Left(right) Semi-uninorm And Implication On A Complete Lattice

Uninorms are special aggregation operators that are important generalization of both t-norms and t-conorms, with the neutral elements lying anywhere in the unit interval [0, 1]. But there are real-life situations when truth functions can not be associative or commutative. By throwing away the associativity and commutativity from the axioms of uninorms, ones introduced the concept of left(right) uninorms, semi-uninorms, left(right) semi-uninorms and so on. In this paper, we study the left(right) semi-uninorms and implications on a complete lattice. This paper is divided into the following four parts:The ?rst chapter introduces the research background of this article and main conclusions in this paper.In Chapter 2, we introduce the concept of the left(right) semi-uninorms and implications on a complete lattice and illustrate these concepts by means of some examples.In Chapter 3, we study the relations between left(right) semi-uninorms and implications on a complete lattice. We ?rstly investigate the left(right) semiuninorms induced by implications. Then, we discuss the residual operations of left(right) semi-uninorms. Finally, we reveal the relationships between strict left(right)-conjunctive left(right) in?nitely ∨-distributive left(right) semi-uninorms and right in?nitely ∧-distributive implications which satisfy the order property.In Chapter 4, we further investigate the constructions of left(right) semiuninorms and implications on a complete lattice. We ?rstly give out the formulas for calculating the upper and lower approximation conjunctive left(right) semiuninorms of a binary operation and the upper and lower approximation strict left(right)-conjunctive left(right) semi-uninorms of a binary operation. Then, we lay bare the formulas for calculating the upper and lower approximation implications,which satisfy the neutrality principle and the order property, respectively, of a binary operation. Finally, we study the relations between the upper approximation conjunctive left(right) semi-uninorms and the lower approximation implications which satisfy the neutrality principle and investigate the relations between the upper approximation strict left(right)-conjunctive left(right) semi-uninorms and the lower approximation implications which satisfy the order property.