On regular anti-congruence in anti-ordered semigroups

Abstract

For an anti-congruence q we say that it is regular anti-congruence on semigroup (S,=, _=, ·, α) ordered under anti-order α if there exists an antiorder θ 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 σ on S under α such that q = σ ∪ σ−1. Besides, for regular anti-congruence q on S, a construction of the maximal quasi-antiorder relation under α with respect to q is shown.