In
mathematics, a quotient algebra is the result of
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
ing 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 ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
s of
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
,
quotient groups of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, the
quotient spaces of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices ...
and the
quotient module
In algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is very similar to that of a quotient vector space. It differs from analogous quotient constructions of rings and groups by ...
s of
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
into a common framework.
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
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
, 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 in
(where the superscript simply denotes that it is an operation in
, and the subscript
enumerates the functions in
and their arities) define
as
, 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
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from
to
mapping every element to its equivalence class. In fact, every homomorphism ''h'' determines a congruence relation via the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
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
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
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
In mathematics, a homogeneous relation (also called endorelation) over a set ''X'' is a binary relation over ''X'' and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on ''X''" or "a (binary) 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 closure of a binary relation is a congruence and thus depends on the operations in
, not just on the carrier set. Now define
as
.
For every algebra
,
with the two operations defined above forms a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
, called the ''congruence lattice'' of
.
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
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
is said to be congruence-permutable if all its members are
congruence-permutable algebras.
In 1954,
Anatoly Maltsev Anatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate – 7 June 1967, Novosibirsk) was born in Misheronsky, near Moscow, an ...
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,
quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ...
s (take ,
complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''& ...
s,
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
s etc. Furthermore, every congruence-permutable algebra is congruence-modular, i.e. its lattice of congruences is
modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self- dual condition,
;Modular law: implies
where are arbitrary elements in the lattice, ≤ is the partial order, and & ...
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. In 1967
Bjarni Jónsson
Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory. He was emeritus distinguished professor of mathematics at Vanderb ...
found the
conditions for varieties having congruence lattices that are distributive (thus called congruence-distributive varieties), while in 1969 Alan Day did the same for varieties having congruence lattices that are modular.
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.
See also
*
Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
*
Congruence lattice problem In mathematics, the congruence lattice problem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed by Robert P. Dilworth, and for many years it was one of the most ...
*
Lattice of subgroups
In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion.
In this lattice, the join of two subgroups is the subgroup generated by their uni ...
Notes
References
*
*
* {{cite book, author=Clifford Bergman, title=Universal Algebra: Fundamentals and Selected Topics, year=2011, publisher=CRC Press, isbn=978-1-4398-5129-6, pages=122–124, 137 (Maltsev varieties)
Universal algebra