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The rank function is compatible with the ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and; 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.
There are several known constructions of rank codes, which are maximum rank distance (or MRD) codes with d = n − k + 1.The easiest one to construct is known as the (generalized) Gabidulin code, it was discovered first by Delsarte (who called it a Singleton system) and later by Gabidulin [2] (and Kshevetskiy [3]).
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .
The covariant 4-position is given by ... For a (0,2) tensor, [1] ... General rank. For a tensor of order n, ...
A set is independent if and only if its rank equals its cardinality, and dependent if and only if it has greater cardinality than rank. [3] A nonempty set is a circuit if its cardinality equals one plus its rank and every subset formed by removing one element from the set has equal rank. [3]
If X is a set and G = F(X) is the free group with free basis X then rank(G) = |X|. If a group H is a homomorphic image (or a quotient group) of a group G then rank(H) ≤ rank(G). If G is a finite non-abelian simple group (e.g. G = A n, the alternating group, for n > 4) then rank(G) = 2.
Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on"), and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the order type of the set.
1. An urelement, something that is not a set but allowed to be an element of a set 2. An element of a poset such that any two elements smaller than it are compatible. 3. A set of positive measure such that every measurable subset has the same measure or measure 0 atomic An atomic formula (in set theory) is one of the form x=y or x∈y axiom