arXiv subjects
Publications and source records attributed to Niovi Kehayopulu.
This note is written to show that the definition of the ${\cal L}{\cal A}$-hypergroupoids in [5] should be corrected and that it is not enough to replace the multiplication "$\cdot$" of an ${\cal L}{\cal A}$-groupoid by the hyperoperation "$\circ$" to pass from an ${\cal L}{\cal A}$-groupoid to an ${\cal L}{\cal A}$-hypergroupoid.
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.
The concept of a $Γ$-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 $Γ$-semigroups techniques to the Green's Theorem in an attempt to show the way we pass from semigroups to $Γ$-semigroups.
We wrote this paper as an example to show the way we pass from ordered semigroups to ordered hypersemigroups.
The aim of writing this paper is given in the title. The results on semigroups can be easily transferred to hyper-semigroups in the way indicated in the present paper.
This paper serves as an example to show the way we pass from semigroups to $Γ$-semigroups and to hypersemigroups.
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.
We prove that an hypersemigroup $H$ is regular if and only, for any fuzzy subset $f$ of $H$, we have $f\preceq f\circ 1\circ f$ and it is intra-regular if and only if, for any fuzzy subset $f$ of $H$, we have $f\preceq 1\circ f\circ f\circ 1$. An hypersemigroup $H$ is left (resp. right) quasi-regular if and only if, for any fuzzy subset $f$ of $S$ we have $f\preceq 1\circ f\circ 1\circ f$ (resp. $f\preceq f\circ 1\circ f\circ 1)$ and it is semisimple if and only if, for any fuzzy subset $f$ of $S$ we have $f\preceq 1\circ f\circ 1\circ f\circ 1$. The characterization of regular and intra-regular hypersemigroups in terms of fuzzy subsets are very useful for applications.
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).
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 $H$.
We keep the definition of intra-regularity (left regularity) of $po$-$Γ$-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 $po$-$Γ$-semigroup into simple components. Then we prove that a $po$-$Γ$-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 $po$-$Γ$-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 $po$-$Γ$-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 $po$-$Γ$-semigroup into left simple components has been also given.
We prove that a nonempty subset $B$ of a regular hypersemigroup $H$ is a bi-ideal of $H$ if and only if it is represented in the form $B=A*C$ where $A$ is a right ideal and $C$ a left ideal of $H$. We also show that an hypersemigroup $H$ is regular if and only if the right and the left ideals of $H$ are idempotent, and for every right ideal $A$ and every left ideal $B$ of $H$, the product $A*B$ is a quasi-ideal of $H$. Our aim 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. We never work directly on an hypersemigroup. If we want to get a result on an hypersemigroup, then we have to prove it first for a semigroup and transfer its proof to hypersemigroup. But there is further interesting information concerning this structure as well, we will deal with at another time.
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.
This is about the paper by Thawhat Changphas and Nawamin Phaipong in Quasigroups and Related Systems 22 (2014), 193--200.
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.
We characterize the fuzzy left (resp. right) ideals, the fuzzy ideals and the fuzzy prime (resp. semiprime) ideals of an ordered $Γ$-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 $μ$ and $σ$ are fuzzy left (resp. right) ideals, ideals, fuzzy prime or fuzzy semiprime ideals of $M$, then the nonempty level subsets $(μ\timesσ)_t$ are so.
The results on fuzzy ordered semigroups (or on fuzzy semigroups) can be transferred to fuzzy ordered gamma (or to fuzzy gamma) semigroups. We show the way we pass from fuzzy ordered semigroups to fuzzy ordered gamma semigroups.