# Differences

This shows you the differences between two versions of the page.

— |
definable_principal_congruences [2010/08/20 20:21] (current) jipsen created |
||
---|---|---|---|

Line 1: | Line 1: | ||

+ | =====Definable principle congruences===== | ||

+ | |||

+ | A (quasi)variety $\mathcal{K}$ of algebraic structures has \emph{first-order definable principal (relative) congruences} (DP(R)C) if | ||

+ | there is a first-order formula $\phi(u,v,x,y)$ such that for all | ||

+ | $\mathbf{A}\in\mathcal{K}$ we have $\langle x,y\rangle\in\mbox{Cg}_{\mathcal{K}}(u,v)\iff \mathbf{A}\models \phi(u,v,x,y)$. | ||

+ | |||

+ | Here | ||

+ | $\theta=\mbox{Cg}_{\mathcal{K}}(u,v)$ denotes the smallest (relative) congruence that identifies the elements | ||

+ | $u,v$, where "relative" means that $\mathbf{A}//\theta\in\mathcal{K}$. | ||

+ | |||

+ | === Properties that imply DP(R)C === | ||

+ | |||

+ | [[Equationally def. pr. cong.|Equationally definable principal (relative) congruences]] | ||

+ | |||

+ | === Properties implied by DP(R)C === | ||

Trace: