In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order.
first studied order dimension; for a more detailed treatment of this subject than provided here, see .
The dimension of a poset P is the least integer t for which there exists a family
of linear extensions of P so that, for every x and y in P, x precedes y in P if and only if it precedes y in all of the linear extensions. That is,
An alternative definition of order dimension is the minimal number of total orders such that P embeds into their product with componentwise ordering i.e. if and only if for all i (, ).
A family of linear orders on X is called a realizer of a poset P = (X,
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Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. A partial order is a reflexive, transitive and antisymmetric relation.
In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . The dimension of a poset P is the least integer t for which there exists a family of linear extensions of P so that, for every x and y in P, x precedes y in P if and only if it precedes y in all of the linear extensions.
In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.
Covers relations, sequences, and posets, emphasizing properties like anti-symmetry and transitivity, and introduces arithmetic and geometric progressions.