General interval-valued overlap functions and interval-valued overlap indices

General Complex-Valued Grouping Functions

“…Proposition 1. Let g: D n ⟶ D be an n-ary mapping, then (1) If it is an n-dimensional complex-valued grouping function, then it is an n-dimensional complex-valued 0-grouping and 1-grouping function (2) If it is an n-dimensional complex-valued 0-grouping (or 1-grouping) function, then it is also a general complex-valued grouping function is relation between complex-valued grouping functions is similar to that between interval-valued overlap functions [9]. Now, we give several examples to demonstrate their relations of complex-valued grouping functions.…”

“…In the literature, one can find many works on realvalued grouping functions. e concepts of general grouping functions [8] and interval-valued grouping functions [9,10] have been proposed. Some properties incluing migrativity, homogeneity, idempotency, and distributivity of grouping functions have been studied [1,[11][12][13][14].…”

General quasi-overlap functions on lattices

“…The operation / on the set X is known as right division (that is, 16]). If (X, * ) is a quasigroup which is associative, then (X, * ) necessarily has a unique identity element 4 .…”

“…Here, a groupoid is an algebraic structure (X, * ) where X is a non-empty set and * is a binary function defined on X. 4 An element e is a left (right) identity element for quasigroup (X, * ) means that Le(x) = x (Re(x) = x) ∀x ∈ X.…”

“…As one kind of binary general overlap functions, in [13] the authors extended the notion of BL-algebras, which are the algebraic counterpart of the a type of fuzzy logic modeled by Peter Hájek. This class of functions offers a promising field of research [2,4,8,14,15]. In [14], Paiva et al introduced the concept of quasioverlap functions on bounded lattices and investigated some vital properties of them.…”