On Regular Anti-congruence in Anti-ordered Semigroups

Daniel Romano

For an anti-congruence $q$ we say that it is regular anti-congruence on semigroup $(S,=,\neq,\cdot,\alpha)$ ordered under anti-order $\alpha$ if there exists an anti-order $\theta$ on $S/q$ such that the natural epimorphism is a reverse isotone homomorphism of semigroups. Anti-congruence $q$ is regular if there exists a quasi-antiorder $\sigma$ on $S$ under $\alpha$ such that $q=\sigma\cup\sigma^{-1}$. Besides, for regular anti-congruence $q$ on $S$, a construction of the maximal quasi-antiorder relation under $\alpha$ with respect to $q$ is shown.