Generalized t-norm structures

Some methods to obtain t-norms and t-conorms on bounded lattices

“…Definition 2.9. (Drossos and Navara [13], Drossos [14]) Let (L, ≤, 0, 1) be a bounded lattice. A mapping int : L → L is called an interior operator on L if it satisfies the following properties:…”

“…Based on the constructions of t-norms and t-conorms on the unit interval [0, 1], the study of the constructions of t-norms and t-conorms defined on bounded lattices has recently become important. In the papers [13,14,10,17,24,25], t-norms and t-conorms defined on bounded lattices were investigated and several methods for constructing these operators were introduced. Furthermore, by using the existence of a priori given t-norm and t-conorm, some ordinal sum constructions for t-norms and t-conorms on bounded lattices were presented in [10,17,24,25].…”

“…There are many initiatives for this construction method extended for bounded lattices based on Goguen's proposal [18] that considers fuzzy sets with membership values from bounded lattices. In [13,14], t-norms and tconorms defined as operations on a bounded poset or lattice were investigated. In the meantime, a class of t-norms and t-conorms on any bounded lattice was generated by use of interior operators and closure operators, correspondingly.…”

Ambiguous representations as fuzzy relations between sets

“…The simplest such " * " is the lattice meet "∧". See also [1] for more information on such operations.…”

Adjoints and Monads Related to Compact Lattices and Compact Lawson Idempotent Semimodules

“…If * is also commutative, then * is a generalized triangular norm (t-norm) [4] on L. Nevertheless, we do not need the commutativity of * in this chapter.…”