Completely-simple semi-group

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.

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)

Comments

A semi-group is called simple (-simple) if it has no proper ideals (respectively, if its only proper ideal is and ) (cf. Simple semi-group). More accurately, a primitive idempotent is a non-zero idempotent such that for any non-zero idempotent , only if ( "e is not an identity for any f≠ e" ).

A Rees semi-group of matrix type is often called a Rees matrix semi-group.

References

[a1]	A.K. Suschkewitsch, "Ueber die endlichen Gruppen ohne das Gesetz der eindeutigen Umkehrbarkeit" Math. Ann. , 99 (1928) pp. 30–50
[a2]	D. Rees, "On semi-groups" Proc. Cambridge Phil. Soc. , 36 (1940) pp. 387–400