Idempotents, semi-group of

idempotent semi-group

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 $S$ is considered with respect to its natural partial order, then $ab$ is the greatest lower bound of the elements $a,b\in S$. Every semi-lattice is a subdirect product of two-element semi-lattices. A semi-group $S$ is said to be singular if $S$ satisfies one of the identities $xy=x$, $xy=y$; in the first case $S$ 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 $xyx=x$ (this term is sometimes used in a wider sense, see [1]). The following conditions are equivalent for a semi-group $S$: 1) $S$ is rectangular; 2) $S$ is an ideally-simple idempotent semi-group (see Simple semi-group); 3) $S$ is a completely-simple semi-group of idempotents; and 4) $S$ is isomorphic to a direct product $L\times R$, where $L$ is a left-singular and $R$ 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 $\mathfrak V$ of all idempotent semi-groups has been described completely in [4]–[6]; it is countable and distributive, and every subvariety of $\mathfrak V$ is defined by one identity. See the figure for the diagram of this lattice; also indicated in this figure are the identities giving in $\mathfrak V$ the varieties on some of the lower "floors" .

Figure: i050090a

References

[1]	A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967)
[2]	A.D. McLean, "Idempotent semigroups" Amer. Math. Monthly , 61 : 2 (1954) pp. 110–113
[3]	N. Kimura, "The structure of idempotent semigroups" Pacific J. Math. , 8 (1958) pp. 257–275
[4]	A.P. Biryukov, "Varieties of idempotent semigroups" Algebra and Logic , 9 : 3 (1970) pp. 153–164 Algebra i Logika , 9 : 3 (1970) pp. 255–273
[5]	J. Gerhard, "The lattice of equational classes of idempotent semigroups" J. of Algebra , 15 : 2 (1970) pp. 195–224
[6]	C. Fennemore, "All varieties of bands I, II" Math. Nachr. , 48 : 1–6 (1971) pp. 237–252; 253–262