The sum game finishes when there are no moves left in either of the two parallel games, at which point (in normal play) the last player to move wins.
Disjunctive sums arise 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.
Based on these properties, the class of combinatorial games may be thought of as having the structure of an abelian group, although with a proper class of elements rather than (as is more standard for groups) a set of elements.
For an important subclass of the games called the surreal numbers, there exists a multiplication operator that extends this group to a field.
For impartial misère play games, an analogous theory of sums can be developed, but with fewer of these properties: these games form a commutative monoid with only one nontrivial invertible element, called star (*), of order two.