The adjunction of a zero to an ordered groupoid-semigroup. (English. Russian original) Zbl 1075.06007

Kehayopulu, N.; Tsingelis, M.

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)

Keywords:

adjunction of zero; ordered groupoid; ordered semigroup; divisors of zero; nilpotent elements