Characterization of \((m,n)\)-high-ideals of posemigroups when they are \(0^{\ast}\)-minimal

M. M. Shamivand, H. Doostie, H. Rasouli

Abstract


A semigroup \(S\) is called a posemigroup if \(S\) is equipped with a partial ordering relation ``\(\leq\)'' such that \(a \leq b\) in \(S\) implies \(xa \leq xb\) and \(ax \leq bx\), for all \(x\in S\). In this paper, we define an equivalence relation \(\textbf{B}^{\ast}\) on a posemigroup \(S\) and introduce the concept of \((m,n)\)-high-ideals by generalizing the concept of \((m,n)\)-ideals in a posemigroup, for two non-negative integers \(m\) and \(n\). As a result of this definition we get a relationship between \(0^{\ast}\)-minimal \((m,n)\)-high-ideals and \((m,n)\)-regular posemigroups. A necessary and sufficient condition for a posemigroup \(S\) to be an \((m,n)\)-regular posemigroup is given as well.

Keywords


Posemigroup; \((m,n)\)-regular; High-bi-ideal; \(0^{\ast}\)-minimal; \((m,n)\)-high-ideal

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References


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DOI: http://dx.doi.org/10.26713%2Fcma.v8i1.452

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