Ordinal sum combinations of continuous t-norms and their related ordered algebraic structures

Abstract

Recently, Vemuri et al. (2023) [15] proposed two operators for constructing new continuous t-norms with the intent of integrating the ordinal sum structures of given ones. Unfortunately, these two operators on non-trivial ordinal sum t-norms are not well-defined so that almost all the results in that part as well as some ones of the following papers are incorrect. The purpose of this paper is to revisit such topic, to redefine a pair of combination operations and to study their related ordered algebraic structures. It is proved that our redefined operations both satisfy the laws of commutativity, associativity, idempotency, and finer absorption, and take the minimum t-norm as the absorption and neutral elements, respectively. Since these operations will finerize in general the subintervals of the underlying t-norms, it is impossible to make the set of all continuous t-norms a lattice under these two operations, and then we turn to find the sufficient and necessary conditions for these operations to be lattice operations and get finally many distributive sublattices of continuous t-norms. Among four possible pairs of ordinal sum combination operations on continuous t-norms, the pair defined in the text matches every well and induces the richest distributive sublattice structures. Lastly, our redefined operations are applied to the contexts of $(S,N)$-implications and residual implications with continuous t-conorms and t-norms, respectively.