Discrete Mathematics
WEN-CHING LIEN Department of Mathematics National Cheng Kung University
2008
7.3: Partial orders: Hasse Diagrams
Example (7.35)
Define R on A= {1,2,3,4}by xRy if x|y−that is (exactly) divides y.
Then R = {(1,1), (2,2), (3,3), (4,4), (1,2), (1,3), (1,4), (2,4)}
is a partial order, and(A,R)is a poset.
7.3: Partial orders: Hasse Diagrams
Example (7.35)
Define R on A= {1,2,3,4}by xRy if x|y−that is (exactly) divides y.
Then R = {(1,1), (2,2), (3,3), (4,4), (1,2), (1,3), (1,4), (2,4)}
is a partial order, and(A,R)is a poset.
7.3: Partial orders: Hasse Diagrams
Example (7.37)
Consider the digraph for the partial order in Example 7.35.Fig.
7.17(a) is the graphical representation of R.In part (b) of the figure, we have a somewhat simpler diagram, which is called the Hasse diagram for R.
When we know that a relation R is a partial order on a set A, we can eliminate the loops at the vertices of its digraph .Since R is also transitive , having the edges(1,2)and(2,4)is enough to insure the existence of edge(1,4), so we need not include that edges.In this way we obtain the diagram in Fig. 7.17(b), where we have not lost the directions on the edges- the directions are assumed to go from the bottom to the top.
7.3: Partial orders: Hasse Diagrams
Example (7.37)
Consider the digraph for the partial order in Example 7.35.Fig.
7.17(a) is the graphical representation of R.In part (b) of the figure, we have a somewhat simpler diagram, which is called the Hasse diagram for R.
When we know that a relation R is a partial order on a set A, we can eliminate the loops at the vertices of its digraph .Since R is also transitive , having the edges(1,2)and(2,4)is enough to insure the existence of edge(1,4), so we need not include that edges.In this way we obtain the diagram in Fig. 7.17(b), where we have not lost the directions on the edges- the directions are assumed to go from the bottom to the top.
7.3: Partial orders: Hasse Diagrams
Example (7.37)
Consider the digraph for the partial order in Example 7.35.Fig.
7.17(a) is the graphical representation of R.In part (b) of the figure, we have a somewhat simpler diagram, which is called the Hasse diagram for R.
When we know that a relation R is a partial order on a set A, we can eliminate the loops at the vertices of its digraph .Since R is also transitive , having the edges(1,2)and(2,4)is enough to insure the existence of edge(1,4), so we need not include that edges.In this way we obtain the diagram in Fig. 7.17(b), where we have not lost the directions on the edges- the directions are assumed to go from the bottom to the top.
7.3: Partial orders: Hasse Diagrams
Example (7.37)
Consider the digraph for the partial order in Example 7.35.Fig.
7.17(a) is the graphical representation of R.In part (b) of the figure, we have a somewhat simpler diagram, which is called the Hasse diagram for R.
When we know that a relation R is a partial order on a set A, we can eliminate the loops at the vertices of its digraph .Since R is also transitive , having the edges(1,2)and(2,4)is enough to insure the existence of edge(1,4), so we need not include that edges.In this way we obtain the diagram in Fig. 7.17(b), where we have not lost the directions on the edges- the directions are assumed to go from the bottom to the top.
7.3: Partial orders: Hasse Diagrams
Example (7.37)
Consider the digraph for the partial order in Example 7.35.Fig.
7.17(a) is the graphical representation of R.In part (b) of the figure, we have a somewhat simpler diagram, which is called the Hasse diagram for R.
When we know that a relation R is a partial order on a set A, we can eliminate the loops at the vertices of its digraph .Since R is also transitive , having the edges(1,2)and(2,4)is enough to insure the existence of edge(1,4), so we need not include that edges.In this way we obtain the diagram in Fig. 7.17(b), where we have not lost the directions on the edges- the directions are assumed to go from the bottom to the top.
Rule:
In general, if R is a partial order on a finite set A, we construct a Hasse diagram for R on A by drawing a line segment from x up to y,if x,y ∈A with xRy and, most important, if there is no other element z ∈A such that xRz and zRy.( So there is nothing ”in between” x and y.) If we adopt the convention of reading the diagram from bottom to top, then it is not necessary to direct any edges.
Rule:
In general, if R is a partial order on a finite set A, we construct a Hasse diagram for R on A by drawing a line segment from x up to y, if x,y ∈A with xRy and, most important, if there is no other element z ∈A such that xRz and zRy.( So there is nothing ”in between” x and y.) If we adopt the convention of reading the diagram from bottom to top, then it is not necessary to direct any edges.
