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As a continuation of the paper "Adjunction Identity to Hypersemigroup" in Turk J Math 2022; 46 (7): 2834-2853, it has been proved here that the adjunction of a greatest element to an ordered hypersemigroup is actually an embedding problem. The concept of pseudoideal has been introduced and has been proved that for each ordered hypersemigroup S an ordered hypersemigroup V having a greatest element (poe-hypersemigroup) can be constructed in such a way that there exists a pseudoideal T of S such that S is isomorphic to T. If S does not have a greatest element, then this can be regarded as the embedding of an ordered hypersemigroup in an ordered semigroup with greatest element. [ABSTRACT FROM AUTHOR]

This is from Birkhoff, the "father of lattice theory" in Trends in Lattice Theory. Van Nostrand 1970: "Lattices can do things for you, no matter what kind of mathematician you are!". The aim of this paper is to show that the le-semigroups (lattice ordered semigroups possessing a greatest element) play the main role in studying the ordered hypersemigroups. From many results on lattice ordered semigroups corresponding results on ordered semigroups can be obtained. The converse is also possible but the beauty and simplicity of "order" makes it easier to investigate the lattice ordered semigroup at first. After getting the results on ordered semigroups, by an easy modification corresponding results on ordered hypersemigroups can be obtained. [ABSTRACT FROM AUTHOR]

It is shown that some embedding problems on hypersemigroups are actually problems of adjunction. According to the theorem of this paper, for every hypersemigroup S which does not have identity element, an hypersemigroup T having identity element can be constructed in such a way that S is an ideal of T. Moreover, if S is regular, intra-regular, right (left) regular, right (left) quasi-regular or semisimple, then so is T. If A is an ideal, subidempotent bi-ideal or quasi-ideal of S, then it is an ideal, bi-ideal, quasi-ideal of T as well. Illustrative examples are given. [ABSTRACT FROM AUTHOR]

Let T(X) be the full transformation semigroup on the set X . For a fixed nonempty subset Y of X, let PGY (X) = {α ∈ T(X) : α|Y ∈ G(Y )} where G(Y ) is the permutation group on Y . It is known that P GY (X) is a regular subsemigroup of T(X). In this paper, we give a simpler description of Green’s relations and characterize the ideals of P GY (X). Moreover, we prove some isomorphism theorems for PGY (X). For finite sets, we investigate the cardinalities of PGY (X) and of its subsets of idempotents, and we also calculate their ranks. [ABSTRACT FROM AUTHOR]

In this paper we mainly deal with I(q) c -convergence. In particular we study bounded multipliers of bounded I(q) c -convergent sequences. We also give some I-core results and characterize the inclusion K-core {Ax} ⊆ I-core {x} for bounded sequences x = (xn). [ABSTRACT FROM AUTHOR]

As we have already seen in Turkish Journal of Mathematics (2019) 43: 2592--2601 many results on hypersemigroups do not need any proof as they can be obtained from lattice ordered semigroups. The present paper goes a step further, to show that many results on Γ-hypersemigroups as well can be obtained from lattice ordered semigroups. It can be instructive to prove them directly, but even in that case the proofs go along the lines of lattice ordered semigroups (or poe-semigroups). In the investigation, we faced the problem to correct the definition of Γ-hypersemigroups given in the existing bibliography. [ABSTRACT FROM AUTHOR]

In this paper, we consider fuzzy notion of a Γ-near ring, introduce the notion of a fuzzy coset and obtained some related important fundamental isomorphism theorems. [ABSTRACT FROM AUTHOR]

In this paper we define a θ-Euclidean level subset and a θ-Euclidean level ideal. We also give some properties of a θ-Euclidean level subset. [ABSTRACT FROM AUTHOR]