Semi-Overlap Functions on Complete Lattices, Semi-Θ-Ξ Functions, and Inflationary MTL Algebras

As new kinds of aggregation functions, overlap functions and semi overlap functions are widely applied to information fusion, approximation reasoning, data classification, decision science, etc. This paper extends the semi-overlap function on [0, 1] to the function on complete lattices and investigates the residual implication derived from it; then it explores the construction of a semi-overlap function on complete lattices and some fundamental properties. Especially, this paper introduces a more generalized concept of the ‘semi-Θ-Ξ function’, which innovatively unifies the semi-overlap function and semi-grouping function. Additionally, it provides methods for constructing and characterizing the semi-Θ-Ξ function. Furthermore, this paper characterizes the semi-overlap function on complete lattices and the semi-Θ-Ξ function on [0, 1] from an algebraic point of view and proves that the algebraic structures corresponding to the inflationary semi-overlap function, the inflationary semi-Θ-Ξ function, and residual implications derived by each of them are inflationary MTL algebras. This paper further discusses the properties of inflationary MTL algebra and its relationship with non-associative MTL algebra, and it explores the connections between some related algebraic structures.