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A general method for constructing left-continuous t-norms. (English) Zbl 1020.03020
Summary: A new method for constructing (left-continuous) t-norms is introduced and analyzed in this paper. We construct via embedding a left-continuous t-norm from any countable residuated totally and densely ordered commutative integral monoid. Moreover, we can construct a left-continuous t-norm from any countable, totally ordered, commutative integral monoid which is not necessarily densely ordered and residuated. A special case, the embedding of such monoids on lexicographic product spaces, is investigated in detail, and several examples are demonstrated. The results shed some light on Chang’s MV-algebras, on a recently proposed ‘extraordinary’ t-norm, and on the standard semantics of the recently introduced logic $$\Pi$$-MTL of Hájek.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness
##### Keywords:
residuated lattice; conjunction; left-continuous t-norm
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##### References:
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