The disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. This is extended to disjunctive sums of any number of games by associativity by allowing each player to move in just one of the games per turn. Combinatorial game theory (CGT) is a mathematical theory that studies a certain kind of game. ... In mathematics, associativity is a property that a binary operation can have. ...

This is the fundamental operation that is used in the Sprague-Grundy theorem for impartial games and which led to the field of Combinatorial game theory for partisan games. In combinatorial game theory, the Sprague–Grundy theorem states that every impartial game is equivalent to a nimber. ... In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. ... Combinatorial game theory (CGT) is a mathematical theory that studies a certain kind of game. ... In combinatorial game theory, a game is partisan or partizan if it is not impartial. ...

The importance of disjunctive sums arises in games that naturally break up into components or regions that do not interact except in that each player in turn must choose just one component to play in. Examples of such games are Go, Nim, Sprouts, and Domineering. Jump to: navigation, search Go is a strategic, two-player board game originating in ancient China between 2000 BC and 200 BC. Go is a popular game in East Asia. ... Jump to: navigation, search Nim is a two-player mathematical game of strategy in which players take turns removing objects from heaps, one or more objects at a time but only from a single heap. ... Sprouts is a pencil-and-paper game with interesting mathematical properties. ... Domineering is a mathematical game played on a sheet of graph paper, with any set of designs traced out. ...

By analyzing each component, it is possible to find simplifications of the component that do not affect its outcome or the outcome of its disjunctive sum with other games. In addition, the components can be combined by taking the disjuctive sum of two games at a time, combining them into a single game.

The disjuctive sum is a fairly well-studied tool for analysis of normal play games, in which a player who is unable to play loses. Some progress has been made in analyzing impartial games in misère play, where a player unable to play wins. In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. ... A misère version of a game is a game that is played according to its conventional rules, except that it is played to lose; that is, the winner is the one who loses according to the normal game rules. ...

Mathematially, the disjunctive sum imposes an Abelian group structure on games, that can be extended to a field for an important subclass of games called the Surreal numbers. Impartial misère play games form an Abelian monoid with only one invertible element, called star (*), of order two. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... Jump to: navigation, search In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similiar to superreal numbers and hyperreal numbers. ... A misère version of a game is a game that is played according to its conventional rules, except that it is played to lose; that is, the winner is the one who loses according to the normal game rules. ... In mathematics, an abelian group is a commutative group, i. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... Star, written as * or *1, is the value given to the combinatorial game {0 | 0}, where zero is the zero game. ...