In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-parallel partial orders by two simple composition operations.
The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension at most two. They include weak orders and the reachability relationship in directed trees and directed series–parallel graphs. The comparability graphs of series-parallel partial orders are cographs.
Series-parallel partial orders have been applied in job shop scheduling, machine learning of event sequencing in time series data, transmission sequencing of multimedia data, and throughput maximization in dataflow programming.
Series-parallel partial orders have also been called multitrees; however, that name is ambiguous: multitrees also refer to partial orders with no four-element diamond suborder and to other structures formed from multiple trees.
Consider P and Q, two partially ordered sets. The series composition of P and Q, written P; Q, P * Q, or P ⧀ Q,is the partially ordered set whose elements are the disjoint union of the elements of P and Q. In P; Q, two elements x and y that both belong to P or that both belong to Q have the same order relation that they do in P or Q respectively. However, for every pair x, y where x belongs to P and y belongs to Q, there is an additional order relation x ≤ y in the series composition. Series composition is an associative operation: one can write P; Q; R as the series composition of three orders, without ambiguity about how to combine them pairwise, because both of the parenthesizations (P; Q); R and P; (Q; R) describe the same partial order. However, it is not a commutative operation, because switching the roles of P and Q will produce a different partial order that reverses the order relations of pairs with one element in P and one in Q.
The parallel composition of P and Q, written P Q, P + Q, or P ⊕ Q, is defined similarly, from the disjoint union of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q having the same order as they do in P or Q respectively.
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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 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.
In graph theory, a cograph, or complement-reducible graph, or P4-free graph, is a graph that can be generated from the single-vertex graph K1 by complementation and disjoint union. That is, the family of cographs is the smallest class of graphs that includes K1 and is closed under complementation and disjoint union. Cographs have been discovered independently by several authors since the 1970s; early references include , , , and . They have also been called D*-graphs, hereditary Dacey graphs (after the related work of James C.
Introduces equivalence relations, partitions, partial orderings, and total ordering concepts with examples and definitions.
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