Lattice induced threshold functions and Boolean functions

TL;DR

This paper explores the relationship between lattice induced threshold functions and Boolean functions, proving their equivalence and providing representations for lattice valued up-sets on finite Boolean lattices.

Contribution

It establishes the equivalence between isotone Boolean functions and lattice induced threshold functions, and characterizes their representations using cuts and closure systems.

Findings

Every isotone Boolean function is a lattice induced threshold function.

Lattice valued up-sets can be represented via cuts and threshold functions.

Necessary and sufficient conditions for representing lattice valued up-sets are provided.

Abstract

Lattice induced threshold function is a Boolean function determined by a particular linear combination of lattice elements. We prove that every isotone Boolean function is a lattice induced threshold function and vice versa. We also represent lattice valued up-sets on a finite Boolean lattice in the framework of cuts and lattice induced threshold functions. In terms of closure systems we present necessary and sufficient conditions for a representation of lattice valued up-sets on a finite Boolean lattice by linear combinations of elements of the co-domain lattice.