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

Abstract

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 $$\mathfrak {m}$$ m of the monoid of all linear endomorphism of a lattice $$\mathcal {L}$$ L in order to give a more general approach and apply our results in the theory of modules. We also show that $$\mathfrak {m}$$ m -Rickart and $$\mathfrak {m}$$ m -Baer lattices can be characterized by the annihilators in $$\mathfrak {m}$$ m generated by idempotents as in the case of modules.