to the Green's Theorem in an attempt to show the way we pass from semigroups to $\Gamma$-semigroups

On involution lele-semigroups

We deal with involution ordered semigroups possessing a greatest element, we introduce the concepts of $*$-regularity, $*$-intra-regularity, $*$-bi-ideal element and $*$-quasi-ideal element in this type of semigroups and, using the right and left ideal elements, we give relations between the regularity and $*$-regularity, between intra-regularity and $*$-intra-regularity. Finally, we prove that in an involution $*$-regular $\vee e$-semigroup every $*$-bi-ideal element can be considered as a product of a right and a left ideal element, we describe the form of the filter generated by an element of an involution $*$-intra-regular $poe$-semigroup $S$, showing that every $\cal N$-class of $S$ has a greatest element

Decomposition of intra-regular popo-Γ\Gamma-semigroups into simple components

We keep the definition of intra-regularity (left regularity) of $po$-$\Gamma$-semigroups introduced in arXiv: 1511.00679 which is absolutely necessary for the investigation. Being able to describe the form of the elements of the principal filter by using this definition, we study the decomposition of an intra-regular $po$-$\Gamma$-semigroup into simple components. Then we prove that a $po$-$\Gamma$-semigroup $M$ is intra-regular and the ideals of $M$ form a chain if and only if $M$ is a chain of simple semigroups. Moreover, a $po$-$\Gamma$-semigroup $M$ is intra-regular and the ideals of $M$ form a chain if and only if the ideals of $M$ are prime. Finally, for an intra-regular $po$-$\Gamma$-semigroup $M$, the set $\{(x)_{\cal N} \mid x\in M\}$ coincides with the set of all maximal simple subsemigroups of $M$. A decomposition of left regular and left duo $po$-$\Gamma$-semigroup into left simple components has been also given

On Fuzzy Ideals and Level Subsets of Ordered Γ\Gamma-Groupoids

We characterize the fuzzy left (resp. right) ideals, the fuzzy ideals and the fuzzy prime (resp. semiprime) ideals of an ordered $\Gamma$-groupoid $M$ in terms of level subsets and we prove that the cartesian product of two fuzzy left (resp. right) ideals of $M$ is a fuzzy left (resp. right) ideal of $M\times M$, and the cartesian product of two fuzzy prime (resp. semiprime) ideals of $M$ is a fuzzy prime (resp. semiprime) ideal of $M\times M$. As a result, if $\mu$ and $\sigma$ are fuzzy left (resp. right) ideals, ideals, fuzzy prime or fuzzy semiprime ideals of $M$, then the nonempty level subsets $(\mu\times\sigma)_t$ are so

Comment on "Filtres in ordered Γ\Gamma-semigroups"

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

Fuzzy sets in ≤\le-hypergroupoids

This paper serves as an example to show the way we pass from ordered groupoids (ordered semigroups) to ordered hypergroupoids (ordered hypersemigroups), from groupoids (semigroups) to hypergroupoids (hypersemigroups). The results on semigroups (or on ordered semigroup) can be transferred to hypersemigroups (or to ordered hypersemigroups) in the way indicated in the present paper