Generating methods of some classes of uninorms and associated structures

Abstract

Uninorms are an important class of fuzzy logic connectives that have, so far, neither generating methods nor algebraic structures. In this work, we attempt to propose some novel generating methods of some well established classes of uninorms (with a fixed but arbitrary neutral element e) and investigate the algebraic structures that these generating methods can impose on those specific classes of uninorms. Towards this, we show that the operations point-wise maximum (∨) and the point-wise minimum (∧) on the set of idempotent uninorms are closed and make it a distributive lattice. Also, we show that the operations ∧ and ∨ on some sub-classes of idempotent uninorms with other classes of uninorms generate uninorms. Since the operations ∨ and ∧ on the set of uninorms are not closed in general, we propose some novel generating methods of uninorms that come from various classes such as representable uninorms, pseudo-continuous uninorms, uninorms continuous in the open unit square, uninorms with underlying continuous Archimedean t-norms and t-conorms and for each class of these uninorms, we investigate the algebraic structure imposed on it by the proposed generating methods with the help of results obtained recently in Vemuri et al. (2022) [51].