A mixed logic with binary operators

TL;DR

This paper introduces weak mixed algebras with binary operators of possibility and sufficiency, develops a corresponding logic system, and proves an embedding theorem addressing an open problem from previous betweenness algebra research.

Contribution

It extends betweenness algebra theory to a larger class of algebras and establishes a sound and complete logic system with an embedding theorem.

Findings

Development of weak mixed algebras with new binary operators

A sound and complete logic system for these algebras

An embedding theorem solving an open problem

Abstract

In previous work "Betweenness algebras" we introduced and examined the class of betweenness algebras. In the current paper we study a larger class of algebras with binary operators of possibility and sufficiency, the weak mixed algebras. Furthermore, we develop a system of logic with two binary modalities, sound and complete with respect to the class of frames closely related to the aforementioned algebras, and we prove an embedding theorem which solves an open problem from "Betweenness algebras".