# Lattice of closure endomorphisms of a Hilbert algebra

## TL;DR

This paper investigates the structure of closure endomorphisms in Hilbert algebras, revealing their lattice properties, isomorphisms with filter lattices, and implications for algebraic extensions.

## Contribution

It establishes the isomorphism between the lattice of closure endomorphisms and certain filter lattices, and explores conditions for algebraic isomorphisms and extensions.

## Key findings

- The lattice of closure endomorphisms is distributive and isomorphic to filter lattices.
- Compact elements of the closure endomorphism lattice correspond to the adjoint semilattice.
- Conditions for isomorphism of Hilbert algebras via their adjoint semilattices are characterized.

## Abstract

A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, anti-isomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.