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A natural partial order for semigroups

Author: H. Mitsch
Journal: Proc. Amer. Math. Soc. 97 (1986), 384-388
MSC: Primary 20M10; Secondary 06F05
MathSciNet review: 840614
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Abstract: A partial order on a semigroup $ (S, \cdot )$ is called natural if it is defined by means of the multiplication of $ S$. It is shown that for any semigroup $ (S, \cdot )$ the relation $ a \leq b$ iff $ a = xb = by$, $ xa = a$ for some $ x$, $ y \in {S^1}$, is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup $ ({T_X}, \circ )$ of all maps on the set $ X$ are investigated.

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Article copyright: © Copyright 1986 American Mathematical Society

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