$\mathfrak{m}$-Baer and $\mathfrak{m}$-Rickart lattices

TL;DR

This paper introduces and explores the concepts of Rickart and Baer lattices, extending module theory techniques to lattice theory, and characterizes these lattices via annihilators generated by idempotents.

Contribution

It defines $rak{m}$-Rickart and $rak{m}$-Baer lattices, connecting lattice properties with module theory through linear morphisms and annihilator characterizations.

Findings

$rak{m}$-Rickart and $rak{m}$-Baer lattices are characterized by annihilators generated by idempotents.

The theory of Rickart and Baer modules can be understood using lattice techniques.

A general approach using a submonoid $rak{m}$ of endomorphisms is developed.

Abstract

In this paper we introduce the notions of Rickart and Baer lattices and their duals. We show that part of the theory of Rickart and Baer modules can be understood just using techniques from the theory of lattices. For, we use linear morphisms introduced by T. Albu and M. Iosif. We focus on a submonoid with zero $\mathfrak{m}$ of the monoid of all linear endomorphism of a lattice $\mathcal{L}$ in order to give a more general approach and apply our results in the theory of modules. We also show that $\mathfrak{m}$-Rickart and $\mathfrak{m}$-Baer lattices can be characterized by the annihilators in $\mathfrak{m}$ generated by idempotents as in the case of modules.