PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE (BEOGRAD) (N.S.) Vol. 53(67), pp. 3744 (1993) 

Completely regular and orthodox congruences on regular semigroupsBranka P. Alimpi\'c and Dragica N. Krgovi\'cMatematicki fakultet, Beograd, Yugoslavia and Matematicki institut SANU, Beograd, YugoslaviaAbstract: Let $S$ be a regular semigroup and $E(S)$ the set of all idempotents of $S$. Let $\operatorname{Con}} S$ be the congruence lattice of $S$, and let $T$, $K$, $U$ and $V$ be equivalences on $\operatorname{Con}} S$ defined by $\rho T\xi \Leftrightarrow \tr\rho = \tr\xi$,\ $\rho K\xi \Leftrightarrow \ker \rho = \ker \xi$, $\rho U\xi \Leftrightarrow \rho \cap \leq = \xi\, \cap \leq$ and $V = U\cap K$, where $\tr\rho = \rho \mid_{E(S)}$,\enskip $\ker \rho = E(S)\rho$, and $\leq$ is the natural partial order on $E(S)$. It is known that $T$, $U$ and $V$ are complete congruences on $\operatorname{Con}} S$ and $T$, $K$, $U$ and $V$classes are intervals $[\rho_T,\rho^T]$, $[\rho_K,\rho^K]$, $[\rho_U,\rho^U]$, and $[\rho_V,\rho^V]$, respectively ([13], [10], [9]). In this paper $U$classes for which $\rho^U$ is a semilattice congruence, and $V$classes for which $\rho^V$ is an inverse congruence are considered. It turns out that the union of all such $U$classes is the lattice CR$\operatorname{Con}} S$ of all completely regular congruences on $S$, and the union of all such $V$classes is the lattice O$\operatorname{Con}} S$ of all orthodox congruences on $S$. Also, some complete epimorphisms of the form $\rho \to \rho^U$ and $\rho \to \rho^V$ are obtained. Classification (MSC2000): 20M17 Full text of the article:
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