Trice-valued fuzzy sets: Mathematical model for three-way decisions

Abstract

Under the idea to develop a mathematical model for three-way decisions, the aim of the paper is to study trice-valued fuzzy sets, i.e., mappings from a set to a structure called a trice. A trice is a triple semilattice satisfying roundabout absorption laws, suitable for representing multi-dimensional orders, which appear in complex movements in a plane or in a space. Our approach is cutworthy, namely, we investigate cuts of such fuzzy sets and prove theorems of decomposition and synthesis. This new notion provides a possibility to capture vague triangular situations. Therefore, a motivation for our research is to provide a new algebraic and order-theoretic model for three-way decisions, as this topic has been introduced recently for solving particular human problems and for information processing.