Algebraic systems, variety of

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

$$ ( \forall x _ {1} ) \dots ( \forall x _ {s} ) P ( f _ {1} \dots f _ {n} ) , $$

where $ P $ is some predicate symbol from $ \Omega $ or the equality sign, while $ f _ {1} \dots f _ {n} $ are terms of the signature $ \Omega $ in the object variables $ x _ {1} \dots x _ {s} $. A variety of algebraic systems is also known as an equational class, or a primitive class. A variety of signature $ \Omega $ can also be defined (Birkhoff's theorem) as a non-empty class of $ \Omega $- systems closed with respect to subsystems, homomorphic images and Cartesian products.

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

Let $ {\mathcal L} $ be a class of $ \Omega $- systems, let $ S {\mathcal L} $ be the class of subsystems of systems of $ {\mathcal L} $, let $ H {\mathcal L} $ be the class of homomorphic images of the systems from $ {\mathcal L} $, and let $ \Pi {\mathcal L} $ be the class of isomorphic copies of Cartesian products of the systems of $ {\mathcal L} $. The following relation [1], [2] is valid for an arbitrary non-empty class $ \mathfrak K $ of $ \Omega $- systems:

$$ \mathop{\rm var} \mathfrak K = H S \Pi \mathfrak K . $$

A variety is said to be trivial if the identity $ x = y $ is true in each one of its systems. Any non-trivial variety $ \mathfrak M $ has free systems $ F _ {m} ( \mathfrak M ) $ of any rank $ m $ and $ \mathfrak M = \mathop{\rm var} F _ {\aleph _ {0} } ( \mathfrak M ) $[1], [2].

Let $ S $ be a set of identities of the signature $ \Omega $ and let $ KS $ be the class of all $ \Omega $- systems in which all the identities of $ S $ are true. If the equality $ \mathfrak M = KS $ is satisfied for a variety $ \mathfrak M $ of signature $ \Omega $, $ S $ is known as a basis for $ \mathfrak M $. A variety $ \mathfrak M $ is known as finitely baseable if it has a finite basis $ S $. For any system $ \mathbf A $, a basis of the variety $ \mathop{\rm var} \mathbf A $ is also known as a basis of identities of the system $ \mathbf A $. If $ \mathfrak M $ is a finitely-baseable variety of algebras of a finite signature and if all algebras of $ \mathfrak M $ have distributive congruence lattices, then each finite algebra $ \mathbf A $ of $ \mathfrak M $ has a finite basis of identities [10]. In particular, any finite lattice $ \langle \mathbf A , \lor, \wedge \rangle $ 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 $ \Omega $- systems contained in some fixed variety $ \mathfrak M $ of signature $ \Omega $ constitute under inclusion a complete lattice $ L ( \mathfrak M ) $ with a zero and a unit, known as the lattice of subvarieties of the variety $ \mathfrak M $. The zero of this lattice is the variety with the basis $ x = y $, $ P ( x _ {1} \dots x _ {n} ) $( $ P \in \Omega $), while its unit is the variety $ \mathfrak M $. If the variety $ \mathfrak M $ is non-trivial, the lattice $ L ( \mathfrak M ) $ is anti-isomorphic to the lattice of all fully-characteristic congruences (cf. Fully-characteristic congruence) of the system $ F _ {\aleph _ {0} } ( \mathfrak M ) $ of countable rank which is free in $ \mathfrak M $[1]. The lattice $ L _ \Omega $ of all varieties of signature $ \Omega $ is infinite, except for the case when the set $ \Omega $ is finite and consists of predicate symbols only. The exact value of the cardinality of the infinite lattice $ L _ \Omega $ 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 $ L _ \Omega $ of all varieties of signature $ \Omega $ are known as minimal varieties of signature $ \Omega $. Every variety with a non-unit system contains at least one minimal variety. If the $ \Omega $- system $ \mathbf A $ is finite and of finite type, then the variety $ \mathop{\rm var} \mathbf A $ contains only a finite number of minimal subvarieties [1].

Let $ \mathfrak A , \mathfrak B $ be subvarieties of a fixed variety $ \mathfrak M $ of $ \Omega $- systems. The Mal'tsev product $ \mathfrak A _ {\mathfrak M} \circ \mathfrak B $ denotes the class of those systems $ \mathbf A $ of $ \mathfrak M $ with a congruence $ \theta $ such that $ ( \mathbf A / \theta ) \in \mathfrak B $, and all cosets $ a / \theta $( $ a \in \mathbf A $), which are systems in $ \mathfrak M $, belong to $ \mathfrak A $. If $ \mathfrak M $ is the variety of all groups and if $ \mathfrak A $ and $ \mathfrak B $ are subvarieties of it, then the product $ \mathfrak A _ {\mathfrak M} \circ \mathfrak B $ is identical with the Neumann product [3]. The product of varieties of semi-groups need not be a variety. A variety $ \mathfrak M $ of $ \Omega $- systems is called polarized if there exists a term $ e (x) $ of the signature $ \Omega $ such that in each system from $ \mathfrak M $ the identities $ e(x) = e(y) $, $ F(e(x) \dots e(x)) = e (x) $( $ F \in \Omega $) are true. If $ \mathfrak M $ is a polarized variety of algebras and the congruences in all algebras in $ \mathfrak M $ are commutable, then the Mal'tsev product $ \mathfrak A _ {\mathfrak M} \circ \mathfrak B $ of any subvarieties $ \mathfrak A $ and $ \mathfrak B $ in $ \mathfrak M $ is a variety. One may speak, in particular, of the groupoid $ G _ {I} ( \mathfrak M ) $ of subvarieties of an arbitrary variety $ \mathfrak M $ of groups, rings, etc. If $ \mathfrak M $ is the variety of all groups or all Lie algebras over a fixed field $ P $ of characteristic zero, then $ G _ {I} ( \mathfrak M ) $ is a free semi-group [1].

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

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