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A natural partial order for semigroups. (English) Zbl 0596.06015
A partial order on a semigroup (S,$$\cdot)$$ is called natural if it is defined by means of the multiplication of S. For regular semigroups, such a natural partial order was found by R. E. Hartwig [Math. Jap. 25, 1-13 (1980; Zbl 0442.06006)] and by K. S. S. Nambooripad [Proc. Edinb. Math. Soc., II. Ser. 23, 249-260 (1980; Zbl 0459.20054)].
In this paper, the following natural partial order on a semigroup (S,$$\cdot)$$ is defined: $$a\leq b$$ iff $$a=xb=by$$, $$a=xa$$ for some $$x,y\in S^ 1$$ (here $$S^ 1=S$$ if S has identity and $$S^ 1=S\cup \{1\}$$ if not). In the case of regular semigroups, this order coincides with that of Hartwig and Nambooripad. If (S,$$\cdot)$$ is a semigroup and $$\leq$$ is the natural order on S defined above, it is shown that the following are equivalent: for a,b$$\in S:$$ i) $$a\leq b$$, ii) $$a=wb=bz$$, $$az=a$$ for some $$w,z\in S^ 1$$, iii) $$a=xb=by$$, $$xa=ay=a$$ for some $$x,y\in S^ 1$$.
Reviewer: M.Deaconescu

##### MSC:
 06F05 Ordered semigroups and monoids 20M10 General structure theory for semigroups
##### Keywords:
regular semigroups; natural partial order
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##### References:
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