From free idempotent monoids to free multiplicatively idempotent rigs

TL;DR

This paper investigates free multiplicatively idempotent rigs (mirigs), proving their finiteness when finitely generated and calculating their size, motivated by applications in decentralized social networks like Mastodon.

Contribution

It establishes the finiteness of free mirigs on finitely many generators and provides explicit size computations, a novel contribution in the study of idempotent algebraic structures.

Findings

Free mirigs on finitely many generators are finite.

Explicit sizes of free mirigs are computed.

Motivated by applications in decentralized social networks.

Abstract

A multiplicatively idempotent rig (which we abbreviate to mirig) is a rig satisfying the equation $r^2 = r$. We show that a free mirig on finitely many generators is finite and compute its size. This work was originally motivated by a collaborative effort on the decentralized social network Mastodon to compute the size of the free mirig on two generators.