# Variety of universal algebras

$\newcommand{\parens}{\mathopen{}\left(#1\right)\mathclose{}}$ $\newcommand{\braces}{\mathopen{}\left\{#1\right\}\mathclose{}}$ 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 $X$ and $Y$ are bases of the same free algebra in a non-trivial variety and $X$ is infinite, then $X$ and $Y$ 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 $K$ consists of all quotient algebras of subdirect products of algebras in $K$. 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 $M$ of signature $\Omega$ commute if and only if there exists a ternary term $f$ of the signature $\Omega$ such that

$$f \parens{x, x, y} = y = f \parens{y, x, x}$$

for all algebras in $M$. In similar fashion one can characterize varieties of universal algebras whose algebras have modular or distributive congruence lattices (cf. , , , ).

In a variety $M$, an $n$-ary operation $f$ is called trivial if for every algebra in $M$ the identity $f \parens{x_1, \dots, x_n} = f \parens{y_1, \dots, y_n}$ holds. E.g. in the variety of rings with zero multiplication the operation of multiplication is trivial. Every trivial operation $f$ may be replaced by the $0$-ary operation $\nu_f$ defined by the equation $\nu_f = f \parens{x_1, \dots, x_n}$. Suppose that the signatures $\Omega$, $\Omega'$ of two varieties of universal algebras $M$, $M'$, respectively, do not contain trivial operations. A mapping $\Phi$ from $\Omega$ into the set $W \parens{\Omega'}$ of terms of $\Omega'$ is called admissible if the arities of $f$ and $\Phi \parens{f}$ coincide for all $f \in \Omega$. An admissible mapping $\Phi$ can be extended to a mapping from $W \parens{\Omega}$ to $W \parens{\Omega'}$, still denoted by $\Phi$, in a natural fashion. The varieties $M$ and $M'$ are said to be rationally equivalent if there exist admissible mappings $\Phi : \Omega \to W \parens{\Omega'}$ and $\Phi' : \Omega' \to W \parens{\Omega}$ such that $f = \Phi' \parens{\Phi \parens{f}}$ for all $f \in \Omega$, $f' = \Phi \parens{\Phi' \parens{f'}}$ for all $f' \in \Omega'$, and if for every defining identity $u = v$ (respectively, $u' = v'$) of $M$ (respectively, $M'$) the identity $\Phi \parens{u} = \Phi \parens{v}$ (respectively, $\Phi' \parens{u'} = \Phi' \parens{v'}$) holds for all algebras in $M'$ (in $M$). The last requirement is equivalent to the fact that every algebra $A$ in $M$ ($A'$ in $M'$) corresponds to an algebra in $M'$ (in $M$), where each $n$-ary operation $f'$ in $\Omega'$ ($f$ in $\Omega$) is defined by the equation $f' \parens{x_1, \dots, x_n} = \Phi' \parens{f'} \parens{x_1, \dots, x_n}$ (respectively, $f \parens{x_1, \dots, x_n} = \Phi \parens{f} \parens{x_1, \dots, x_n}$). 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 $\Omega$, with defining identities

$$\braces{ u_{\iota} \parens{x} = v_{\iota} \parens{x} : \iota \in \mathfrak{J} },$$

is rationally equivalent to the variety of all left $R$-polygons (cf. Polygon (over a monoid)), where $R$ is the quotient monoid of the free monoid generated by $\Omega$ by the congruence generated by the pairs $\braces{ \parens{u_{\iota}, v_{\iota}} : \iota \in \mathfrak{J} }$. A variety of universal algebras $M$ is rationally equivalent to the variety of all right modules over some associative ring if and only if the congruences on any algebra in $M$ commute, if finite free products (cf. Free product) in $M$ coincide with direct products (cf. Direct product) and if there exist $0$-ary derived operations forming a distinguished subalgebra. The first two conditions may be replaced by the requirement: Every subalgebra of any algebra in $M$ is the class of a certain congruence and every congruence of any algebra in $M$ 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=40189
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article