ON REGULAR ANTI-CONGRUENCE
In Anti-Ordered Semigroups, Daniel A. Romano · 2007
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 anti- order θ on S/q such that the natural epimorphism is a reverse isotone ho- momorphism 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. This short investigation in Bishop's Constructive Algebra is a continuation of (9) and (10). Bishop's Constructive Mathematics is developed on Constructive Logic (11) - logic without the Law of Excluded Middle P ∨¬ P. Let us note that in the Constructive Logic the 'Double Negation Law' P ⇔¬ ¬P does not hold, but the following implication P ⇒¬ ¬P holds even in the Minimal Logic. We have to note that 'the crazy axiom' ¬P ⇒ (P ⇒ Q) is included in the Constructive Logic. In the Constructive Logic 'Weak Law of Excluded Middle' ¬P ∨¬ ¬P does not hold, too. It is interesting, that in the Constructive Logic the following deduction principle A ∨ B, AB holds, but this is impossible to prove without 'the crazy axiom'. Bishop's Constructive Mathematics is consistent with the Classical Mathematics. Relational structure (S, =, � where the relationis a binary relation on S, which satisfies the following properties: