Perception of BCC-algebras under the Bishops principled-philosophical orientation: BCC-algebra with apartness

Abstract

In this paper it is given a short introduction in a reconsideration about BCC-algebras under the light of Bishop's principled-philosophical orientation. At the first, it is introduced the concept of BCCalgebras into this orientation. Additionally, the consequences of the selected axioms in the determination of BCC-algebras with apartness are analyzed. Also, some substructures in the BCC-algebras with apartness that have no counterparts in the classical case and which appear as products of the chosen logical environment such as co-ideals are analyzed. At the end, two different results that can be viewed as the isomorphism theorem in for BCC-algebras are exposed. Definition 1.1 ([6]). By a BCC-algebra is a non-empty set X together with a binary internal operation ' •' and a distinguished element ' 0' such that the following axioms are satisfied: (BCC1) (∀x, y, z ∈ X)((y • z) • ((x • y) • (x • z)) = 0); (BCC2) (∀x ∈ X)(x • x = 0); (BCC3) (∀x ∈ X)(x • 0 = 0); (BCC4) (∀x ∈ X)(0 • x = x); and (BCC5) (∀x, y ∈ X)((x • y = 0 ∧ y • x = 0) =⇒ x = y). Remark 1.2. Let us observe that their form is coherent with the interpretation in which • stay for =⇒, 0 for and = for the semantical equivalence usually denoted by ≡.