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) |
Algebraic systems, variety of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_systems,_variety_of&oldid=17213