Finite Atomized Semilattices

TL;DR

This paper introduces atomized semilattices, a novel algebraic structure that represents finite semilattices using atoms forming a hypergraph, enabling computational modeling and applications in machine learning.

Contribution

It proves the existence and uniqueness of atomized semilattices for finite semilattices and demonstrates their use as computational tools for modeling and semantic analysis.

Findings

Every finite semilattice has a unique atomized representation.

Atomized semilattices can be used to construct models, subalgebras, and products.

Applications include machine learning and semantic embeddings.

Abstract

We show that every finite semilattice can be represented as an atomized semilattice, an algebraic structure with additional elements (atoms) that extend the semilattice's partial order. Each atom maps to one subdirectly irreducible component, and the set of atoms forms a hypergraph that fully defines the semilattice. An atomization always exists and is unique up to "redundant atoms". Atomized semilattices are representations that can be used as computational tools for building semilattice models from sentences, as well as building its subalgebras and products. Atomized semilattices can be applied to machine learning and to the study of semantic embeddings into algebras with idempotent operators.