Quotient (universal algebra)

In mathematics, a quotient algebra is the result of partitioning the elements of an algebraic structure using a congruence relation.
Quotient algebras are also called factor algebras. Here, the congruence relation must be an equivalence relation that is additionally "compatible" with all the operations of the algebra, in the formal sense described below.
Its equivalence classes partition the elements of the given algebraic structure. The quotient algebra has these classes as its elements, and the compatibility conditions are used to give the classes an algebraic structure.
The idea of the quotient algebra abstracts into one common notion the quotient structure of quotient rings of ring theory, quotient groups of group theory, the quotient spaces of linear algebra and the quotient modules of representation theory into a common framework.
Compatible relation.
Let "A" be the set of the elements of an algebra formula_1, and let "E" be an equivalence relation on the set "A". The relation "E" is said to be "compatible" with (or have the "substitution property" with respect to) an "n"-ary operation "f", if formula_2 for formula_3 implies formula_4 for any formula_5 with formula_3. An equivalence relation compatible with all the operations of an algebra is called a congruence with respect to this algebra.
Quotient algebras and homomorphisms.
Any equivalence relation "E" in a set "A" partitions this set in equivalence classes. The set of these equivalence classes is usually called the quotient set, and denoted "A"/"E". For an algebra formula_1, it is straightforward to define the operations induced on the elements of "A"/"E" if "E" is a congruence. Specifically, for any operation formula_8 of arity formula_9 in formula_1 (where the superscript simply denotes that it is an operation in formula_1, and the subscript formula_12 enumerates the functions in formula_1 and their arities) define formula_14 as formula_15, where formula_16 denotes the equivalence class of formula_17 generated by "E" (""x" modulo "E"").
For an algebra formula_18, given a congruence "E" on formula_1, the algebra formula_20 is called the "quotient algebra" (or "factor algebra") of formula_1 modulo "E". There is a natural homomorphism from formula_1 to formula_23 mapping every element to its equivalence class. In fact, every homomorphism "h" determines a congruence relation via the kernel of the homomorphism, formula_24.
Given an algebra formula_1, a homomorphism "h" thus defines two algebras homomorphic to formula_1, the image h(formula_1) and formula_28 The two are isomorphic, a result known as the "homomorphic image theorem" or as the first isomorphism theorem for universal algebra. Formally, let formula_29 be a surjective homomorphism. Then, there exists a unique isomorphism "g" from formula_28 onto formula_31 such that "g" composed with the natural homomorphism induced by formula_32 equals "h".
Congruence lattice.
For every algebra formula_1 on the set "A", the identity relation on A, and formula_34 are trivial congruences. An algebra with no other congruences is called "simple".
Let formula_35 be the set of congruences on the algebra formula_1. Because congruences are closed under intersection, we can define a meet operation: formula_37 by simply taking the intersection of the congruences formula_38.
On the other hand, congruences are not closed under union. However, we can define the closure of any binary relation "E", with respect to a fixed algebra formula_1, such that it is a congruence, in the following way: formula_40. Note that the closure of a binary relation is a congruence and thus depends on the operations in formula_1, not just on the carrier set. Now define formula_42 as formula_43.
For every algebra formula_1, formula_45 with the two operations defined above forms a lattice, called the "congruence lattice" of formula_1.
Maltsev conditions.
If two congruences "permute" (commute) with the composition of relations as operation, i.e. formula_47, then their join (in the congruence lattice) is equal to their composition: formula_48. An algebra is called "congruence-permutable" if every pair of its congruences permutes; likewise a variety is said to be congruence-permutable if all its members are 
congruence-permutable algebras.
In 1954, Anatoly Maltsev established the following characterization of congruence-permutable varieties: a variety is congruence permutable if and only if there exist a ternary term such that ; this is called a Maltsev term and varieties with this property are called Maltsev varieties. Maltsev's characterization explains a large number of similar results in groups (take ), rings, quasigroups (take , complemented lattices, Heyting algebras etc. Furthermore, every congruence-permutable algebra is congruence-modular, i.e. its lattice of congruences is modular lattice as well; the converse is not true however.
After Maltsev's result, other researchers found characterizations based on conditions similar to that found by Maltsev but for other kinds of properties, e.g. in 1967 Bjarni Jónsson found the conditions for varieties having congruence lattices that are distributive (thus called congruence-distributive varieties). Generically, such conditions are called Maltsev conditions.
This line of research led to the Pixley–Wille algorithm for generating Maltsev conditions associated
with congruence identities.