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Strong NMV-algebras, commutative basic algebras and naBL-algebras

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Mathematica Slovaca

Abstract

The aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that

  1. (1)

    a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid)

  2. (2)

    a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra.

Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.

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Authors and Affiliations

  1. Department of Mathematics College of Arts and Sciences, Shanghai Maritime University, Shanghai, 201306, P. R. China

    Xiaohong Zhang

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  1. Xiaohong Zhang
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Correspondence to Xiaohong Zhang.

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Communicated by Jiří Rachůnek

This work was supported by National Natural Science Foundation of China (Grant No. 61175044) and Innovation Program of Shanghai Municipal Education Commission (No. 13ZZ122).

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Zhang, X. Strong NMV-algebras, commutative basic algebras and naBL-algebras. Math. Slovaca 63, 661–678 (2013). https://doi.org/10.2478/s12175-013-0126-1

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  • DOI: https://doi.org/10.2478/s12175-013-0126-1

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