Algebraic systems, variety of

A class of algebraic systems (cf. Algebraic systems, class of) of a fixed signature , axiomatizable by identities, i.e. by formulas of the type

where is some predicate symbol from or the equality sign, while are terms of the signature in the object variables . A variety of algebraic systems is also known as an equational class, or a primitive class. A variety of signature can also be defined (Birkhoff's theorem) as a non-empty class of -systems closed with respect to subsystems, homomorphic images and Cartesian products.

The intersection of all varieties of signature which contain a given (not necessarily abstract) class of -systems is called the equational closure of the class (or the variety generated by the class ), and is denoted by . In particular, if the class consists of a single -system , its equational closure is denoted by . If the system is finite, all finitely-generated systems in the variety are also finite [1], [2].

Let be a class of -systems, let be the class of subsystems of systems of , let be the class of homomorphic images of the systems from , and let be the class of isomorphic copies of Cartesian products of the systems of . The following relation [1], [2] is valid for an arbitrary non-empty class of -systems:

A variety is said to be trivial if the identity is true in each one of its systems. Any non-trivial variety has free systems of any rank and [1], [2].

Let be a set of identities of the signature and let be the class of all -systems in which all the identities of are true. If the equality is satisfied for a variety of signature , is known as a basis for . A variety is known as finitely baseable if it has a finite basis . For any system , a basis of the variety is also known as a basis of identities of the system . If is a finitely-baseable variety of algebras of a finite signature and if all algebras of have distributive congruence lattices, then each finite algebra of has a finite basis of identities [10]. In particular, any finite lattice has a finite basis of identities. Any finite group has a finite basis of identities [3]. On the other hand, there exists a six-element semi-group [5] and a three-element groupoid [6] without a finite basis of identities.

The varieties of -systems contained in some fixed variety of signature constitute under inclusion a complete lattice with a zero and a unit, known as the lattice of subvarieties of the variety . The zero of this lattice is the variety with the basis , (), while its unit is the variety . If the variety is non-trivial, the lattice is anti-isomorphic to the lattice of all fully-characteristic congruences (cf. Fully-characteristic congruence) of the system of countable rank which is free in [1]. The lattice of all varieties of signature is infinite, except for the case when the set is finite and consists of predicate symbols only. The exact value of the cardinality of the infinite lattice is known [1]. The lattice of all lattice varieties is distributive and has the cardinality of the continuum [7], [8]. The lattice of all group varieties is modular, but it is not distributive [3], [4]. The lattice of varieties of commutative semi-groups is not modular [9].

Atoms of the lattice of all varieties of signature are known as minimal varieties of signature . Every variety with a non-unit system contains at least one minimal variety. If the -system is finite and of finite type, then the variety contains only a finite number of minimal subvarieties [1].

Let be subvarieties of a fixed variety of -systems. The Mal'tsev product denotes the class of those systems of with a congruence such that , and all cosets (), which are systems in , belong to . If is the variety of all groups and if and are subvarieties of it, then the product is identical with the Neumann product [3]. The product of varieties of semi-groups need not be a variety. A variety of -systems is called polarized if there exists a term of the signature such that in each system from the identities , () are true. If is a polarized variety of algebras and the congruences in all algebras in are commutable, then the Mal'tsev product of any subvarieties and in is a variety. One may speak, in particular, of the groupoid of subvarieties of an arbitrary variety of groups, rings, etc. If is the variety of all groups or all Lie algebras over a fixed field of characteristic zero, then is a free semi-group [1].

References

[1]	A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[2]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[3]	H. Neumann, "Varieties of groups" , Springer (1967)
[4]	G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[5]	P. Perkins, "Bases of equational theories of semigroups" J. of Algebra , 11 : 2 (1968) pp. 298–314
[6]	V.L. Murskii, "The existence in three-valued logic of a closed class with finite basis, not having a finite system of identities" Soviet Math. Dokl. , 6 : 4 (1965) pp. 1020–1024 Dokl. Akad. Nauk SSSR , 163 : 4 (1965) pp. 815–818
[7]	B. Jónsson, "Algebras whose congruence lattices are distributive" Math. Scand. , 21 (1967) pp. 110–121
[8]	K.A. Baker, "Equational classes of modular lattices" Pacific J. Math. , 28 (1969) pp. 9–15
[9]	R. Schwabauer, "A note on commutative semi-groups" Proc. Amer. Math. Soc. , 20 (1969) pp. 503–504
[10]	K.A. Baker, "Primitive satisfaction and equational problems for lattices and other algebras" Trans. Amer. Math. Soc. , 190 (1974) pp. 125–150

Comments

A categorical characterization of varieties of algebraic systems was introduced by F.W. Lawvere [a1]; for a detailed account of this approach see [a2].

References

[a1]	F.W. Lawvere, "Functional semantics of algebraic theories" Proc. Nat. Acad. Sci. USA , 50 (1963) pp. 869–873
[a2]	E.G. Manes, "Algebraic theories" , Springer (1976)