On triangular norm based axiomatic extensions of the weak nilpotent minimum logic
In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic
(WNM) and its t‐norm based axiomatic extensions. We consider the algebraic counterpart of
WNM, the variety of WNM‐algebras (𝕎ℕ𝕄) and prove that it is locally finite, so all its
subvarieties are generated by finite chains. We give criteria to compare varieties generated
by finite families of WNM‐chains, in particular varieties generated by standard WNM‐chains,
or equivalently t‐norm based axiomatic extensions of WNM, and we study their standard …
(WNM) and its t‐norm based axiomatic extensions. We consider the algebraic counterpart of
WNM, the variety of WNM‐algebras (𝕎ℕ𝕄) and prove that it is locally finite, so all its
subvarieties are generated by finite chains. We give criteria to compare varieties generated
by finite families of WNM‐chains, in particular varieties generated by standard WNM‐chains,
or equivalently t‐norm based axiomatic extensions of WNM, and we study their standard …
Abstract
In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic (WNM) and its t‐norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM‐algebras (𝕎ℕ𝕄) and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM‐chains, in particular varieties generated by standard WNM‐chains, or equivalently t‐norm based axiomatic extensions of WNM, and we study their standard completeness properties. We also characterize the generic WNM‐chains, i. e. those that generate the variety 𝕎ℕ𝕄, and we give finite axiomatizations for some t‐norm based extensions of WNM. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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