The monoids of the patience sorting algorithm

Abstract

The left patience sorting ([Formula: see text][Formula: see text]PS) monoid, also known in the literature as the Bell monoid, and the right patient sorting ([Formula: see text]PS) 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 [Formula: see text]PS monoid is shown to have polynomial growth and to satisfy a nontrivial identity (dependent on its rank), while the infinite rank [Formula: see text]PS monoid does not satisfy any nontrivial identity. Each [Formula: see text][Formula: see text]PS monoid of finite rank has exponential growth and does not satisfy any nontrivial identity. The complexity of the insertion algorithms is discussed. [Formula: see text]PS monoids of finite rank are shown to be automatic and to have recursive complete presentations. When the rank is [Formula: see text] or [Formula: see text], they are also biautomatic. [Formula: see text][Formula: see text]PS monoids of finite rank are shown to have finite complete presentations and to be biautomatic.