“Quotient algebra”的意思、由来-开放百科全书

In mathematics, a quotient algebra, (where algebra means algebraic structure in the sense of universal algebra), also called a factor algebra,^[1] is obtained by partitioning the elements of an algebra into equivalence classes given by a congruence relation, that is an equivalence relation that is additionally compatible with all the operations of the algebra, in the formal sense described below.

Compatible relation

Let A be the set of the elements of an algebra

, 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

for

implies

for any

with

. 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

, it is straightforward to define the operations induced on the elements of A/E if E is a congruence. Specifically, for any operation

of arity

(where the superscript simply denotes that it is an operation in

, and the subscript

enumerates the functions in

and their arities) define

, where

denotes the equivalence class of

generated by E ("x modulo E").

For an algebra

, given a congruence E on

, the algebra

is called the quotient algebra (or factor algebra) of

modulo E. There is a natural homomorphism from

mapping every element to its equivalence class. In fact, every homomorphism h determines a congruence relation via the kernel of the homomorphism,

Given an algebra

, a homomorphism h thus defines two algebras homomorphic to

, the image h(

) and

The two are isomorphic, a result known as the homomorphic image theorem or as the first isomorphism theorem for universal algebra. Formally, let

be a surjective homomorphism. Then, there exists a unique isomorphism g from

onto

such that g composed with the natural homomorphism induced by

equals h.

Congruence lattice

For every algebra

on the set A, the identity relation on A, and

are trivial congruences. An algebra with no other congruences is called simple.

Let

be the set of congruences on the algebra

. Because congruences are closed under intersection, we can define a meet operation:

by simply taking the intersection of the congruences

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

, such that it is a congruence, in the following way:

. Note that the (congruence) closure of a binary relation depends on the operations in

, not just on the carrier set. Now define

For every algebra

with the two operations defined above forms a lattice, called the congruence lattice of

. It's a distributive lattice.

Maltsev conditions

If two congruences permute (commute) with the composition of relations as operation, i.e.

, then their join (in the congruence lattice) is equal to their composition:

. 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

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 {{nowrap|q(x, y, z)}} such that {{nowrap|q(x, y, y) ≈ x ≈ q(y, y, x)}}; 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 {{nowrap|1=q = xy⁻¹z}}), rings, quasigroups (take {{nowrap|1=q = (x / (y \\ y))(y \\ z))}}, 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

Compatible relation

Quotient algebras and homomorphisms

Congruence lattice

Maltsev conditions

See also

Notes

References