The monoids of the patience sorting algorithm

TL;DR

This paper introduces and analyzes the algebraic structures called monoids derived from patience sorting algorithms, exploring their growth, identities, and automatic properties.

Contribution

It defines the left and right patience sorting monoids, provides presentations, and studies their algebraic properties including growth, identities, and automaticity.

Findings

Finite-rank rPS monoids have polynomial growth and satisfy non-trivial identities.

Infinite-rank rPS monoid does not satisfy any non-trivial identity.

Finite-rank lPS monoids have exponential growth and are biautomatic.

Abstract

The left patience sorting (lPS) monoid, also known in the literature as the Bell monoid, and the right patient sorting (rPS) monoid are introduced by defining certain congruences on words. Such congruences are constructed using insertion algorithms based on the concept of decreasing subsequences. Presentations for these monoids are given. Each finite-rank rPS monoid is shown to have polynomial growth and to satisfy a non-trivial identity (dependent on its rank), while the infinite rank rPS monoid does not satisfy a non-trivial identity. The lPS monoids of finite rank have exponential growth and thus do not satisfy non-trivial identities. The complexity of the insertion algorithms is discussed. rPS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is $1$ or $2$, they are also biautomatic. lPS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.