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A poe-semigroup is a semigroup S at the same time an ordered set having a greatest element “e” in which the multiplication is compatible with the ordering. A ∨e-semigroup is a semigroup S at the same time an upper semilattice with a greatest element “e” such that a(b ∨ c) = ab ∨ ac and (a ∨ b)c = ac ∨ bc for every a, b, c ∈ S. If S is not only an upper semi-lattice but a lattice, then it is called le-semigroup. From many results on le-semigroups, ∨e-semigroups or poe-semigroups, corresponding results on ordered semigroups (without greatest element) can be obtained. Related results on hypersemigroups or ordered hypersemigroups follow as application. An example is presented in the present note; the same can be said for every result on these structures. So order-lattices play an essential role in studying the hypersemigroups and the ordered hypersemigroups.

All the results on semigroups obtained using only sets, can be written in an abstract form in a more general setting. Let us consider a recent paper to justify what we say. The bi-interior ideals of semigroups introduced and studied by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. in 2018, follow for more general statements about ordered semigroups. The same holds for every result of this sort on semigroups based on right (left) ideals, bi-ideals, quasi-ideals, interior ideals etc. for which we use sets. As a result, we have an abstract formulation of the results on semigroups obtained by sets that is in the same spirit with the abstract formulation of general topology (the so-called topology without points) initiated by Koutsk ́y, Nöbeling and, even earlier, by Chittenden, Terasaka, Nakamura, Monteiro and Ribeiro. As a consequence, results on ordered Γ-hypersemigroups and on similar simpler structures can be obtained.

A $$poe$$ -semigroup $$(S,\cdot,\leq)$$ equipped with an order preserving involution ‘‘ $$*$$ ’’ is called an involution ordered semigroup. An $$le$$ -semigroup is a lattice ordered semigroup possessing a greatest element. The concepts of $$*$$ -regularity, $$*$$ -intra-regularity, $$*$$ -bi-ideal element and $$*$$ -quasi-ideal element in this type of semigroups have been introduced and, using the right and left ideal elements, relationships among them are given. The $$*$$ -regular (resp. $$*$$ -intra-regular) $$poe$$ -semigroups are regular (resp. intra-regular). It is shown, among others, that in an involution $$*$$ -regular $$\vee e$$ -semigroup every $$*$$ -bi-ideal element can be represented as the product of a right and a left ideal element. Necessary and sufficient conditions are given under which an involution $$le$$ -semigroup is $$*$$ -regular, regular, $$*$$ -intra-regular or intra-regular. An involution $$poe$$ -semigroup $$S$$ is $$*$$ -intra-regular if and only if the ideal elements of $$S$$ are $$*$$ -semiprime; and an analogous result for involution $$*$$ -left or $$*$$ -right $$poe$$ -semigroups also holds. It is also shown that in an involution $$*$$ -intra-regular $$poe$$ -semigroup $$S$$ , the filter generated by an element $$x$$ of $$S$$ has a very simple form that is useful for the investigation. As a consequence, in this type of semigroups the element $$ex^{*}e$$ is contained in the $$(x^{*})_{\mathcal{N}}$$ -class (for any $$x$$ ) and it is the greatest element of the class $$(x)_{\mathcal{N}}$$ .

It is shown that an hypersemigroup (S, ◦) is regular if and only if the set of all quasi-ideals of S with the operation “∗” is a von Neumann regular semigroup. It is both regular and intra-regular if and only if the set of all quasi-ideals of S with the operation “∗” is a band.

This is about the paper “Bi-interior ideals of semigroups” by M. Murali Krishna Rao in Discuss. Math. Gen. Algebra Appl. 38 (2018) 69–78. According to Theorem 3.11 (also Theorem 3.3(8)) of the paper, the intersection of a bi-interior ideal [Formula: see text] of a semigroup [Formula: see text] and a subsemigroup [Formula: see text] of [Formula: see text] is a bi-interior ideal of [Formula: see text]. Regarding to Theorem 3.6, every bi-interior ideal of a regular semigroup is an ideal of [Formula: see text]. We give an example that the above two results are not true for semigroups. According to the same paper, if [Formula: see text] is a regular semigroup then, for every bi-interior ideal [Formula: see text], every ideal [Formula: see text] and every left ideal [Formula: see text] of [Formula: see text], we have [Formula: see text]. The proof is wrong, we provide the corrected proof. In most of the results of the paper the assumption of unity is not necessary. Care should be taken about the proofs in the paper.

Giving the proper citations, it is shown that, except of Lemma 2.4 and Theorem 2.6, almost all the results of the paper in the title have been previously published for ordered hypersemigroups in Eur. J. Pure Appl. Math. and they are not new. There are also two results obtained from ordered semigroups just putting a ``$\circ$" instead of ``$\cdot$" (that isn't a correct way to work), without reference to ordered semigroups on which the results on ordered hypersemigroups are based. One of them can be obtained as corollary to a theorem in Eur. J. Pure Appl. Math. as well, and it is not new.

