Uninorms via two comparable closure operators on bounded lattices
Zhenyu Xiu, Xu Zheng
TL;DR
This paper generalizes uninorm construction to arbitrary bounded lattices by employing two comparable closure operators or two comparable interior operators, under necessary and sufficient conditions. The main approach uses a t-conorm on the top region $[e,1]$ (or a t-norm on $[0,e]$ in dual forms) to define a piecewise uninorm $U$ across lattice regions, with the operators controlling behavior in cross-regions; associativity and monotonicity are ensured only under specific operator constraints. The authors present two primary construction schemes and their duals, include degenerate single-operator variants, and show that the resulting uninorms can lie outside the previously studied classes $ mathcal{U}_{min}^{*}rac{U}{min}^{1}$ or $ mathcal{U}_{max}^{*}rac{U}{max}^{0}$, thereby expanding the design space for aggregation on lattices. They also connect certain instances to known results and discuss conditions under which the new constructions recover or relate to existing methods, suggesting further applications and extensions to ZX23 and other aggregation settings.
Abstract
In this paper, we propose novel methods for constructing uninorms using two comparable closure operators or, alternatively, two comparable interior operators on bounded lattices. These methods are developed under the necessary and sufficient conditions imposed on these operators. Specifically, the construction of uninorms for $(x ,y )\in ]0 ,e [\times]e ,1 [ \cup ]e ,1 [\times]0 ,e [$ depends not only on the structure of the bounded lattices but also on the chosen closure operators (or interior operators). Consequently, the resulting uninorms do not necessarily belong to $\mathcal{U}_{min}^{*}\cup \mathcal{U}_{min}^{1}$ (or $\mathcal{U}_{max}^{*}\cup\mathcal{U}_{max}^{0}$). Moreover, we present the degenerate cases of the aforementioned results, which are constructed using only a single closure operator or a single interior operator. Some of these cases correspond to well-known results documented in the literature.