Verbal congruence

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 $\mathbf{F}/\eta$.

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