On ordinal sums of triangular norms on bounded lattices

Authors:
Susanne Saminger
Affiliations:
Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, A-4040 Linz, Austria
Venue:
Fuzzy Sets and Systems
Year:
2006

Citing 5
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Abstract

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. By such 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.