Extending states on finite concrete logics

TL;DR

This paper investigates conditions for extending states in finite concrete logics, providing new extension theorems and insights relevant to quantum logic theory.

Contribution

It offers new criteria for state extensions in finite concrete logics and introduces an extension theorem for subadditive states.

Findings

Extensions exist in even-element subset logics under certain conditions

Signed extensions can be characterized in specific concrete logics

An extension theorem for subadditive states is established

Abstract

In this note we collect several observations on state extensions. They may be instrumental to anyone who pursues the theory of quantum logics. In particular, we find out when extensions (resp. signed extensions) exist in the "concrete" concrete logic of all even-element subsets of an even-element set. We also mildly add to the study of difference-closed logics by finding an extension theorem for subadditive states.