The concept of a fuzzy set, introduced by Zadeh, was first applied by Rosenfeld to groups, then by Kuroki to semigroups. In the present paper it is applied to hypersemigroups, and some properties of fuzzy hypersemigroups are discussed.

We wrote this note to show that the definition of Γ-hypersemigroups in [2] should be corrected, and that it is not enough to replace the hyperoperation ◦ of the hypersemigroup by Γ to pass from a hypersemigroup to a Γ-hypersemigroup. Care should be taken about the definitions of m, n -quasi-Γ-hyperideal, the m-left Γ-hyperideal and the n-right Γ-hyperideal as well.

An $le$-semigroup, is a semigroup $S$ at the same time a lattice with a greatest element $e$ ($e\ge a$ for every $a\in S$) such that $a(b\vee c)=ab\vee ac$ and $(a\vee b)c=ac\vee bc$ for all $a,b,c\in S$. If $S$ is not a lattice but only an upper semilattice ($\vee$-semilattice), then is called $\vee e$-semigroup. A $poe$-semigroup is a semigroup $S$ at the same time an ordered set with a greatest element $e$ such that $a\le b$ implies $ac\le bc$ and $ca\le cb$ for all $c\in S$. Every $\vee e$-semigroup is a $poe$-semigroup. If $S$ is a semigroup or a $\Gamma$-semigroup, then the set ${\cal P}(S)$ of all subsets of $S$ is an $le$-semigroup. If $S$ is an hypersemigroup, then the set ${\cal P^*}(S)$ of all nonempty subsets of $S$ is an $le$-semigroup. So all the results of $le$-semigroups, $\vee e$-semigroups and $poe$-semigroups based on ideal elements, automatically hold for semigroups, $\Gamma$-semigroups and hypersemigroups. This is not the case for ordered $\Gamma$-semigroups or ordered hypersemigroups; however the main idea, even in these cases, comes from the $le$ ($\vee e$)-semigroups. As an example, we study the weakly prime ideal elements of a $\vee e$-semigroup and their role to the different type of semigroups mentioned above.

For a proper subset $A$ of an ordered semigroup $S$, we denote by $H_A(S)$ the subset of $S$ defined by $H_A(S):=\{h\in S \mbox { such that if } s\in S\backslash A, \mbox { then } s\notin (shS]\}$. We prove, among others, that if $A$ is a right ideal of $S$ and the set $H_A(S)$ is nonempty, then $H_A(S)$ is an ideal of $S$; in particular it is a semiprime ideal of $S$. Moreover, if $A$ is an ideal of $S$, then $A\subseteq H_A(S)$. Finally, we prove that if $A$ and $I$ are right ideals of $S$, then $I\subseteq H_A(S)$ if and only if $s\notin (sI]$ for every $s\in S\backslash A$. We give some examples that illustrate our results. Our results generalize the Theorem 2.4 in Semigroup Forum 96 (2018), 523--535.

We give some information concerning the Green's relations $\cal R$ and $\cal L$ in hypergroupoids extending the concepts of right (left) consistent or intra-consistent groupoids in case of hypergroupoids. We prove, for example, that if an hypergroupoid $H$ is right (left) consistent or intra-consistent, then the Green's relations $\cal R$ and $\cal L$ are equivalence relations on $H$ and give some conditions under which in consistent commutative hypergroupoids the relation $\cal R$ (= $\cal L$) is a semilattice congruence. A commutative hypergroupoid is right consistent if and only if it is left consistent and if an hypergroupoid is commutative and right (left) consistent, then it is intra-consistent. A characterization of right (left) consistent (or intra-consistent) right (left) simple hypergroupoids has been also given. Illustrative examples are given.

This note defines the left magnifying and the strongly left magnifying elements in an ordered groupoid and discusses their properties. It is shown that in an ordered semigroup, every left magnifying element is of infinite order. The concept of factorizable ordered semigroups has been also introduced and, using the infinite order property, it is shown that every ordered semigroup having a strongly left magnifying element is factorizable.

We prove that if $H$ is an hypersemigroup (resp. ordered hypersemigroup) and $\sigma$ is a semilattice congruence (resp. complete semilattice congruence) on $H$, then there exists a family $\cal A$ of proper prime ideals of $H$ such that $\sigma$ is the intersection of the semilattice congruences $\sigma_I$, $I\in\cal A$ ($\sigma_I$ is the known relation defined by $a\sigma_I b$ $\Leftrightarrow$ $a,b\in I$ or $a,b\notin I$). Furthermore, we study the relation between the semilattices of an ordered semigroup and the ordered hypersemigroup derived by the hyperoperations $a\circ b=\{ab\}$ and $a\circ b:=\{t\in S \mid t\le ab\}$. We introduce the concept of a pseudocomplete semilattice congruence as a semilattice congruence $\sigma$ for which $\le\subseteq\sigma$ and we prove, among others, that if $(S,\cdot,\le)$ is an ordered semigroup, $(S,\circ,\le)$ the hypersemigroup defined by $t\in a\circ b$ if and only if $t\le ab$ and $\sigma$ is a pseudocomplete semilattice congruence on $(S,\cdot,\le)$, then it is a complete semilattice congruence on $(S,\circ,\le)$. Illustrative examples are given.

