Namespaces
Variants
Actions

Verbal congruence

From Encyclopedia of Mathematics
Revision as of 19:23, 12 December 2015 by Richard Pinch (talk | contribs) (→‎References: expand bibliodata)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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=36896
This article was adapted from an original article by D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article