We give " kG " the structure of a cocommutative Hopf algebra by defining the coproduct, counit, and antipode to be the linear extensions of the following maps defined on " G ":
22.
The interval dimension of a partial order can be defined as the minimal number of interval order extensions realizing this order, in a similar way to the definition of the order dimension which uses linear extensions.
23.
Let " G " be a algebra, its product is defined by linear extension of the group composition in " G ", with multiplicative unit the identity in " G "; this product is also known as convolution.
24.
As with the case above for Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying " E " [ " H " 2 ?" t " ] < ".
25.
The family of topological orderings of a DAG is the same as the family of linear extensions of the reachability relation for the DAG, so any two graphs representing the same partial order have the same set of topological orders.
26.
The gold partition conjecture would also imply that a partial order with " E " linear extensions can be sorted in at most log ? " E " comparisons; the name of the conjecture is derived from this connection with the golden ratio.
27.
A relation having the property of " totality " means that any pair of elements in the set of the relation are reflexivity, not totality . ) An extension of a given partial order to a total order is called a linear extension of that partial order.
28.
A "'linearization "'of a partial order plan is a total order plan derived from the particular partial order plan; in other words, both order plans consist of the same actions, with the order in the linearization being a linear extension of the partial order in the original partial order plan.
29.
The order dimension of a partial order is the minimum cardinality of a set of linear extensions whose intersection is the given partial order; equivalently, it is the minimum number of linear extensions needed to ensure that each critical pair of the partial order is reversed in at least one of the extensions.
30.
The order dimension of a partial order is the minimum cardinality of a set of linear extensions whose intersection is the given partial order; equivalently, it is the minimum number of linear extensions needed to ensure that each critical pair of the partial order is reversed in at least one of the extensions.
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