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\iteman{io-port 01604170}
\itemau{Ershad, M.}
\itemti{Semigroups over which no automaton has proper essential congruences.}
\itemso{Ann. Univ. Sci. Budap. Rolando E\H{o}tv\H{o}s, Sect. Math. 42, 9-12 (1999).}
\itemab
Summary: Let $S$ be a semigroup and $A$ an $S$-automaton (throughout the paper, automata are understood as right automata). A congruence $\sigma$ on $A$ is said to be essential if for every congruence $\alpha\ne\iota$ (the identity relation) on $A$ we have $\alpha\cap\sigma\ne\iota$. It follows from the definitions that an $S$-automaton $B$ is an essential extension of an $S$-automaton $A$ if and only if the Rees congruence of $B$ defined by $A$ is essential. In the present paper we restrict our consideration to the case where the Rees congruences are replaced by all congruences, and prove that, for a semigroup $S$, no $S$-automaton has proper essential right congruences if and only if $S$ is a finite chain considered as a semilattice.
\itemrv{~}
\itemcc{}
\itemut{automata; semigroups; essential congruences; Rees congruences; finite chains}
\itemli{}
\end