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In mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or; a poset with the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or; a poset in which all maximal chains have the same ...
The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1. The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset.
In this poset, 60 is an upper bound (though not a least upper bound) of the subset {,,,}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a ...
A k-Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels, [1] or, equivalently, the poset has a maximum k-family consisting of k rank levels. [2] A strict Sperner poset is a graded poset in which all maximum antichains are rank levels. [2]
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
The poset of positive integers has deviation 0: every descending chain is finite, so the defining condition for deviation is vacuously true. However, its opposite poset has deviation 1. Let k be an algebraically closed field and consider the poset of ideals of the polynomial ring k[x] in one variable. Since the deviation of this poset is the ...
Poset games are impartial games, meaning that every move available to Alice would also be available to Bob if Alice were allowed to pass, and vice versa. Therefore, by the Sprague–Grundy theorem , every position in a poset game has a Grundy value, a number describing an equivalent position in the game of Nim.
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