Generation of continuous T-norms through latticial operations

Abstract

It is well known that the usual point-wise ordering over the set T of t-norms makes it a poset but not a lattice, i.e., the point-wise maximum or minimum of two t-norms need not always be a t-norm again. In this work, we propose, two binary operations ▪ on the set TCA of continuous Archimedean t-norms and obtain, via these binary operations, a partial order relation ⊑, different from the usual point-wise order ≤, on the set TCA. As an interesting outcome of this structure, some stronger versions of some existing results dealing with the upper and lower bounds of two continuous Archimedean t-norms with respect to the point-wise order ≤ are also obtained. Finally, with the help of the operations ▪ on the set TCA, two binary operations ⊕,⊗ on the set TC of continuous t-norms are proposed and showed that (TC,⊕,⊗) is a lattice. Thus we have both a way of generating continuous t-norms from continuous t-norms and also obtain an order on them.