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In [[mathematics]], a '''quotient algebra''', (where algebra is used in the sense of [[universal algebra]]), also called a '''factor algebra''', is obtained by [[Partition of a set|partition]]ing the elements of an algebra into [[equivalence class]]es given by a [[congruence relation]], that is an [[equivalence relation]] that is additionally ''compatible'' with all the [[Operation (mathematics)|operations]] of the algebra, in the formal sense described below.
 
== Compatible relation ==
Let ''A'' be a set (of the elements of an algebra <math>\mathcal{A}</math>), 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 all <math>a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in A</math> whenever <math>(a_1, b_1) \in E, (a_2, b_2) \in E, \ldots, (a_n, b_n) \in E</math> implies <math>(f (a_1, a_2, \ldots, a_n), f (b_1, b_2, \ldots, b_n)) \in E</math>. An equivalence relation compatible with all the operations of an algebra is called a congruence.
 
== Congruence lattice ==
For every algebra <math>\mathcal{A}</math> on the set ''A'', the [[identity relation]] on A, and <math>A \times A</math> are trivial congruences. An algebra with no other congruences is called ''simple''.
 
Let <math>\mathrm{Con}(\mathcal{A})</math> be the set of congruences on the algebra <math>\mathcal{A}</math>. Because congruences are closed under intersection, we can define a [[Meet (mathematics)|meet operation]]: <math> \wedge : \mathrm{Con}(\mathcal{A}) \times \mathrm{Con}(\mathcal{A}) \to \mathrm{Con}(\mathcal{A})</math> by simply taking the intersection of the congruences <math>E_1 \wedge E_2 = E_1\cap E_2</math>.
 
On the other hand, congruences are not closed under union. However, we can define the [[Closure operator|closure]] of any [[binary relation]] ''E'', with respect to a fixed algebra <math>\mathcal{A}</math>, such that it is a congruence, in the following way: <math> \langle E \rangle_{\mathcal{A}} = \bigcap \{ F \in \mathrm{Con}(\mathcal{A}) | E \subseteq F \}</math>. Note that the (congruence) closure of a binary relation depends on the operations in <math>\mathcal{A}</math>, not just on the carrier set. Now define <math> \vee: \mathrm{Con}(\mathcal{A}) \times \mathrm{Con}(\mathcal{A}) \to \mathrm{Con}(\mathcal{A})</math> as <math>E_1 \vee E_2 = \langle E_1\cup E_2 \rangle_{\mathcal{A}} </math>.
 
For every algebra <math>\mathcal{A}</math>, <math>(\mathcal{A}, \wedge, \vee)</math> with the two operations defined above forms a [[Lattice (order)|lattice]], called the ''congruence lattice'' of <math>\mathcal{A}</math>.
 
== Quotient algebras and homomorphisms ==
A set ''A'' can be partitioned in [[equivalence class]]es given by an equivalence relation ''E'', and usually called a [[quotient set]], and denoted ''A''/''E''. For an algebra <math>\mathcal{A}</math>, it is straightforward to define the operations induced on ''A''/''E'' if ''E'' is a congruence. Specifically, for any operation <math>f^{\mathcal{A}}_i</math> of [[arity]] <math>n_i</math> in <math>\mathcal{A}</math> (where the superscript simply denotes that it's an operation in <math>\mathcal{A}</math>) define <math>f^{\mathcal{A}/E}_i : (A/E)^{n_i} \to A/E</math> as <math>f^{\mathcal{A}/E}_i ([a_1]_E, \ldots, [a_{n_i}]_E) = [f^{\mathcal{A}}_i(a_1,\ldots, a_{n_i})]_E</math>, where <math>[a]_E</math> denotes the equivalence class of ''a'' modulo ''E''.
 
For an algebra <math>\mathcal{A} = (A, (f^{\mathcal{A}}_i)_{i \in I})</math>, given a congruence ''E'' on <math>\mathcal{A}</math>, the algebra <math>\mathcal{A}/E = (A/E, (f^{\mathcal{A}/E}_i)_{i \in I})</math> is called the ''quotient algebra'' (or ''factor algebra'') of <math>\mathcal{A}</math> modulo ''E''. There is a natural [[homomorphism]] from <math>\mathcal{A}</math> to <math>\mathcal{A}/E</math> mapping every element to its equivalence class. In fact, every homomorphism ''h'' determines a congruence relation; the [[Kernel (algebra)#Universal algebra|kernel]] of the homomorphism, <math> \mathop{\mathrm{ker}}\,h = \{(a,a') \in A \times A | h(a) = h(a')\}</math>.
 
Given an algebra <math>\mathcal{A}</math>, a homomorphism ''h'' thus defines two algebras homomorphic to <math>\mathcal{A}</math>, the [[Image (mathematics)|image]] h(<math>\mathcal{A}</math>) and <math>\mathcal{A}/\mathop{\mathrm{ker}}\,h</math> The two are [[isomorphic]], a result known as the ''homomorphic image theorem''. Formally, let <math> h : \mathcal{A} \to \mathcal{B} </math> be a [[surjective]] homomorphism. Then, there exists a unique isomorphism ''g'' from <math>\mathcal{A}/\mathop{\mathrm{ker}}\,h</math> onto <math>\mathcal{B} </math> such that ''g'' [[function composition|composed]] with the natural homomorphism induced by <math>\mathop{\mathrm{ker}}\,h</math> equals ''h''.
 
== See also ==
* [[quotient ring]]
 
== References ==
* {{cite book|author1=Klaus Denecke|author2=Shelly L. Wismath|title=Universal algebra and coalgebra|url=http://books.google.com/books?id=NgTAzhC8jVAC&pg=PA14|year=2009|publisher=World Scientific|isbn=978-981-283-745-5|pages=14–17}}
* {{cite book|author=Purna Chandra Biswal|title=Discrete mathematics and graph theory|url=http://books.google.com/books?id=hLX6OG1U5W8C&pg=PA215|year=2005|publisher=PHI Learning Pvt. Ltd.|isbn=978-81-203-2721-4|page=215}}
 
[[Category:Universal algebra]]

Latest revision as of 22:22, 18 April 2013

In mathematics, a quotient algebra, (where algebra is used in the sense of universal algebra), also called a factor algebra, 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 a set (of the elements of an algebra 𝒜), 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 all a1,a2,,an,b1,b2,,bnA whenever (a1,b1)E,(a2,b2)E,,(an,bn)E implies (f(a1,a2,,an),f(b1,b2,,bn))E. An equivalence relation compatible with all the operations of an algebra is called a congruence.

Congruence lattice

For every algebra 𝒜 on the set A, the identity relation on A, and A×A are trivial congruences. An algebra with no other congruences is called simple.

Let Con(𝒜) be the set of congruences on the algebra 𝒜. Because congruences are closed under intersection, we can define a meet operation: :Con(𝒜)×Con(𝒜)Con(𝒜) by simply taking the intersection of the congruences E1E2=E1E2.

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 𝒜, such that it is a congruence, in the following way: E𝒜={FCon(𝒜)|EF}. Note that the (congruence) closure of a binary relation depends on the operations in 𝒜, not just on the carrier set. Now define :Con(𝒜)×Con(𝒜)Con(𝒜) as E1E2=E1E2𝒜.

For every algebra 𝒜, (𝒜,,) with the two operations defined above forms a lattice, called the congruence lattice of 𝒜.

Quotient algebras and homomorphisms

A set A can be partitioned in equivalence classes given by an equivalence relation E, and usually called a quotient set, and denoted A/E. For an algebra 𝒜, it is straightforward to define the operations induced on A/E if E is a congruence. Specifically, for any operation fi𝒜 of arity ni in 𝒜 (where the superscript simply denotes that it's an operation in 𝒜) define fi𝒜/E:(A/E)niA/E as fi𝒜/E([a1]E,,[ani]E)=[fi𝒜(a1,,ani)]E, where [a]E denotes the equivalence class of a modulo E.

For an algebra 𝒜=(A,(fi𝒜)iI), given a congruence E on 𝒜, the algebra 𝒜/E=(A/E,(fi𝒜/E)iI) is called the quotient algebra (or factor algebra) of 𝒜 modulo E. There is a natural homomorphism from 𝒜 to 𝒜/E mapping every element to its equivalence class. In fact, every homomorphism h determines a congruence relation; the kernel of the homomorphism, kerh={(a,a)A×A|h(a)=h(a)}.

Given an algebra 𝒜, a homomorphism h thus defines two algebras homomorphic to 𝒜, the image h(𝒜) and 𝒜/kerh The two are isomorphic, a result known as the homomorphic image theorem. Formally, let h:𝒜 be a surjective homomorphism. Then, there exists a unique isomorphism g from 𝒜/kerh onto such that g composed with the natural homomorphism induced by kerh equals h.

See also

References

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