Difference between revisions of "Verbal congruence"
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| − | A congruence on an algebra | + | {{TEX|done}} |
| + | 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> | + | <table> |
| − | + | <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> | |
| − | + | </table> | |
====Comments==== | ====Comments==== | ||
| − | Cf. also [[ | + | 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
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