On some questions concerning the axiomatisation of WNM-algebras and their subvarieties

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Authors:
Stefano Aguzzoli
Matteo Bianchi
Title: On some questions concerning the axiomatisation of WNM-algebras and their subvarieties
Journal: Fuzzy Sets and Systems
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Number
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Year: 2014
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Abstract

In a seminal paper Esteva and Godo introduced monoidal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle t} -norm-based logic MTL and some of its prominent extensions such as NM and WNM. We notice that NM is axiomatisable from IMTL, and hence MTL, with one-variable axioms, by instantiating the WNM axiom over one variable. This observation leads us here to study the logic axiomatised by extending MTL by this one-variable axiom. We shall refer to its equivalent algebraic semantics as the variety of GHP-algebras, for those algebras will be shown to form the largest variety of MTL-algebras such that the falsum-free reducts of the positive cones of their chains are the most general totally ordered Gödel hoops. Among other results we obtain a general description of GHP standard algebras, and use the latter to characterise those extensions of WNM that can be obtained from GHP via the same set of extending axioms.