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On derived algebras and subvarieties of implication zroupoids. (English) Zbl 1381.06004

Summary: In [Sci. Math. Jpn. 75, No. 1, 21–50 (2012; Zbl 1279.06009)], the second author introduced and studied 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 [Algebra Univers. 77, No. 2, 125–146 (2017; Zbl 1421.06002); Stud. Log. 104, No. 3, 417–453 (2016; Zbl 1392.06011); Soft Comput. 20, No. 8, 3139–3151 (2016; Zbl 1373.06015)] 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: \((\mathrm{I}_{2,0})x'' \approx x\), (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 [loc. cit.].

MSC:

06D75 Other generalizations of distributive lattices
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
03G25 Other algebras related to logic
08B05 Equational logic, Mal’tsev conditions
08B15 Lattices of varieties

Software:

Mace4; Prover9
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Full Text: DOI

References:

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