Implication in finite posets with pseudocomplemented sections

TL;DR

This paper extends the concept of pseudocomplemented lattices to posets with top elements, introducing an 'unsharp' implication that generalizes intuitionistic logic semantics and explores its properties and structure.

Contribution

It introduces a new 'unsharp' implication for posets with top elements, generalizing pseudocomplementation and providing a basis for a more general intuitionistic logic.

Findings

The 'unsharp' implication is not necessarily unique but is maximal.

The implication determines the structure of the poset.

Introduction of the operator  leads to an 'unsharply' residuated poset.

Abstract

It is well-known that relatively pseudocomplemented lattices can serve as an algebraic semantics of intuitionistic logic. To extend the concept of relative pseudocomplementation to non-distributive lattices, the first author introduced so-called sectionally pseudocomplemented lattices, i.e. lattices with top element 1 where for every element y the interval [y,1], the so called section, is pseudocomplemented. We extend this concept to posets with top element. Our goal is to show that such a poset can be considered as an algebraic semantics for a certain kind of more general intuitionistic logic provided an implication is introduced as shown in the paper. We prove some properties of such an implication. This implication is "unsharp" in the sense that the value for given entries need not be a unique element, but may be a subset of the poset in question. On the other hand, all of these values are as high as possible. We show that this implication even determines the poset and if a new operator \odot is introduced in an "unsharp" way, such structure forms an "unsharply" residuated poset.