## parallel partial orders

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• Series-parallel partial orders have order dimension at most two.
• The class of series-parallel partial orders is the set of partial orders that can be built up from single-element partial orders using these two operations.
• More strongly, although a partial order may have many different conjugates, every conjugate of a series parallel partial order must itself be series parallel.
• The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension at most two.
• 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.
• 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.
• Conversely, every cograph is the comparability graph of a series-parallel partial order.
• Separable permutations also closely related to series-parallel partial orders, the partially ordered sets whose comparability graphs are the cographs.
• It follows immediately from this (although it can also be proven directly) that any nonempty restriction of a series-parallel partial order is itself a series-parallel partial order.
• If a series-parallel partial order is represented as an expression tree describing the series and parallel composition operations that formed it, then the elements of the partial order may be represented by the leaves of the expression tree.
• However, unlike series-parallel partial orders, PQ trees allow the linear ordering of any Q node to be reversed.
• As with the cographs and separable permutations, the series-parallel partial orders may also be characterized by four-element forbidden suborders.
• A weak order is the series parallel partial order obtained from a sequence of composition operations in which all of the parallel compositions are performed first, and then the results of these compositions are combined using only series compositions.
• They use the formula for computing the number of linear extensions of a series-parallel partial order as the basis for analyzing multimedia transmission algorithms.
• The comparability graphs of series-parallel partial orders are cographs.
• Although the problem of counting the number of linear extensions of an arbitrary partial order is #P-complete, it may be solved in polynomial time for series-parallel partial orders.
• It is NP-complete to test, for two given series-parallel partial orders P and Q, whether P contains a restriction isomorphic to Q.
• 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.
• 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.
• Directed trees and (two-terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallel partial orders may be used to represent reachability relations in directed trees and series parallel graphs.