Generation of continuous T-norms through latticial operations

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 (sic), (sic) on the set T-CA of continuous Archimedean t-norms and obtain, via these binary operations, a partial order relation (sic), different from the usual point-wise order <=, on the set T-CA. 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 (R), (R) on the set T-CA, two binary operations circle plus, circle times on the set T-C of continuous t-norms are proposed and showed that (T-C, circle plus, circle times) 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. (c) 2022 Elsevier B.V. All rights reserved.