Combinatorial Route to Algebra: The Art of Composition & Decomposition

TL;DR

This paper introduces a general framework for composition and decomposition of objects, leading to algebraic structures like multiplication laws, with a focus on combinatorial origins and properties.

Contribution

It presents a generic scheme that connects combinatorial composition and decomposition to algebraic structures, providing foundational insights.

Findings

Establishes multiplication and co-multiplication laws from combinatorial principles.

Provides a unified algebraic framework for combinatorial classes.

Serves as an introductory guide to algebraic structures derived from combinatorics.

Abstract

We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.