BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874, ISSN (o) 2303-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 7(2017), 293-303 DOI: 10.7251/BIMVI1702293R Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN 0354-5792 (o), ISSN 1986-521X (p) WEAKLY ISOTONE AND STRONGLY REVERSE ISOTONE MAPPINGS OF RELATIONAL SYSTEMS Daniel Abraham Romano Abstract. The setting of this article is Classical algebra and Bishop’s con- structive algebra (the algebra based on the Intuitionistic logic). The Esakia’s concept in the classical mathematics of strongly isotone mapping between or- dered sets is extended onto two different concepts of mappings: on the concept of weakly isotone and the concept of strongly reverse isotone mapping of rela- tional systems. Some characterizations of those mappings are given and some application of those mappings in ordered semigroup theory are given. 1. Introduction This investigation in Classical algebra and in Bishop’s constructive mathemat- ics (in sense of well-known books [1, 2, 3, 7, 8, 13] and papers [9, 10, 11]) is a continuation of the author’s paper [12]. Bishop’s constructive mathematics is develop on Constructive Logic / Intuition- istic Logic - logic without the Law of Excluded Middle P ∨¬P . Let us note that in the Intuitionistic Logic the ’Double Negation Law’ P ⇐⇒ ¬¬P does not hold, but the following implication P = ⇒ ¬¬P holds even in the Minimal Logic. Since the Intuitionistic Logic is a part of the Classical Logic, these results in the Constructive mathematics are compatible with suitable results in the Classical mathematics. Let us recall that the following deduction principle A ∨ B, ¬B ⊢ A is acceptable in the Intuitionistic Logic. Let (A, =, ̸=) be a set in the sense of books [1, 2, 3, 7, 8, 13] , where ’̸=’ is a binary relation on A which satisfies the following properties: 2010 Mathematics Subject Classification. 03F65, 08A02, 20M99. Key words and phrases. Constructive mathematics, relational system, set with apartness, co-quasiorder, (weakly reverse) isotone mapping. 293