Equivalences and Congruences on Infinite

The increasing use of games as a convenient metaphor for modeling interac-tions has spurred the growth of a broad variety of game definitions in Computer Science. Furthermore, in the presentation of games many related concepts are used, e.g. move, position, play, turn, winning condition, payoff function, strat-egy, etc. None has a unique definition. Some, but not always the same, are taken as primitive while the others are reduced to them. And many more properties need to be specified before the kind of game one is interested in is actually pinned down, e.g.: perfect knowledge, zero-sum, chance, number of players, finiteness, determinacy, etc. All this together with the wide gamut of games arising in real life calls for a unifying foundational approach to games. In [HL09], we started such a programme using very unbiased foundational tools, namely algebras and coalgebras. We build upon Conway’s notion of game. It provides an elementary but sufficiently abstract notion of game amenable to a rich algebraic-coalgebraic treatment because of the special role that sums of games play in this theory