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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
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