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Difference between revisions of "Verbal congruence"

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A congruence on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965601.png" /> which is expressible as the intersection of all congruences on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965602.png" /> whose factor algebras belong to some fixed variety of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965603.png" />-algebras. A congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965604.png" /> on an arbitrary algebraic system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965605.png" /> is said to be verbal if there exists a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965607.png" />-systems for which the canonical mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965608.png" /> is universal amongst the morphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v0965609.png" /> to algebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656010.png" />. A verbal congruence is a [[Fully-characteristic congruence|fully-characteristic congruence]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656011.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656012.png" />-system in some variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656013.png" />, then, conversely, any fully-characteristic congruence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656015.png" /> is a verbal congruence with respect to the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656016.png" /> generated by the factor system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096560/v09656017.png" />.
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A congruence on an algebra $\mathbf{A}$ which is expressible as the intersection of all congruences on $\mathbf{A}$ whose factor algebras belong to some fixed variety of $\Omega$-algebras. A congruence $\theta$ on an arbitrary algebraic system $(A,\Omega)$ is said to be verbal if there exists a variety $\mathfrak{M}$ of $\Omega$-systems for which the canonical mapping $\mathbf{A} \rightarrow \mathbf{A}/\theta$ is universal amongst the morphisms from $\mathbf{A}$ to algebras in $\mathfrak{M}$. A verbal congruence is a [[fully-characteristic congruence]]. If $\mathbf{F}$ is a free $\Omega$-system in some variety $\mathfrak{B}$, then, conversely, any fully-characteristic congruence $\eta$ in $\mathbf{F}$ is a verbal congruence with respect to the variety $\mathfrak{M}$ generated by the factor system $\mathbf{F}/\eta$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,  "Algebraic systems" , Springer  (1973)  (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.I. Mal'tsev,  "Algebraic systems" , Die Grundlehren der mathematischen Wissenschaften '''192''', Springer  (1973)  (Translated from Russian) {{ZBL|0266.08001}}</TD></TR>
 
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</table>
  
 
====Comments====
 
====Comments====
Cf. also [[Universal property|Universal property]]; [[Congruence (in algebra)|Congruence (in algebra)]].
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Cf. also [[Universal property]]; [[Congruence (in algebra)]].

Latest revision as of 19:23, 12 December 2015

A congruence on an algebra $\mathbf{A}$ which is expressible as the intersection of all congruences on $\mathbf{A}$ whose factor algebras belong to some fixed variety of $\Omega$-algebras. A congruence $\theta$ on an arbitrary algebraic system $(A,\Omega)$ is said to be verbal if there exists a variety $\mathfrak{M}$ of $\Omega$-systems for which the canonical mapping $\mathbf{A} \rightarrow \mathbf{A}/\theta$ is universal amongst the morphisms from $\mathbf{A}$ to algebras in $\mathfrak{M}$. A verbal congruence is a fully-characteristic congruence. If $\mathbf{F}$ is a free $\Omega$-system in some variety $\mathfrak{B}$, then, conversely, any fully-characteristic congruence $\eta$ in $\mathbf{F}$ is a verbal congruence with respect to the variety $\mathfrak{M}$ generated by the factor system $\mathbf{F}/\eta$.

References

[1] A.I. Mal'tsev, "Algebraic systems" , Die Grundlehren der mathematischen Wissenschaften 192, Springer (1973) (Translated from Russian) Zbl 0266.08001

Comments

Cf. also Universal property; Congruence (in algebra).

How to Cite This Entry:
Verbal congruence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Verbal_congruence&oldid=13547
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article