View source for Idempotents, semi-group of

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''idempotent semi-group''

A semi-group each element of which is an [[Idempotent|idempotent]]. An idempotent semi-group is also called a band (this is consistent with the concept of a [[Band of semi-groups|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 [[#References|[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|Simple semi-group]]); 3) $S$ is a [[Completely-simple semi-group|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|Clifford semi-group]] and splits into a semi-lattice of rectangular semi-groups (see [[Band of semi-groups|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 [[#References|[4]]]–[[#References|[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" .

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i050090a.gif" />

Figure: i050090a

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.H. Clifford,   G.B. Preston,   "Algebraic theory of semi-groups" , '''1–2''' , Amer. Math. Soc.  (1961–1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.D. McLean,   "Idempotent semigroups"  ''Amer. Math. Monthly'' , '''61''' :  2  (1954)  pp. 110–113</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Kimura,   "The structure of idempotent semigroups"  ''Pacific J. Math.'' , '''8'''  (1958)  pp. 257–275</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Gerhard,   "The lattice of equational classes of idempotent semigroups"  ''J. of Algebra'' , '''15''' :  2  (1970)  pp. 195–224</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  C. Fennemore,   "All varieties of bands I, II"  ''Math. Nachr.'' , '''48''' :  1–6  (1971)  pp. 237–252; 253–262</TD></TR></table>