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.[1]
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.
Let A be the set of the elements of an algebra
l{A}
(ai, bi)\inE
1\lei\len
(f(a1,a2,\ldots,an),f(b1,b2,\ldots,bn))\inE
ai, bi\inA
1\lei\len
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
l{A}
fl{A
ni
l{A}
l{A}
i\inI
l{A}
fl{A/E}i:
ni | |
(A/E) |
\toA/E
fl{A/E}i([a1]E,\ldots,
[a | |
ni |
]E)=[fl{A
[x]E\inA/E
x\inA
For an algebra
l{A}=(A,(fl{A
l{A}
l{A}/E=(A/E,(fl{A/E}i)i)
l{A}
l{A}
l{A}/E
kerh=\{(a,a')\inA2|h(a)=h(a')\}\subseteqA2
Given an algebra
l{A}
l{A}
l{A}
l{A}/kerh
h:l{A}\tol{B}
l{A}/kerh
l{B}
kerh
For every algebra
l{A}
A x A
Let
Con(l{A})
l{A}
\wedge:Con(l{A}) x Con(l{A})\toCon(l{A})
E1\wedgeE2=E1\capE2
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
l{A}
\langleE\ranglel{A
l{A}
\vee:Con(l{A}) x Con(l{A})\toCon(l{A})
E1\veeE2=\langleE1\cupE2\ranglel{A
For every algebra
l{A}
(Con(l{A}),\wedge,\vee)
l{A}
If two congruences permute (commute) with the composition of relations as operation, i.e.
\alpha\circ\beta=\beta\circ\alpha
\alpha\circ\beta=\alpha\vee\beta
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. In 1967 Bjarni Jónsson found the conditions for varieties having congruence lattices that are distributive[2] (thus called congruence-distributive varieties), while in 1969 Alan Day did the same for varieties having congruence lattices that are modular.[3] Generically, such conditions are called Maltsev conditions.
This line of research led to the Pixley–Wille algorithm for generating Maltsev conditions associatedwith congruence identities.[4]