On derived algebras and subvarieties of implication zroupoids

Abstract

In 2012, the second author introduced and studied in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012) the variety I of algebras, called implication zroupoids, that generalize De Morgan algebras. An algebra A= ⟨ A, → , 0 ⟩ , where → is binary and 0 is a constant, is called an implication zroupoid (I-zroupoid, for short) if A satisfies: (x→y)→z≈[(z′→x)→(y→z)′]′ and 0 ′ ′≈ 0 , where x′: = x→ 0. The present authors devoted the papers, Cornejo and Sankappanavar (Alegbra Univers, 2016a; Stud Log 104(3):417–453, 2016b. doi:10.1007/s11225-015-9646-8; and Soft Comput: 20:3139–3151, 2016c. doi:10.1007/s00500-015-1950-8), to the investigation of the structure of the lattice of subvarieties of I, and to making further contributions to the theory of implication zroupoids. This paper investigates the structure of the derived algebras Am: = ⟨ A, ∧ , 0 ⟩ and Amj: = ⟨ A, ∧ , ∨ , 0 ⟩ of A∈ I, where x∧y:=(x→y′)′ and x∨y:=(x′∧y as well as the lattice of subvarieties of I. The varieties I2 , 0, RD, SRD, C, CP, A, MC, and CLD are defined relative to I, respectively, by: (I2 , 0) x′ ′≈ x, (RD) (x→ y) → z≈ (x→ z) → (y→ z) , (SRD) (x→ y) → z≈ (z→ x) → (y→ z) , (C) x→ y≈ y→ x, (CP) x→ y′≈ y→ x′, (A) (x→ y) → z≈ x→ (y→ z) , (MC) x∧ y≈ y∧ x, (CLD) x→ (y→ z) ≈ (x→ z) → (y→ x). The purpose of this paper is two-fold. Firstly, we show that, for each A∈ I, Am is a semigroup. From this result, we deduce that, for A∈ I2 , 0∩ MC, the derived algebra Amj is a distributive bisemilattice and is also a Birkhoff system. Secondly, we show that CLD⊂ SRD⊂ RD and C⊂CP∩A∩MC∩CLD, both of which are much stronger results than were announced in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012).