On some operators and dimensions in modular meet-continuous lattices

Abstract

Given a complete modular meet-continuous lattice $A$, an inflator on $A$ is a monotone function $d\colon A\rightarrow A$such that $a\leq d(a)$ for all $a\in A$. If $I(A)$ is the set of all inflators on $A$, then $I(A)$ is a complete lattice. Motivated by preradical theory we introduce two operators, the totalizer and the equalizer. We obtain some properties of these operators and see how they are related to the structure of the lattice $A$ and with the concept of dimension.