SEMILATTICE ORDERED SEMI-RINGS WITH APARTNESS Daniel Abraham Romano Abstract. The logical environment of this article are the Intuitionistic logic - the logic without the principle TND and the Bishop’s constructive algebra. In this paper, we introduce and analyses the concept of semilattice ordered semi-rings with apartness. In addition we introduce and analyse concepts of co-ideals and co-filters in such semi-rings. The construction of these concepts, as duals of ideals and filters in semi-rings, enable the cancellativity of co- quasiorder relation in such semi-ring with respect to the apartness. 1. Introduction Our setting is Bishop’s constructive mathematics ([1], [2], [3], [4], [7] and [21]), mathematics developed with Constructive logic (or Intuitionistic logic [21]) - logic without the Law of Excluded Middle P ∨¬P [TND]. We have to note that ’the crazy axiom’ ¬P = ⇒ (P = ⇒ Q) is included in the Constructive logic. Precisely, in Constructive logic the ’Double Negation Law’ P ⇐⇒ ¬¬P does not hold, but the following implication P = ⇒ ¬¬P holds even in Minimal logic. In Constructive logic ’Weak Law of Excluded Middle’ ¬P ∨ ¬¬P does not hold as well. It is interesting, in Constructive logic the following deduction principle A ∨ B, ¬A ⊢ B holds, but this is impossible to prove without ’the crazy axiom’. Dual of the equality relations ’=’ in a set A is diversity relation ’̸=’. This last relation is extensive in terms of equality in the following sense: = ◦̸ = ⊈ = and ̸= ◦ = ⊈ =. It is obvious that the following connection between these relations is valid: = ⊆¬̸ =. In this case for relations = and ̸= we say that they are associate. So, it’s quite natural to ask the question: Is there the maximal relation ’̸=’ such that it is associated with equality ’=’ ? Generally speaking: Let S be a subset of set (A, =, ̸=) determined by a pred- icate P . The first task is to construct a dual T of the set S so that the subsets 2010 Mathematics Subject Classification. 03F65; 16Y60. Key words and phrases. Bishop’s constructive mathematics, Intuitionistic logic, semilattice- orderes semiring with apartness. 1