# 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].

#### 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