Residuation in modular lattices and posets

TL;DR

This paper explores how complemented modular lattices and certain modular posets can be transformed into operator left residuated structures, revealing connections between lattice residuation and poset operators.

Contribution

It introduces methods to convert complemented modular lattices and modular posets into operator left residuated structures, expanding the understanding of residuation in these contexts.

Findings

Complemented modular lattices can be turned into left residuated lattices.

Strongly and strictly modular posets can be organized into operator left residuated posets.

Connections between poset operators and lattice residuation are established.

Abstract

We show that every complemented modular lattice can be converted into a left residuated lattice where the binary operations of multiplication and residuum are term operations. The concept of an operator left residuated poset was introduced by the authors recently. We show that every strongly modular poset with complementation as well as every strictly modular poset with complementation can be organized into an operator left residuated poset in such a way that the corresponding operators M(x,y) and R(x,y) can be expressed by means of the operators L and U in posets. We describe connections between the operator left residuation in these posets and the residuation in their lattice completion. We also present examples of strongly modular and strictly modular posets.