Partial Residuated Implications Derived from Partial Triangular Norms and Partial Residuated Lattices

TL;DR

This paper explores the connections between fuzzy logic and quantum logic by studying partial residuated implications derived from partial t-norms and lattices, expanding theoretical understanding of these algebraic structures.

Contribution

It introduces the concepts of partial residuated implications, partial fuzzy implications, and partial residuated lattices, establishing their properties and relationships with lattice effect algebras.

Findings

Partial operations in commutative quasiresiduated lattices are partial t-norms.

Each partial residuated implication is a partial fuzzy implication.

The paper defines well partial residuated lattices and studies their properties.

Abstract

In this paper, we reveal some relations between fuzzy logic and quantum logic, and mainly study the partial residuated implications (PRIs) derived from partial triangular norms (partial t-norms) and partial residuated lattices (PRLs), and expand some results in the article "material implication in lattice effect algebra". Firstly, according to the concept of partial triangular norms given by Borzooei, we introduce the connection between lattice effect algebra and partial t-norms, and prove that partial operations in any commutative quasiresiduated lattice are partial t-norms. Secondly, we give the general form of partial residuated implications and the concept of partial fuzzy implications (PFIs), and the condition that partial residuated implication is a fuzzy implication is given. We also prove that each partial residuated implication is a partial fuzzy implication. Thirdly, we propose the partial residuated lattice and study their basic properties, to discuss the corresponding relationship between PRLs and lattice effect algebras (LEAs), to further reveal the relationship between LEAs and residuated partial algebras. In addition, like the definition of partial t-norms, we also propose the concepts of partial triangular conorms (partial t-conorms) and corresponding partial co-residuated lattices (PcRLs). Finally, based on partial residuated lattices, we give the definition of well partial residuated lattices (wPRLs), study the filter of well partial residuated lattices, and then construct quotient structure of partial residuated lattices.