Some well known results on ordered semigroups are examined in case of ordered hypersemigroups. Following the paper in Semigroup Forum 44 (1992), 341--346, we prove the following: The ideals of an ordered hypergroupoid$H$ are idempotent if and only if for any two ideals $A$ and $B$ of $H$, we have $A\cap B=(A*B]$. Let now $H$ be an ordered hypersemigroup. Then, the ideals of $H$ are idempotent if and only if $H$ is semisimple. The ideals of $H$ are weakly prime if and only if they are idempotent and they form a chain. The ideals of $H$ are prime if and only if they form a chain and $H$ is intra-regular. The paper serves as an example to show how we pass from ordered semigroups to ordered hypersemigroups.

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.

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., arXiv admin note: text overlap with arXiv:1606.00428

The aim is to show that from every example of a regular, intraregular, left (right) regular, left (right) quasiregular, semisimple, left (right) simple, simple, or strongly simple ordered semigroup given by a table of multiplication and an order, a corresponding example of regular, intraregular, left (right) regular, left (right) quasiregular, semisimple, left (right) simple, simple, or strongly simple ordered hypersemigroup can be constructed having the same left (right) ideals, bi-ideals, quasi-ideals, or interior ideals. On this occasion, some further related results have also been given.

This note is written to show that the definition of the LA- Γ -hypersemi-group and the definition of the ordered LA- Γ -hypersemigroup in [2] should be corrected and that it is not enough to replace the “Γ” of the ordered LA-Γ-semigroup by “◦ Γ ◦” to pass from an ordered LA- Γ -semigroup to an ordered LA- Γ -hypersemigroup. The definition of the (m,n)- Γ -hyperideal and the strange symbols used throughout the paper have no sense as well.

We characterize the set of regular elements of an ordered semigroup S which is a left (right) ideal, an ideal, a bi-ideal or a quasi-ideal of S using the left (resp. right) ideal and the quasi-ideal LE(S)∩RE(S) of S, where E(S) is the set of elements of S for which e ≤ e2. Illustrative example is given.

Two well known results on semigroups, one of them clue to Calais, the other to Lajos, are generalized in case of hypersemigroups in the way indicated in the present paper. 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 hyper semigroup 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.

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. © 2018, Pleiades Publishing, Ltd.

We characterize the ordered semigroups that are archimedean and contain an intra-regular element, showing that they are exactly nil extensions of simple ordered semigroups, also the ordered semigroups that are both archimedean and $$\pi $$ -semisimple or both archimedean and $$\pi $$ -quasi-semisimple.

We prove that if an ordered semigroup is a nil extension of a left strongly simple ordered semigroup, then it is left strongly archimedean, but, in contrast to the unordered case, the converse does not hold in general. However, a left strongly archimedean ordered semigroup is a nil extension of a simple ordered semigroup.

We study the decomposition of left regular ordered semigroups into left regular components and the decomposition of intra-regular ordered semigroups into simple or intra-regular components, adding some additional information to the results considered in [KEHAYOPULU, N.: On left regular ordered semigroups, Math. Japon. 35 (1990), 1057–1060] and [KEHAYOPULU, N.: On intra-regular ordered semigroups, Semigroup Forum 46 (1993), 271–278]. We prove that an ordered semigroup S is left regular if and only if it is a semilattice (or a complete semilattice) of left regular semigroups, equivalently, it is a union of left regular subsemigroups of S. Moreover, S is left regular if and only if it is a union of pairwise disjoint left regular subsemigroups of S. The right analog also holds. The same result is true if we replace the words “left regular” by “intraregular”. Moreover, an ordered semigroup is intra-regular if and only if it is a semilattice (or a complete semilattice) of simple semigroups. On the other hand, if an ordered semigroup is a semilattice (or a complete semilattice) of left simple semigroups, then it is left regular, but the converse statement does not hold in general. Illustrative examples are given.

In an attempt to show the way we pass from ordered semigroups to ordered hypersemigroups, we examine some well known results of regular and intraregular ordered semigroups in case of ordered hypersemigroups. The corresponding results on hypersemigroups (without order) can also be obtained as application of the results of the present paper. The sets we use in our investigation shows the pointless character of the results.

We characterize the ordered semigroups which are decomposable into simple and regular components. We prove that each ordered semigroup which is both regular and intra-regular is decomposable into simple and regular semigroups, and the converse statement also holds. We also prove that an ordered semigroup S is both regular and intra-regular if and only if every bi-ideal of S is an intra-regular (resp. semisimple) subsemigroup of S. An ordered semigroup S is both regular and intra-regular if and only if the left (resp. right) ideals of S are right (resp. left) quasi-regular subsemigroups of S. We characterize the chains of simple and regular semigroups, and we prove that S is a complete semilattice of simple and regular semigroups if and only if S is a semilattice of simple and regular semigroups. While a semigroup which is both π-regular and intra-regular is a semilattice of simple and regular semigroups, this does not hold in ordered semigroups, in general.