The Semiring Properties of Boolean Propositional Algebras

TL;DR

This paper explores the algebraic structure of boolean propositional algebras through the lens of semirings, analyzing homomorphisms, subalgebras, and partial orderings to deepen understanding of their properties.

Contribution

It establishes conditions for representing boolean propositional subalgebras as equivalent to the original algebra and characterizes homomorphic images in this context.

Findings

Conditions for boolean subalgebra representation

Characterization of homomorphic images

Necessary and sufficient conditions for onto-order preserving homomorphisms

Abstract

This paper illustrates the relationship between boolean propositional algebra and semirings, presenting some results of partial ordering on boolean propositional algebras, and the necessary conditions to represent a boolean propositional subalgebra as equivalent to a corresponding boolean propositional algebra. It is also shown that the images of a homomorphic function on a boolean propositional algebra have the relationship of boolean propositional algebra and its subalgebra. The necessary and sufficient conditions for that homomorphic function to be onto-order preserving, and also an extension of boolean propositional algebra, are explored.