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The adjunction of a zero to an ordered groupoid-semigroup. (English. Russian original) Zbl 1075.06007
Russ. Math. 47, No. 9, 25-32 (2003); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2003, No. 9, 28-35 (2003).
Summary: For an ordered groupoid $$S$$ without zero, we denote by $$S^0$$ the ordered groupoid arising from $$S$$ by the adjunction of a zero element. The adjunction of a zero to an ordered groupoid is unique up to isomorphism. In this paper we prove that an ordered groupoid $$S$$ is isomorphic to an ordered groupoid $$T$$ if and only if the ordered groupoids $$S^0$$ and $$T^0$$ arising from $$S,T$$ by the adjunction of a zero to $$S$$ and $$T$$, are isomorphic (in symbols $$S\cong T$$ if and only if $$S^0\cong T^0)$$. Furthermore, we characterize the 0-simple ordered semigroups $$S$$ for which there exists an ordered semigroup $$T$$ such that $$S\cong T^0$$. The characterization is by means of left (resp. right) divisors of zero and the nilpotent elements. Finally we show that while adjunction of a zero to an ordered groupoid is unique up to isomorphism, the adjunction of a generalized zero to an ordered groupoid up to isomorphism is not unique.
##### MSC:
 06F05 Ordered semigroups and monoids 20N02 Sets with a single binary operation (groupoids)