The Riesz-Kantorovich formulas for $\mathbb{L}$-vector lattices

TL;DR

This paper establishes Riesz-Kantorovich formulas for order bounded module homomorphisms in the setting of Dedekind complete unital $f$-algebras and $ ext{L}$-vector lattices, expanding the theoretical framework of vector lattice operators.

Contribution

It proves the Riesz-Kantorovich formulas for a broad class of $ ext{L}$-module homomorphisms under mild conditions, extending existing results in vector lattice theory.

Findings

Riesz-Kantorovich formulas are valid for order bounded $ ext{L}$-module homomorphisms.

The formulas hold in Dedekind complete $ ext{L}$-vector lattices with mild additional conditions.

The results generalize classical formulas to a wider algebraic and lattice-theoretic context.

Abstract

Let $\mathbb{L}$ be a Dedekind complete unital $f$-algebra. We prove the Riesz-Kantorovich formulas for order bounded $\mathbb{L}$-module homomorphisms from a directed partially ordered $\mathbb{L}$-module with the Riesz Decomposition Property into a Dedekind complete $\mathbb{L}$-vector lattice satisfying an additional mild condition.