On a Generalization of Heyting Algebras I

TL;DR

This paper introduces $ abla$-algebras, a broad generalization of Heyting algebras, and explores their algebraic properties, including subdirect irreducibility, closure properties, and representations, laying groundwork for future logical and duality studies.

Contribution

It characterizes the structure of $ abla$-algebras, proves their closure under Dedekind-MacNeille completion, and develops canonical and Kripke representations, advancing the algebraic theory of these structures.

Findings

Characterization of subdirectly irreducible and simple $ abla$-algebras

Closure of $ abla$-algebra varieties under Dedekind-MacNeille completion

Development of canonical and Kripke representations for $ abla$-algebras

Abstract

$\nabla$-algebra is a natural generalization of Heyting algebra, unifying many algebraic structures including bounded lattices, Heyting algebras, temporal Heyting algebras and the algebraic presentation of the dynamic topological systems. In a series of two papers, we will systematically study the algebro-topological properties of different varieties of $\nabla$-algebras. In the present paper, we start with investigating the structure of these varieties by characterizing their subdirectly irreducible and simple elements. Then, we prove the closure of these varieties under the Dedekind-MacNeille completion and provide the canonical construction and the Kripke representation for $\nabla$-algebras by which we establish the amalgamation property for some varieties of $\nabla$-algebras. In the sequel of the present paper, we will complete the study by covering the logics of these varieties and their corresponding Priestley-Esakia and spectral duality theories.