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Difference between revisions of "Congruence (in algebra)"

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An equivalence relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248701.png" /> on a [[Universal algebra|universal algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248702.png" /> commuting with all operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248703.png" />, that is, an equivalence relation such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248704.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248705.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248707.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248708.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c0248709.png" />-ary operation. Congruences in algebraic systems are defined in a similar way. Thus, the equivalence classes modulo a congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487010.png" /> form a universal algebra (algebraic system) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487011.png" /> of the same type as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487012.png" />, called the quotient algebra (or quotient system) modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487013.png" />. The natural mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487014.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487015.png" /> (which takes an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487016.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487017.png" />-class containing it) is a surjective homomorphism. Conversely, every homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487018.png" /> defines a unique congruence, whose classes are the pre-images of the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487019.png" />.
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An [[equivalence relation]] $\pi$ on a [[universal algebra]] $\mathcal{A} = (A,\Omega)$ commuting with all operations in $\Omega$, that is, an equivalence relation such that $(a_1,\ldots, a_n) \omega \,\pi\, (b_1,\ldots,b_n)\omega$ whenever $a_i \,\pi\, b_i$, where $a_i, b_i \in A$, $i=1,\ldots,n$, and $\omega$ is an $n$-ary operation. Congruences in algebraic systems are defined in a similar way. Thus, the equivalence classes modulo a congruence $\pi$ form a universal algebra (algebraic system) $\mathcal{A}/\pi$ of the same type as $\mathcal{A}$, called the quotient algebra (or quotient system) modulo $\pi$. The natural mapping from $A$ onto $A/\pi$ (which takes an element $a \in A$ to the $\pi$-class containing it) is a surjective homomorphism. Conversely, every homomorphism $\phi:A \rightarrow B$ defines a unique congruence, whose classes are the pre-images of the elements of $B$ (cf. [[Kernel of a function]]).
  
The intersection of a family of congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487021.png" />, in the lattice of relations on a universal algebra (algebraic system) is a congruence. In general, a union of congruences in the lattice of relations is not a congruence. The product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487022.png" /> of two congruences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487024.png" /> is a congruence if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487026.png" /> commute, i.e. if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024870/c02487027.png" />.
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The intersection of a family of congruences $\pi_i$, $i \in I$, in the lattice of relations on a universal algebra (algebraic system) is a congruence. In general, a union of congruences in the lattice of relations is not a congruence. The product $\pi_1\pi_2$ of two congruences $\pi_1$ and $\pi_2$ is a congruence if and only if $\pi_1$ and $\pi_2$ commute, i.e. if and only if $\pi_1 \pi_2 = \pi_2 \pi_1$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972)  (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Higher algebra" , MIR  (1972)  (Translated from Russian into German) {{ZBL|0237.13001}}</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" (rev. ed.), Reidel  (1981) {{ZBL|0461.08001}}</TD></TR>
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</table>
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{{TEX|done}}

Revision as of 17:39, 12 November 2016

An equivalence relation $\pi$ on a universal algebra $\mathcal{A} = (A,\Omega)$ commuting with all operations in $\Omega$, that is, an equivalence relation such that $(a_1,\ldots, a_n) \omega \,\pi\, (b_1,\ldots,b_n)\omega$ whenever $a_i \,\pi\, b_i$, where $a_i, b_i \in A$, $i=1,\ldots,n$, and $\omega$ is an $n$-ary operation. Congruences in algebraic systems are defined in a similar way. Thus, the equivalence classes modulo a congruence $\pi$ form a universal algebra (algebraic system) $\mathcal{A}/\pi$ of the same type as $\mathcal{A}$, called the quotient algebra (or quotient system) modulo $\pi$. The natural mapping from $A$ onto $A/\pi$ (which takes an element $a \in A$ to the $\pi$-class containing it) is a surjective homomorphism. Conversely, every homomorphism $\phi:A \rightarrow B$ defines a unique congruence, whose classes are the pre-images of the elements of $B$ (cf. Kernel of a function).

The intersection of a family of congruences $\pi_i$, $i \in I$, in the lattice of relations on a universal algebra (algebraic system) is a congruence. In general, a union of congruences in the lattice of relations is not a congruence. The product $\pi_1\pi_2$ of two congruences $\pi_1$ and $\pi_2$ is a congruence if and only if $\pi_1$ and $\pi_2$ commute, i.e. if and only if $\pi_1 \pi_2 = \pi_2 \pi_1$.

References

[1] A.G. Kurosh, "Higher algebra" , MIR (1972) (Translated from Russian into German) Zbl 0237.13001


Comments

References

[a1] P.M. Cohn, "Universal algebra" (rev. ed.), Reidel (1981) Zbl 0461.08001
How to Cite This Entry:
Congruence (in algebra). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Congruence_(in_algebra)&oldid=12653
This article was adapted from an original article by V.S. Malakhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article