Mex (mathematics)

In combinatorial game theory, the mex, or "minimum excludant", of a set of ordinals denotes the smallest ordinal not contained in the set.

Some examples:

$\mbox{mex}(\left \{1, 2, 3\right \}) = 0$

$\mbox{mex}(\left \{0, 1, 4, 7, 12\right \}) = 2$

$\mbox{mex}(\left \{0, 1, 2, 3, \ldots\right \}) = \omega$

$\mbox{mex}(\left \{0, 2, 4, 6, \ldots\right \}) = 1$

$\mbox{mex}(\left \{0, 1, 2, 3, \ldots, \omega\right \}) = \omega+1$

where ω is the limit ordinal for the natural numbers.

In the Sprague-Grundy theory the minimum excluded ordinal plays a dominant role in determining the Nimbers of combinatorial games.

References

• Conway, John H. (2001). On Numbers and Games (2nd ed.). A.K. Peters. p. 124. ISBN 1-56881-127-6.