Right group

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A semi-group which is right simple (cf. Simple semi-group) and satisfies the left cancellation law. Every right group is a completely-simple semi-group. The property that a semi-group $S$ is a right group is equivalent to any of the following conditions: a) $S$ is right simple and contains an idempotent element; b) $S$ is regular (cf. Regular semi-group) and satisfies the left cancellation law; c) $S$ can be partitioned into left ideals which are (necessarily isomorphic) groups; and d) $S$ is the direct product of a group and a right zero semi-group (cf. [[Idempotents, semi-group of]). The notion of a left group is similar to that of a right group. Only groups are simultaneously right groups and left groups. Every completely-simple semi-group can be partitioned into right (left) ideals which are (necessarily isomorphic) right (left) groups.


[1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
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Right group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article