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Saminger, Susanne

On ordinal sums of triangular norms on bounded lattices. (English) Zbl 1099.06004

Fuzzy Sets Syst. 157, No. 10, 1403-1416 (2006).
Summary: Ordinal sums have been introduced in many different contexts, e.g., for posets, semigroups, t-norms, copulas, aggregation operators, or quite recently for hoops. In this contribution, we focus on ordinal sums of t-norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets. Necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t-norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t-norms. In this way also the structure of the underlying bounded lattice is investigated. Further, it is shown that up to trivial cases there are no ordinal sum t-norms on product lattices in general.

Cited in 1 Review
Cited in 47 Documents

MSC:

06B05 Structure theory of lattices
03E72 Theory of fuzzy sets, etc.

Keywords:

triangular norm; ordinal sum; horizontal sum; bounded lattice
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\textit{S. Saminger}, Fuzzy Sets Syst. 157, No. 10, 1403--1416 (2006; Zbl 1099.06004)
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