# Variety of semi-groups

A class of semi-groups (cf. Semi-group) defined by a system of identities, or laws (see Algebraic systems, variety of). Every variety of semi-groups is either periodic, i.e. it consists of periodic semi-groups, or overcommutative, i.e. it contains the variety of all commutative semi-groups. Various properties of varieties of semi-groups are classified by singling out certain types of identities. An identity \$ u = v \$ is said to be normal (also homotypical, regular or uniform) if the sets of variables figuring in the words \$ u \$ and \$ v \$ are the same, and anomalous (or heterotypical) otherwise. An identity \$ u = v \$ is said to be balanced if each variable appears in \$ u \$ just as many times as it does in \$ v \$. A special case of a balanced identity is a permutation identity — if \$ u = x _ {1} \dots x _ {m} \$ and \$ v \$ is obtained from \$ u \$ by permuting the variables. A variety of semi-groups is overcommutative if and only if all its identities are balanced. A basis of identities for a variety of semi-groups \$ \mathfrak M \$ is said to be irreducible if any of its proper subsets defines a variety distinct from \$ \mathfrak M \$. Every overcommutative variety of semi-groups has an irreducible basis of identities. There exist varieties of semi-groups which do not have irreducible bases of identities. Examples of varieties of semi-groups with finite bases are: any variety of commutative semi-groups; any periodic variety of semi-groups with a permutation identity; any variety of semi-groups defined by permutation identities. Any semi-group with less than six elements has a finite basis of identities, but there exists a \$ 6 \$- element semi-group that has no finite basis of identities.

The following conditions for a variety of semi-groups \$ \mathfrak M \$ are equivalent: \$ \mathfrak M \$ is defined by normal identities; all identities of \$ \mathfrak M \$ are normal; \$ \mathfrak M \$ contains a two-element semi-lattice. Among the identities of a variety of semi-groups \$ \mathfrak M \$ there is an anomalous one if and only if \$ \mathfrak M \$ is periodic and consists of Archimedean semi-groups (cf. Archimedean semi-group).

The minimal varieties of semi-groups are exhausted by the varieties of all: 1) semi-lattices; 2) semi-groups of left zeros; 3) semi-groups of right zeros (see Idempotents, semi-group of); 4) semi-groups with zero multiplication; 5) Abelian groups of exponent \$ p \$ for any prime number \$ p \$. In the lattice of all varieties of semi-groups, every non-unit element has an element that covers it; a unit element cannot be equal to the union of finitely many non-unit elements. The lattice of all varieties of semi-groups does not satisfy any non-trivial lattice identity and has the cardinality of the continuum. The sublattice of all varieties of nil-semi-groups with the identity \$ x ^ {2} = 0 \$ is also of the cardinality of the continuum, as is the sublattice of all overcommutative varieties. For some varieties of semi-groups \$ \mathfrak M \$, explicit descriptions have been discovered for the lattice \$ L \mathfrak M \$ of subvarieties of it; there are also descriptions of varieties of semi-groups \$ \mathfrak M \$ with certain restrictions on \$ L \mathfrak M \$.

A variety of semi-groups is said to be small if \$ L \mathfrak M \$ is finite. A variety of semi-groups \$ \mathfrak M \$ is called a variety of finite index if the degrees of nilpotency of the nilpotent semi-groups in \$ \mathfrak M \$ are uniformly bounded (equivalent conditions are: every nil-semi-group in \$ \mathfrak M \$ is nilpotent; or: \$ \mathfrak M \$ does not contain the variety of all commutative nil-semi-groups with the identity \$ x ^ {2} = 0 \$). Every small variety of semi-groups is of finite index.

If \$ \mathfrak M \$ is a periodic variety of semi-groups, the following conditions are equivalent : \$ \mathfrak M \$ consists of bands of Archimedean semi-groups; in any semi-group in \$ \mathfrak M \$, every torsion class is a sub-semi-group; \$ \mathfrak M \$ does not contain the Brandt semi-group \$ B _ {2} \$( see Periodic semi-group). These conditions are satisfied by varieties of semi-groups \$ \mathfrak M \$ with a modular lattice \$ L \mathfrak M \$ and varieties of semi-groups of finite index (in particular, small varieties). A small variety of semi-groups is locally finite (i.e. consists of locally finite semi-groups) if and only if the variety of all groups in \$ \mathfrak M \$ is locally finite; the small locally finite varieties of groups are precisely the Cross varieties (see Variety of groups). For other locally finite varieties of semi-groups, see Locally finite semi-group. There is a description of the varieties of semi-groups whose elements are residually-finite semi-groups .

The set of all varieties of semi-groups forms a partial groupoid \$ G \$ relative to the Mal'tsev product. The idempotents of \$ G \$ are known; there are just nine of them. The set of all varieties of semi-groups defined by systems of identities of the type \$ w = 0 \$ is a maximal groupoid in \$ G \$.

Studies have also been conducted on varieties of semi-groups with additional signature operations: varieties of monoids (with an identity, cf. Monoid); varieties of semi-groups with a zero; varieties of inverse semi-groups; etc.

How to Cite This Entry:
Variety of semi-groups. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variety_of_semi-groups&oldid=49130
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article