# Completely-simple semi-group

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One of the more important types of simple semi-groups. A semi-group $S$ is called completely simple (completely $0$-simple) if it is simple ($0$-simple) and contains a primitive idempotent, i.e. a non-zero idempotent that is not an identity for any non-zero idempotent of $S$. If a zero is added to a completely-simple semi-group it becomes a completely $0$-simple semi-group; for this reason, many properties of completely-simple semi-groups may be deduced directly from the corresponding properties of completely $0$-simple semi-groups.

A semi-group $S$ is completely $0$-simple if and only if it is $0$-simple and satisfies one of the following conditions: 1) $S$ has minimal non-zero left and right ideals; or 2) some power of each element of $S$ belongs to a subgroup of $S$. In particular, any periodic (and, a fortiori, finite) $0$-simple semi-group will be a completely $0$-simple semi-group. Any completely $0$-simple semi-group is an O-bisimple regular semi-group and is the union of its $0$-minimal left (right) ideals. A semi-group $S$ is a completely-simple semi-group if and only if it satisfies one of the following conditions: 1) $S$ is a rectangular band of isomorphic groups (cf. Band of semi-groups); or 2) $S$ 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 $0$-simple semi-groups: A semi-group is a completely $0$-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 $0$-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.

#### References

 [1] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) [2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) [3] K. Kapp, H. Schneider, "Completely 0-simple semigroups: an abstract treatment of the lattice of congruences" , Benjamin (1969)

A semi-group $S$ is called simple ($0$-simple) if it has no proper ideals (respectively, if its only proper ideal is $\{0\}$ and $S^2\neq\{0\}$) (cf. Simple semi-group). More accurately, a primitive idempotent is a non-zero idempotent $e\in S$ such that for any non-zero idempotent $f\in S$, $fe=ef=f$ only if $f=e$ ( "e is not an identity for any f≠ e" ).