Semilattices, Canonical Embeddings and Representing Measures

TL;DR

This paper establishes conditions for representing modular functions on semilattices as homomorphic functions on lattices via embeddings, advancing the understanding of algebraic structures and their measure representations.

Contribution

It introduces new criteria for extending modular functions from semilattices to lattices through canonical embeddings, enhancing algebraic and measure-theoretic frameworks.

Findings

Provides necessary and sufficient conditions for homomorphic extension.

Characterizes modular functions compatible with lattice embeddings.

Connects semilattice functions with lattice measures.

Abstract

We provide conditions under which a modular function defined on a semilattice $X$ and with values in a commutative group is homomorphic to a modular function on a lattice $L$ for any embedding $X\hookrightarrow L$.