Abstract
The effect of the existence of a medial or related idempotent in any abundant semigroup is the subject of this paper. The aim is to naturally order any abundant semigroup $S$ which contains an ample multiplicative medial idempotent $u$ in a way that $\mathcal{L}^*$ and $\mathcal{R}^*$ are compatible with the natural order and $u$ is a maximum idempotent. The structure of an abundant semigroup containing an ample normal medial idempotent studied in \cite{item6} will be revisited.
Highlights
A partial order relation on any semigroup S with set of idempotents E is a natural partial order if for any e, f ∈ E:e = ef = f e implies e f
We review some concepts related to general abundant semigroups
If S is an abundant semigroup with an ample medial idempotent u, is a naturally ordered ample semigroup with a maximum idempotent u
Summary
A partial order relation on any semigroup S with set of idempotents E is a natural partial order if for any e, f ∈ E:. The approach to this result is an adaptation of that of [19]. Any undefined notation and terminology be as in [15]
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