Congruences on Regular Semigroups

Abstract

Let $S$ be a regular semigroup and let $\rho$ be a congruence relation on $S$. The kernel of $\rho$, in notation $\ker \rho$, is the union of the idempotent $\rho$-classes. The trace of $\rho$, in notation $\operatorname {tr} \rho$, is the restriction of $\rho$ to the set of idempotents of $S$. The pair $(\ker \rho ,\operatorname {tr} \rho )$ is said to be the congruence pair associated with $\rho$. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple $((\rho \vee \mathcal {L})/\mathcal {L},\ker \rho ,(\rho \vee \mathcal {R})/\mathcal {R})$ is said to be the congruence triple associated with $\rho$. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of $S$. For congruence relations $\rho$ and $\theta$, put $\rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ]$ if and only if $\rho \vee \mathcal {L} = \theta \vee \mathcal {L}\;[\rho \vee \mathcal {R} = \theta \vee \mathcal {R},\operatorname {tr}\rho = \operatorname {tr}\theta ]$. Then ${T_l},{T_r}$ and $T$ are complete congruences on the congruence lattice of $S$ and $T = {T_l} \cap {T_r}$.