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Quotient (universal algebra)
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{{Short description|Result of partitioning the elements of an algebraic structure using a congruence relation}} {{for|quotient associative algebras over a ring|quotient ring}} In [[mathematics]], a '''quotient algebra''' is the result of [[Partition of a set|partition]]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 [[Operation (mathematics)|operations]] of the algebra, in the formal sense described below. Its [[equivalence class]]es 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.<ref>A. G. Kurosh, Lectures on General Algebra, Translated from the Russian edition (Moscow, 1960), Chelsea, New York, 1963.</ref> The idea of the quotient algebra abstracts into one common notion the quotient structure of [[quotient ring]]s of [[ring theory]], [[quotient group]]s of [[group theory]], the [[Quotient space (linear algebra)|quotient space]]s of [[linear algebra]] and the [[quotient module]]s of [[representation theory]] into a common framework. == Compatible relation == Let ''A'' be the 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 <math>(a_i,\; b_i) \in E</math> for <math>1 \le i \le n</math> implies <math>(f (a_1, a_2, \ldots, a_n), f (b_1, b_2, \ldots, b_n)) \in E</math> for any <math>a_i,\; b_i \in A</math> with <math>1 \le i \le n</math>. 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 class]]es. The set of these equivalence classes is usually called the [[quotient set]], and denoted ''A''/''E''. For an algebra <math>\mathcal{A}</math>, it is straightforward to define the operations induced on the elements of ''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 is an operation in <math>\mathcal{A}</math>, and the subscript <math>i \in I</math> enumerates the functions in <math>\mathcal{A}</math> and their arities) 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>[x]_E \in A/E</math> denotes the equivalence class of <math>x \in A</math> generated by ''E'' ("''x'' 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 via the [[Kernel (algebra)#Universal algebra|kernel]] of the homomorphism, <math> \mathop{\mathrm{ker}}\,h = \{(a,a') \in A^2\, |\, h(a) = h(a')\}\subseteq A^2</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'' or as the [[Isomorphism theorem#First Isomorphism Theorem 4|first isomorphism theorem]] for universal algebra. 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''. == 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}) \mid E \subseteq F \}</math>. Note that the closure of a binary relation is a congruence and thus 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>(\mathrm{Con}(\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>. == Maltsev conditions == If two congruences ''permute'' (commute) with the [[composition of relations]] as operation, i.e. <math>\alpha\circ\beta = \beta\circ\alpha</math>, then their join (in the congruence lattice) is equal to their composition: <math>\alpha\circ\beta = \alpha\vee\beta</math>. An algebra is called ''[[congruence permutable]]'' if every pair of its congruences permutes; likewise a [[Variety (universal algebra)|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 {{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''<sup>−1</sup>''z''}}), rings, [[quasigroup]]s (take {{nowrap|1=''q'' = (x / (y \ y))(y \ z))}}, [[complemented lattice]]s, [[Heyting algebra]]s 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 [[Jónsson term|conditions]] for varieties having congruence lattices that are distributive<ref>{{cite journal | url=https://doi.org/10.7146/math.scand.a-10850 | doi=10.7146/math.scand.a-10850 | title=Algebras Whose Congruence Lattices are Distributive | year=1967 | last1=Jonnson | first1=Bjarni | journal=Mathematica Scandinavica | volume=21 | page=110 | doi-access=free }}</ref> (thus called congruence-distributive varieties), while in 1969 Alan Day did the same for varieties having congruence lattices that are modular.<ref>{{cite journal | url=https://doi.org/10.4153/CMB-1969-016-6 | doi=10.4153/CMB-1969-016-6 | title=A Characterization of Modularity for Congruence Lattices of Algebras | year=1969 | last1=Day | first1=Alan | journal=Canadian Mathematical Bulletin | volume=12 | issue=2 | pages=167–173 | s2cid=120602601 | doi-access=free }}</ref> 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.<ref name="KearnesKiss2013">{{cite book|author1=Keith Kearnes|author2=Emil W. Kiss|title=The Shape of Congruence Lattices|year=2013|publisher=American Mathematical Soc.|isbn=978-0-8218-8323-5|page=4}}</ref> == See also == * [[Quotient ring]] * [[Congruence lattice problem]] * [[Lattice of subgroups]] ==Notes== {{reflist}} == References == * {{cite book|author1=Klaus Denecke|author2=Shelly L. Wismath|title=Universal algebra and coalgebra|url=https://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=https://books.google.com/books?id=hLX6OG1U5W8C&pg=PA215|year=2005|publisher=PHI Learning Pvt. Ltd.|isbn=978-81-203-2721-4|page=215}} * {{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)}} [[Category:Universal algebra]]
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