Quasimodules over bounded lattices

TL;DR

This paper introduces the concept of quasimodules over bounded lattices, exploring their structure, properties, and the lattice of closed subquasimodules, extending module theory to non-distributive lattices.

Contribution

It defines quasimodules over bounded lattices, develops their properties, and analyzes the lattice of closed subquasimodules, including orthogonality and splitting concepts.

Findings

The set of all closed subquasimodules forms a complete lattice.

Splitting subquasimodules are always closed, and their orthogonal companions are also splitting.

The paper provides several illustrative examples.

Abstract

We define a quasimodule Q over a bounded lattice L in an analogous way as a module over a semiring is defined. The essential difference is that L need not be distributive. Also for quasimodules there can be introduced the concepts of inner product, orthogonal elements, orthogonal subsets, bases and closed subquasimodules. We show that the set of all closed subquasimodules forms a complete lattice having orthogonality as an antitone involution. Using the Galois connection induced by this orthogonality, we describe important properties of closed subquasimodules. We call a subquasimodule P of a quasimodule Q splitting if the sum of P and its orthogonal companion is the whole set Q and the intersection of P and its orthogonal companion is trivial. We show that every splitting subquasimodule is closed and that its orthogonal companion is splitting, too. Our results are illuminated by several examples.