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A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
Network topology is the arrangement of the elements (links, nodes, etc.) of a communication network. [1] [2] Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, [3] industrial fieldbusses and computer networks.
A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. (A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset.)
In this setting, a subset of a partially ordered set (,) is called a directed subset if it is a directed set according to the same partial order: in other words, it is not the empty set, and every pair of elements has an upper bound. Here the order relation on the elements of is inherited from ; for this reason, reflexivity and transitivity ...
In a partially ordered set (P,≤) an element c is called compact (or finite) if it satisfies one of the following equivalent conditions: For every directed subset D of P, if D has a supremum sup D and c ≤ sup D then c ≤ d for some element d of D. For every ideal I of P, if I has a supremum sup I and c ≤ sup I then c is an element of I.
A power set, partially ordered by inclusion, with rank defined as number of elements, forms a graded poset. In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N of all natural numbers. ρ must satisfy the following two properties:
These elements are also maximal and minimal elements, respectively, of the red subset. In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of .
An optical communication system is any form of communications system that uses light as the transmission medium. Equipment consists of a transmitter, which encodes a message into an optical signal, a communication channel, which carries the signal to its destination, and a receiver, which reproduces the message from the received optical signal.