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 [4]: $ \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 [3].

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.

References

[1]	T. Evans, "The lattice of semigroup varieties" Semigroup Forum , 2 : 1 (1971) pp. 1–43
[2]	A.Ya Aizenshtat, B.K. Boguta, , Semi-group varieties and semi-groups of endomorphisms , Leningrad (1979) pp. 3–46 (In Russian)
[3]	E.A. Golubov, M.V. Sapir, "Varieties of finitely approximable semigroups" Soviet Math. Dokl. , 20 : 4 (1979) pp. 828–832 Dokl. Akad. Nauk SSSR , 247 : 5 (1979) pp. 1037–1041
[4]	M.V. Sapir, E.V. Sukhanov, "On manifolds of periodic semigroups" Soviet Math. Izv. Vyz. , 25 : 4 (1981) pp. 53–63 Izv. Vuzov. Mat. , 25 : 4 (1981) pp. 48–55
[5]	L.N. Shevrin, M.V. Volkov, "Identities of semigroups" Soviet Math. Izv. Vyz. , 29 : 11 (1985) pp. 1–64 Izv. Vuzov. Mat. , 29 : 11 (1985) pp. 3–47
[6]	L.N. Shevrin, E.V. Sukhanov, "Structural aspects of theory of semigroup varieties" Soviet Math. Izv. Vyz. , 33 : 6 (1989) pp. 1–34 Izv. Vuzov. Mat. , 33 : 6 (1989) pp. 3–39
[a1]	E.S. [E.S. Lyapin] Ljapin, "Semigroups" , Amer. Math. Soc. (1978) pp. Chapt. XII (Translated from Russian)
[a2]	N.R. Reilly, "On the lattice of varieties of completely regular semigroups" S.M. Goberstein (ed.) P.M. Higgins (ed.) , Semigroups and Their Applications , Reidel (1987) pp. 153–167