View source for Congruence-permutable algebra

In [[universal algebra]], a '''congruence-permutable algebra''' is an algebra whose [[congruence relation|congruences]] [[commutative property|commute]] under [[composition of relations|composition]]. This symmetry has several equivalent characterizations, which lend to the analysis of such algebras. Many familiar [[variety of algebras|varieties of algebras]], such as the variety of [[group (mathematics)|groups]], consist of congruence-permutable algebras, but some, like the variety of [[lattice (order)|lattices]], have members that are not congruence-permutable.

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== Definition ==
Given an algebra <math>\mathbf{A}</math>, a pair of [[congruence relation|congruences]] <math>\alpha,\beta\in\operatorname{Con}(\mathbf{A})</math> are said to '''permute''' when <math>\alpha\circ\beta=\beta\circ\alpha</math>.{{r|Bergman|p=121}} An algebra <math>\mathbf{A}</math> is called '''congruence-permutable''' when each pair of congruences of <math>\mathbf{A}</math> permute.{{r|Bergman|p=122}} A [[variety (universal algebra)|variety]] of algebras <math>\mathcal{V}</math> is referred to as '''congruence-permutable''' when every algebra in <math>\mathcal{V}</math> is congruence-permutable.{{r|Bergman|p=122}}

== Properties ==
In 1954 [[Anatoly Maltsev|Maltsev]] gave two other conditions that are equivalent to the one given above defining a congruence-permutable variety of algebras. This initiated the study of congruence-permutable varieties.{{r|Bergman|p=122}}

=== Theorem (Maltsev, 1954) ===
Suppose that <math>\mathcal{V}</math> is a variety of algebras. The following are equivalent:

{{ordered list
|1= The variety <math>\mathcal{V}</math> is congruence-permutable.
|2= The [[free object|free algebra]] on <math>3</math> generators in <math>\mathcal{V}</math> is congruence-permutable.
|3= There is a ternary term <math>q</math> such that

:<math>\mathcal{V}\models q(x,y,y)\approx x\approx q(y,y,x)</math>.
}}

Such a term is called a '''Maltsev term''' and congruence-permutable varieties are also known as '''Maltsev varieties''' in his honor.{{r|Bergman|p=122}}

== Examples ==
Most classical varieties in [[abstract algebra]], such as [[group (mathematics)|groups]]{{r|Bergman|p=123}}, [[ring (mathematics)|rings]]{{r|Bergman|p=123}}, and [[Lie algebra|Lie algebras]]{{Citation needed|date=May 2019}} are congruence-permutable. Any variety that contains a group operation is congruence-permutable, and the Maltsev term is <math>xy^{-1}z</math>.{{Citation needed|date=May 2019}}

== Nonexamples ==
Viewed as a lattice the [[chain (order theory)|chain]] with three elements is not congruence-permutable and hence neither is the variety of lattices.{{r|Bergman|p=123}}

== References ==
{{Reflist|refs=
<ref name="Bergman">{{cite book |last1=Bergman |first1=Clifford |title=Universal Algebra: Fundamentals and Selected Topics |date=2011 |publisher=Chapman and Hall/CRC |isbn=978-1-4398-5129-6}}</ref>
}}

[[Category:Universal algebra]]

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