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 \({\mathcal {I}}\) of algebras, called implication zroupoids, that generalize De Morgan algebras. An algebra \({\mathbf {A}} = \langle A, \rightarrow , 0 \rangle \), where \(\rightarrow \) is binary and 0 is a constant, is called an implication zroupoid (\({\mathcal {I}}\)-zroupoid, for short) if \({\mathbf {A}}\) satisfies: \((x \rightarrow y) \rightarrow z \approx [(z' \rightarrow x) \rightarrow (y \rightarrow z)']'\) and \( 0'' \approx 0\), where \(x' : = x \rightarrow 0\). 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 \({\mathcal {I}}\), and to making further contributions to the theory of implication zroupoids. This paper investigates the structure of the derived algebras \(\mathbf {A^{m}} := \langle A, \wedge , 0 \rangle \) and \(\mathbf {A^{mj}} :=\langle A, \wedge , \vee , 0 \rangle \) of \({\mathbf {A}} \in {\mathcal {I}}\), where \(x \wedge y := (x \rightarrow y')'\) and \(x \vee y := (x' \wedge y')'\), as well as the lattice of subvarieties of \({\mathcal {I}}\). The varieties \({\mathcal {I}}_{2,0}\), \({{\mathcal {R}}}{{\mathcal {D}}}\), \(\mathcal {SRD}\), \({\mathcal {C}}\), \({{\mathcal {C}}}{{\mathcal {P}}}\), \({\mathcal {A}}\), \({{\mathcal {M}}}{{\mathcal {C}}}\), and \(\mathcal {CLD}\) are defined relative to \({\mathcal {I}}\), respectively, by: (I\(_{2,0}\)) \(x'' \approx x\), (RD) \((x \rightarrow y) \rightarrow z \approx (x \rightarrow z) \rightarrow (y \rightarrow z)\), (SRD) \((x \rightarrow y) \rightarrow z \approx (z \rightarrow x) \rightarrow (y \rightarrow z)\), (C) \( x \rightarrow y \approx y \rightarrow x\), (CP) \( x \rightarrow y' \approx y \rightarrow x'\), (A) \((x \rightarrow y) \rightarrow z \approx x \rightarrow (y \rightarrow z)\), (MC) \(x \wedge y \approx y \wedge x\), (CLD) \(x \rightarrow (y \rightarrow z) \approx (x \rightarrow z) \rightarrow (y \rightarrow x)\). The purpose of this paper is two-fold. Firstly, we show that, for each \({\mathbf {A}} \in {\mathcal {I}}\), \({\mathbf {A}}^{\mathbf {m}}\) is a semigroup. From this result, we deduce that, for \({\mathbf {A}} \in {\mathcal {I}}_{2,0} \cap {{\mathcal {M}}}{{\mathcal {C}}}\), the derived algebra \(\mathbf {A^{mj}}\) is a distributive bisemilattice and is also a Birkhoff system. Secondly, we show that \(\mathcal {CLD} \subset \mathcal {SRD} \subset {{\mathcal {R}}}{{\mathcal {D}}}\) and \({\mathcal {C}} \subset \ {{\mathcal {C}}}{{\mathcal {P}}} \cap {\mathcal {A}} \cap {{\mathcal {M}}}{{\mathcal {C}}} \cap \mathcal {CLD}\), both of which are much stronger results than were announced in Sankappanavar (Sci Math Jpn 75(1):21–50, 2012).