View source for Variety of universal algebras

A class of universal algebras (cf. [[Universal algebra|Universal algebra]]) defined by a system of identities (cf. [[Algebraic systems, variety of|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|free algebra]] with basis of any cardinality. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963201.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963202.png" /> are bases of the same free algebra in a non-trivial variety and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963203.png" /> is infinite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963205.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963206.png" /> consists of all quotient algebras of subdirect products of algebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963207.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963208.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v0963209.png" /> commute if and only if there exists a ternary [[Term|term]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632010.png" /> of the signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632011.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632012.png" /></td> </tr></table>

for all algebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632013.png" />. In similar fashion one can characterize varieties of universal algebras whose algebras have modular or distributive congruence lattices (cf. [[#References|[1]]]–[[#References|[4]]], [[#References|[7]]], [[#References|[9]]], [[#References|[10]]]).

In a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632014.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632015.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632016.png" /> is called trivial if for every algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632017.png" /> the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632018.png" /> holds. E.g. in the variety of rings with zero multiplication the operation of multiplication is trivial. Every trivial operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632019.png" /> may be replaced by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632020.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632021.png" /> defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632022.png" />. Suppose that the signatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632024.png" /> of two varieties of universal algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632026.png" />, respectively, do not contain trivial operations. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632027.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632028.png" /> into the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632029.png" /> of terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632030.png" /> is called admissible if the arities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632032.png" /> coincide for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632033.png" />. An admissible mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632034.png" /> can be extended to a mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632036.png" />, still denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632037.png" />, in a natural fashion. The varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632039.png" /> are said to be rationally equivalent if there exist admissible mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632041.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632044.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632045.png" />, and if for every defining identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632046.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632047.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632048.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632049.png" />) the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632050.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632051.png" />) holds for all algebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632052.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632053.png" />). The last requirement is equivalent to the fact that every algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632054.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632055.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632057.png" />) corresponds to an algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632058.png" /> (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632059.png" />), where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632060.png" />-ary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632062.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632064.png" />) is defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632065.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632066.png" />). The variety of Boolean rings and that of Boolean algebras (cf. [[Boolean algebra|Boolean algebra]]) are rationally equivalent. The variety of unary algebras (cf. [[Unary algebra|Unary algebra]]) of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632067.png" />, with defining identities

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632068.png" /></td> </tr></table>

is rationally equivalent to the variety of all left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632069.png" />-polygons (cf. [[Polygon (over a monoid)|Polygon (over a monoid)]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632070.png" /> is the quotient monoid of the free monoid generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632071.png" /> by the congruence generated by the pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632072.png" />. A variety of universal algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632073.png" /> is rationally equivalent to the variety of all right modules over some associative ring if and only if the congruences on any algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632074.png" /> commute, if finite free products (cf. [[Free product|Free product]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632075.png" /> coincide with direct products (cf. [[Direct product|Direct product]]) and if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632076.png" />-ary derived operations forming a distinguished subalgebra. The first two conditions may be replaced by the requirement: Every subalgebra of any algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632077.png" /> is the class of a certain congruence and every congruence of any algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632078.png" /> is uniquely determined by the class formed by the subalgebra, [[#References|[3]]], [[#References|[5]]]–[[#References|[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 [[#References|[7]]], [[#References|[8]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Kurosh,   "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.I. Mal'tsev,   "Algebraic systems" , Springer  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L.A. Skornyakov,   "Elements of general algebra" , Moscow  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B. Csakany,   "Abelian properties of primitive classes of universal algebras"  ''Acta Scient. Math.'' , '''25''' :  3–4  (1964)  pp. 202–208  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Grätzer,   "Universal algebra" , Springer  (1979)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.D.H. Smith,   "Mal'cev varieties" , Springer  (1976)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  W. Taylor,   "Characterizing Mal'cev conditions"  ''Algebra Universalis'' , '''3''' :  3  (1973)  pp. 351–397</TD></TR></table>



====Comments====
The term  "variety of universal algebras"  is also used for the [[Category|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632079.png" />, these are exactly the varieties in the category of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632080.png" />-algebras (cf. [[Variety in a category|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|Triple]]) and preserves filtered colimits [[#References|[a1]]], [[#References|[a2]]].

Two varieties are called Morita equivalent if they are equivalent as (abstract) categories; this generalizes the notion of [[Morita equivalence|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|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 [[#References|[a3]]] of those varieties which are Cartesian-closed categories and those which are topoi (cf. [[Topos|Topos]]).

The first volume of an authoritative treatment on universal algebra has appeared [[#References|[a5]]]. The topic of Mal'tsev operations (ternary operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632081.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096320/v09632082.png" />) has been vigorously pursued recently (1991); an exposition of work up to 1989, mainly due to E. Faro and J. Lambek, is in [[#References|[a6]]]. (Cf. [[Mal'tsev product|Mal'tsev product]].)

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.G. Manes,   "Algebraic theories" , Springer  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.C. Wraith,   "Algebraic theories" , ''Lect. notes series'' , '''22''' , Aarhus Univ.  (1975)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.T. Johnstone,   "Collapsed toposes and cartesian closed varieties"  ''J. Algebra'' , '''129'''  (1990)  pp. 446–480</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Freese,   R. McKenzie,   "Commutator theory for congruence modular varieties" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Taylor,   "Algebras, lattices, varieties" , '''1''' , Wadsworth  (1987)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.M. San Luis Fernandez,   "Sobre teorías algebraicas con una operación de Malcev"  ''Alxebra'' , '''55'''  (1989)</TD></TR></table>