2018/08/30 by Thomas Vetterlein, Vetterlein, Thomas · 1 citation
Computer Science · Decision Sciences · #Advanced Algebra and Logic #Fuzzy and Soft Set Theory #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.1808.10324
We present in this paper a universal method of constructing left-continuous triangular norms (l.-c. t-norms). The starting point is an arbitrary, possibly finite, totally ordered monoid fulfilling the conditions that are characteristic for l.-c. t-norms: commutativity, negativity, and quanticity. We show that, under suitable conditions, we can extend this structure by substituting each element for a real interval. The process can be iterated and if the final structure obtained in this way is order-isomorphic to a closed real interval, its monoidal operation can, up to isomorphism, be identified with a l.-c. t-norm. We specify the constituents needed for the construction in an explicit way. We furthermore illustrate the method on the basis of a number of examples.