equivalence
Compatible
Characteristics of Compatible Relationship Diagram
Each node has a ring (determined by reflexivity)
. If there are edges between different nodes, they must appear in pairs (determined by symmetry)
The relationship diagram can be simplified by the above characteristics
Do not draw a ring (determined by reflexivity)
Two directed edges are replaced with one undirected edge (determined by symmetry)
The characteristics of the relation matrix of compatible relations
The main diagonals are all 1 (determined by reflexivity)
symmetrical elements along the main diagonal are equal (determined by symmetry)
The relationship matrix can be simplified by the above characteristics
Lower triangular matrix
Compatible
Compatible class: According to the compatibility relationship classification, if two elements satisfy the compatibility relationship, they are divided into one category
. The compatibility classes in the figure below are:
{a, b}, {a, b, c}, {a, c, d}, {a, b, c, d}, {d, e}, {b, c, f}, {g} etc.\{a,b\},\{a,b,c\} ,\{a,c,d\},\{a,b,c,d\},\{d,e\},\{b,c,f\},\{g\} etc.{
a,b},{
a,b,c},{
a,c,d},{
a,b,c,d},{
d,e},{
b,c,f},{
g }, etc.
Each complete polygon constitutes a compatible class: the
maximum compatible class (a class that is not really included by other compatible classes): the maximum compatible class is to find the largest complete polygon in the graph: {a, b, c, d}, {b, c, f}, {d, e}, {g} r is the compatibility relationship in X, which completely covers C r (X): a set consisting of all the largest compatible classes as elements Maximum compatible class (class not actually included by other compatible classes): \\ The maximum compatible class is to find the largest complete polygon in the graph: \{a,b,c,d\},\{b,c ,f\},\{d,e\},\{g\}\\ r is the compatibility relationship in X, which completely covers Cr(X): \\ is a set consisting of all the largest compatible classes as elementsMost large phase volume type ( not by which he phase content class is true package contains the classes ):Most large relative capacity based on a find FIG in a most big completely full multi- edge shape:{
a,b,c,d},{
b,c,f},{
d,e},{
g}r is X in the phase content off system ,After full covering cap C R & lt ( X- ) :A is has the most large phase volume type of element element configuration to the set co
Partial order≼ (semi-order)
The characteristics of the partial order relationship diagram
Each node has a ring (determined by reflexivity)
. If there are edges between different nodes, they must appear in pairs and only one appears (determined by antisymmetric)
The relationship diagram can be simplified by the above characteristics
Do not draw a circle (determined by reflexivity) The
first element is on the left, the second element is on the right, the first element is on the bottom, and the second element is on the top to indicate the direction (determined by antisymmetric).
There are <a,b>,<b ,c> is <a,c>, but <a,c> is omitted in the figure (transitivity)
Large (small) yuan
Let (P, ≼) be a partially ordered set, A ⊆ P, if a ∈ A, and there is no b (b ≠ a) in A, a ≼ b, then a is the maximal element in A (P ,≼) is a partially ordered set, A \subseteq P, if a\in A, and b(b\neq a), a≼b does not exist in A, then a is called the maximal element in A Suppose ( P ,≼) Is a partially ordered set , A⊆P,Young a∈A,And in A are not stored in the B ( B=a),a≼b,It is called a to A in the pole Da Yuan
Maximum (small) yuan
Let (P, ≼) be a partially ordered set, A ⊆ P, if a ∈ A, ∀ b ∈ A, b ≼ a, then a is the largest element in A. Let (P, ≼) be a partially ordered set, A \subseteq P, if a\in A, \forall b\in A, b≼a, then a is called the largest element in A Suppose ( P ,≼) Is a partially ordered set , A⊆P,Young a∈A,∀b∈A,b≼a,It is called a to A in the most Da Yuan
maximum yuan is also looking at a subset of A, rather than complete works P find, as long as there is only minimax yuan, then it is the minimum and maximum yuan, or does not exist \ tiny largest The element is also searched in subset A, not in the complete set P. As long as there is a unique maximum and minimum element, it is the largest and smallest element, otherwise it does not existMost large Yuan also is in the sub- set of A in seeking to find , and non- full- set P in seeking to find , only to have the only one of a very large pole Xiao Yuan , then it is most big most Xiao Yuan , no it does not exist in
Upper (lower) bounds
Let (P, ≼) be a partially ordered set, A ⊆ P, if a ∈ P, ∀ b ∈ A, b ≼ a, then call a the upper bound in A. Let (P, ≼) be a partially ordered set, A \subseteq P, if a\in P, \forall b\in A, b≼a, then a is the upper bound in A Suppose ( P ,≼) Is a partially ordered set , A⊆P,Young a∈P,∀b∈A,b≼a,It is called a to A in the upper bound
upper bound is looking for the complete works of P \ the tiny community is looking at the P Complete WorksThe industry is in a full set of P in seeking to find
Upper (lower) realm
Let (P, ≼) be a partially ordered set, A ⊆ P, a is an upper bound of A, and any upper bound b of A has a ≼ b, call a the supremum set in A (P, ≼) Is a semi-ordered set, A \subseteq P, a is an upper bound of A, any upper bound b of A has a≼b, call a the supremum in A Suppose ( P ,≼) Is a partially ordered set , A⊆P,a is A is a th upper bound , any meaning A of upper bound b all have a≼b,He said a is A in the upper indeed bound