The Class of Strong Placement Games: Complexes, Values, and Temperature

Abstract

Strong Placement (SP-) games are a class of combinatorial games in which pieces are placed on a board such that the order in which previously placed pieces have been played does not matter. It is known that to each such game one can assign two square-free monomial ideals (the legal and illegal ideal) and two simplicial complexes (the legal and illegal complex). In this work we will show that reverse constructions also exist, in particular when restricting to invariant SP-games. We then use this one-to-one correspondence between games, ideals, and simplicial complexes to study several properties of SP-games. This includes the structure of the game tree of an SP-game, and the set of possible game values. The temperatures of SP-games are also considered. We prove a first general upper bound on the boiling point of a game, and will show through several games that this bound is particularly applicable for SP-games. Motivated by the connection to commutative algebra, we then explore what it could mean for an SP-game to be Cohen-Macaulay, as well as several related properties.