# Variety of universal algebras

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

A class of universal algebras (cf. Universal algebra) defined by a system of identities (cf. Algebraic systems, variety of). A variety of universal algebras may be characterized as a non-empty class of algebras closed under taking quotient algebras, subalgebras and direct products. The last two conditions may be replaced by the requirement of closure under subdirect products. A variety of universal algebras is said to be trivial if it consists of one-element algebras. Every non-trivial variety of universal algebras contains a free algebra with basis of any cardinality. If and are bases of the same free algebra in a non-trivial variety and is infinite, then and are equipotent. The requirement that one of the bases be infinite is essential, but it may be omitted if the variety contains a finite algebra with more than one element.

The variety of universal algebras generated by a class consists of all quotient algebras of subdirect products of algebras in . If a variety of universal algebras is generated by finite algebras, then every finitely-generated algebra in the variety is finite. The congruences of any algebra in a variety of universal algebras of signature commute if and only if there exists a ternary term of the signature such that for all algebras in . In similar fashion one can characterize varieties of universal algebras whose algebras have modular or distributive congruence lattices (cf. , , , ).

In a variety , an -ary operation is called trivial if for every algebra in the identity holds. E.g. in the variety of rings with zero multiplication the operation of multiplication is trivial. Every trivial operation may be replaced by the -ary operation defined by the equation . Suppose that the signatures , of two varieties of universal algebras , , respectively, do not contain trivial operations. A mapping from into the set of terms of is called admissible if the arities of and coincide for all . An admissible mapping can be extended to a mapping from to , still denoted by , in a natural fashion. The varieties and are said to be rationally equivalent if there exist admissible mappings and such that for all , for all , and if for every defining identity (respectively, ) of (respectively, ) the identity (respectively, ) holds for all algebras in (in ). The last requirement is equivalent to the fact that every algebra in ( in ) corresponds to an algebra in (in ), where each -ary operation in ( in ) is defined by the equation (respectively, ). The variety of Boolean rings and that of Boolean algebras (cf. Boolean algebra) are rationally equivalent. The variety of unary algebras (cf. Unary algebra) of signature , with defining identities is rationally equivalent to the variety of all left -polygons (cf. Polygon (over a monoid)), where is the quotient monoid of the free monoid generated by by the congruence generated by the pairs . A variety of universal algebras is rationally equivalent to the variety of all right modules over some associative ring if and only if the congruences on any algebra in commute, if finite free products (cf. Free product) in coincide with direct products (cf. Direct product) and if there exist -ary derived operations forming a distinguished subalgebra. The first two conditions may be replaced by the requirement: Every subalgebra of any algebra in is the class of a certain congruence and every congruence of any algebra in is uniquely determined by the class formed by the subalgebra, , .

The variety of lattices generated by the congruence lattices of all algebras of a certain variety of universal algebras is called a congruence variety. Not every variety of lattices is a congruence variety. There exist congruence varieties which are not modular and differ from the variety of all lattices , .

How to Cite This Entry:
Variety of universal algebras. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variety_of_universal_algebras&oldid=16722
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article