Idempotents, semi-group of
A semi-group each element of which is an idempotent. An idempotent semi-group is also called a band (this is consistent with the concept of a band of semi-groups: An idempotent semi-group is a band of one-element semi-groups). A commutative idempotent semi-group is called a semi-lattice; this term is consistent with its use in the theory of partially ordered sets: If a commutative idempotent semi-group is considered with respect to its natural partial order, then is the greatest lower bound of the elements . Every semi-lattice is a subdirect product of two-element semi-lattices. A semi-group is said to be singular if satisfies one of the identities , ; in the first case is said to be left-singular, or to be a semi-group of left zeros, in the second case it is called right-singular, or a semi-group of right zeros. A semi-group is said to be rectangular if it satisfies the identity (this term is sometimes used in a wider sense, see ). The following conditions are equivalent for a semi-group : 1) is rectangular; 2) is an ideally-simple idempotent semi-group (see Simple semi-group); 3) is a completely-simple semi-group of idempotents; and 4) is isomorphic to a direct product , where is a left-singular and is a right-singular semi-group. Every idempotent semi-group is a Clifford semi-group and splits into a semi-lattice of rectangular semi-groups (see Band of semi-groups). This splitting is the starting point for the study of many properties of idempotent semi-groups. Every idempotent semi-group is locally finite.
Idempotent semi-groups have been studied from various points of view, including that of the theory of varieties. The lattice of all subvarieties of the variety of all idempotent semi-groups has been described completely in –; it is countable and distributive, and every subvariety of is defined by one identity. See the figure for the diagram of this lattice; also indicated in this figure are the identities giving in the varieties on some of the lower "floors" .
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Idempotents, semi-group of. L.N. Shevrin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Idempotents,_semi-group_of&oldid=16549