On some operators and dimensions in modular meet-continuous lattices

TL;DR

This paper explores the properties of inflators on complete modular meet-continuous lattices and introduces new operators, the totalizer and equalizer, to analyze their structure and dimension.

Contribution

It introduces the totalizer and equalizer operators for inflators, linking them to lattice structure and dimension in modular meet-continuous lattices.

Findings

Properties of the totalizer and equalizer operators are established.

Relations between these operators and lattice structure are analyzed.

Connections to the concept of dimension are explored.

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.