Combinatorial Theory

This paper considers combinatorial games with the last move losing (misère play) instead of winning (normal play). Misère games form a pomonoid in which no non-zero elements are invertible; normal-play games, however, form a group with a much richer partial order. To compensate, misère researchers typically weaken the comparison relation by restricting to a subset of games, especially to a "universe" (closed under addition, conjugation, and options). For some well-studied universes (like the dicot and dead-ending universes), there exist finite comparison tests and also complete characterisations of their invertible elements; more recently, the invertible elements of every "parental" universe have been characterised. In this paper, we study the recently-defined universe of "blocking games" (containing both dicots and dead-ending games): we develop a finite comparison test, and we improve on the general invertibility characterisation by showing that, as in dead-ending, a game is invertible modulo blocking if and only if it is free of previous-win subpositions (or is equal to such a form).

Mathematics Subject Classifications: 91A46, 06F05, 20M14, 20K27

Keywords: Combinatorial Game Theory, Ordered semigroups