# Completely-simple semi-group

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One of the more important types of simple semi-groups. A semi-group is called completely simple (completely -simple) if it is simple ( -simple) and contains a primitive idempotent, i.e. a non-zero idempotent that is not an identity for any non-zero idempotent of . If a zero is added to a completely-simple semi-group it becomes a completely -simple semi-group; for this reason, many properties of completely-simple semi-groups may be deduced directly from the corresponding properties of completely -simple semi-groups.
A semi-group is completely -simple if and only if it is -simple and satisfies one of the following conditions: 1) has minimal non-zero left and right ideals; or 2) some power of each element of belongs to a subgroup of . In particular, any periodic (and, a fortiori, finite) -simple semi-group will be a completely -simple semi-group. Any completely -simple semi-group is an O-bisimple regular semi-group and is the union of its -minimal left (right) ideals. A semi-group is a completely-simple semi-group if and only if it satisfies one of the following conditions: 1) is a rectangular band of isomorphic groups (cf. Band of semi-groups); or 2) is regular and all its idempotents are primitive. A special kind of completely-simple semi-groups is the rectangular group which is the direct product of a group and a rectangular band (cf. Idempotents, semi-group of). A right group (left group) is in turn a special case of a rectangular semi-group. Rees' theorem gives an important representation of completely -simple semi-groups: A semi-group is a completely -simple semi-group if and only if it is isomorphic to a regular Rees semi-group of matrix type over a group with zero.
The study of finite completely-simple semi-groups formed the starting point of the development of the theory of semi-groups (cf. Semi-group). Completely -simple and completely-simple semi-groups frequently appear in various theoretical investigations on semi-groups and are one of the most thoroughly studied types of semi-groups.