Variety of universal algebras

$\newcommand{\parens}[1]{\mathopen{}\left(#1\right)\mathclose{}}$ $\newcommand{\braces}[1]{\mathopen{}\left\{#1\right\}\mathclose{}}$ 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. [1]–[4], [7], [9], [10]).

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, [3], [5]–[7].

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 [7], [8].

References

[1]	P.M. Cohn, "Universal algebra" , Reidel (1981)
[2]	A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[3]	A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian)
[4]	L.A. Skornyakov, "Elements of general algebra" , Moscow (1983) (In Russian)
[5]	B. Csakany, "Primitive classes of algebras which are equivalent to classes of semimodules and modules" Acta Scient. Math. , 24 : 1–2 (1963) pp. 157–164 (In Russian)
[6]	B. Csakany, "Abelian properties of primitive classes of universal algebras" Acta Scient. Math. , 25 : 3–4 (1964) pp. 202–208 (In Russian)
[7]	G. Grätzer, "Universal algebra" , Springer (1979)
[8]	B. Jónsson, "Varieties of lattices: some open problems" B. Csákány (ed.) E. Fried (ed.) E.T. Schmidt (ed.) , Universal Algebra (Esztergom, 1977) , Coll. Math. Soc. J. Bolyai , 29 , North-Holland (1982) pp. 421–436
[9]	J.D.H. Smith, "Mal'cev varieties" , Springer (1976)
[10]	W. Taylor, "Characterizing Mal'cev conditions" Algebra Universalis , 3 : 3 (1973) pp. 351–397

Comments

The term "variety of universal algebras" is also used for the category formed by all the algebras in a given variety (in the sense defined above) and all the homomorphisms between them; for algebras in a given signature $\Omega$, these are exactly the varieties in the category of all $\Omega$-algebras (cf. Variety in a category). The categories which occur as varieties may be characterized as those equipped with a forgetful functor to the category of sets which is monadic (cf. Triple) and preserves filtered colimits [a1], [a2].

Two varieties are called Morita equivalent if they are equivalent as (abstract) categories; this generalizes the notion of Morita equivalence for rings. Two varieties are equivalent as concrete categories (that is, there is an equivalence between them which reduces to the identity functor on underlying sets) if and only if they are rationally equivalent, as defined above. Many properties of varieties which are invariant within rational equivalence turn out to be definable in categorical terms. For example, varieties of unary algebras are exactly those for which the underlying-set functor preserves coproducts, and varieties rationally equivalent to varieties of modules are exactly those which are Abelian categories (cf. Abelian category). Note that the second of these classes of varieties is closed under Morita equivalence, although the first is not. Any property of varieties which is expressible in terms of subalgebra lattices or congruence lattices will automatically be invariant within Morita equivalence.

In the opposite direction, one can seek syntactic conditions (that is, conditions on the operations and the equations they satisfy) which correspond to familiar categorical properties of a variety. For example, such characterizations have been given [a3] of those varieties which are Cartesian-closed categories and those which are topoi (cf. Topos).

The first volume of an authoritative treatment on universal algebra has appeared [a5]. The topic of Mal'tsev operations (ternary operations $f$ satisfying $f \parens{x, x, y} = y = f \parens{y, x, x}$) has been vigorously pursued recently (1991); an exposition of work up to 1989, mainly due to E. Faro and J. Lambek, is in [a6]. (Cf. Mal'tsev product.)

References

[a1]	E.G. Manes, "Algebraic theories" , Springer (1976)
[a2]	G.C. Wraith, "Algebraic theories" , Lect. notes series , 22 , Aarhus Univ. (1975)
[a3]	P.T. Johnstone, "Collapsed toposes and cartesian closed varieties" J. Algebra , 129 (1990) pp. 446–480
[a4]	R. Freese, R. McKenzie, "Commutator theory for congruence modular varieties" , Cambridge Univ. Press (1987)
[a5]	W. Taylor, "Algebras, lattices, varieties" , 1 , Wadsworth (1987)
[a6]	A.M. San Luis Fernandez, "Sobre teorías algebraicas con una operación de Malcev" Alxebra , 55 (1989)