Rule:
In general, if R is a partial order on a finite set A, we construct a Hasse diagram for R on A by drawing a line segment from x up to y, if x,y ∈A with xRy and, most important, if there is no other element z ∈A such that xRz and zRy.( So there is nothing ”in between” x and y.) If we adopt the convention of reading the diagram from bottom to top, then it is not necessary to direct any edges.
The pair(A,R)is called a partially ordered set, or poset, if relation R on A is a partial order, or a partial ordering relation.
Definition (7.16)
If(A,R)is a poset, we say that A is totally ordered (or, linearly ordered) if for all x,y ∈A either xRy or yRx.
In this case R is called a total order (or, a linear order).
The pair(A,R)is called a partially ordered set, or poset, if relation R on A is a partial order, or a partial ordering relation.
Definition (7.16)
If(A,R)is a poset, we say that A is totally ordered (or, linearly ordered) if for all x,y ∈A either xRy or yRx.
In this case R is called a total order (or, a linear order).
The pair(A,R)is called a partially ordered set, or poset, if relation R on A is a partial order, or a partial ordering relation.
Definition (7.16)
If(A,R)is a poset, we say that A is totally ordered (or, linearly ordered) if for all x,y ∈A either xRy or yRx.
In this case R is called a total order (or, a linear order).
Definition (7.17)
If(A,R)is a poset, then an element x ∈A is called a maximal element of A if for all a∈A,a6=x ⇒x/Ra.
An element y ∈A is called a minimal element of A if whenever b ∈A,b6=y ⇒b/Ry.
Definition (7.18)
If(A,R)is a poset, then an element x ∈A is called a least element if xRa for all a∈A.
Element y ∈A is called a greatest element if aRy for all a∈A.
Definition (7.17)
If(A,R)is a poset, then an element x ∈A is called a maximal element of A if for all a∈A,a6=x ⇒x/Ra.
An element y ∈A is called a minimal element of A if whenever b ∈A,b6=y ⇒b/Ry.
Definition (7.18)
If(A,R)is a poset, then an element x ∈A is called a least element if xRa for all a∈A.
Element y ∈A is called a greatest element if aRy for all a∈A.
Definition (7.17)
If(A,R)is a poset, then an element x ∈A is called a maximal element of A if for all a∈A,a6=x ⇒x/Ra.
An element y ∈A is called a minimal element of A if whenever b ∈A,b6=y ⇒b/Ry.
Definition (7.18)
If(A,R)is a poset, then an element x ∈A is called a least element if xRa for all a∈A.
Element y ∈A is called a greatest element if aRy for all a∈A.
Definition (7.17)
If(A,R)is a poset, then an element x ∈A is called a maximal element of A if for all a∈A,a6=x ⇒x/Ra.
An element y ∈A is called a minimal element of A if whenever b ∈A,b6=y ⇒b/Ry.
Definition (7.18)
If(A,R)is a poset, then an element x ∈A is called a least element if xRa for all a∈A.
Element y ∈A is called a greatest element if aRy for all a∈A.
Theorem (7.4)
If the poset(A,R)has a greatest(least) element, then that element is unique.
Definition (7.19)
Let(A,R)be a poset with B⊆A.An element x ∈A is called a lower bound of B if xRb for all b∈B.Likewise, an element y ∈A is called an upper bound of B if bRy for all b ∈B.
An element x′ ∈A is called a greatest lower bound (glb) of B if it is a lower bound of B and if for all other lower bounds x′′of B we have x′′Rx′.Similarly, y′ ∈A is a least upper bound (lub) of B if it is an upper bound of B and if y′Ry′′for all other upper bounds y′′of B.
Theorem (7.4)
If the poset(A,R)has a greatest(least) element, then that element is unique.
Definition (7.19)
Let(A,R)be a poset with B⊆A.An element x ∈A is called a lower bound of B if xRb for all b∈B.Likewise, an element y ∈A is called an upper bound of B if bRy for all b ∈B.
An element x′ ∈A is called a greatest lower bound (glb) of B if it is a lower bound of B and if for all other lower bounds x′′of B we have x′′Rx′.Similarly, y′∈A is a least upper bound (lub) of B if it is an upper bound of B and if y′Ry′′for all other upper bounds y′′of B.
Theorem (7.4)
If the poset(A,R)has a greatest(least) element, then that element is unique.
Definition (7.19)
Let(A,R)be a poset with B⊆A.An element x ∈A is called a lower bound of B if xRb for all b∈B.Likewise, an element y ∈A is called an upper bound of B if bRy for all b ∈B.
An element x′ ∈A is called a greatest lower bound (glb) of B if it is a lower bound of B and if for all other lower bounds x′′of B we have x′′Rx′.Similarly, y′ ∈A is a least upper bound (lub) of B if it is an upper bound of B and if y′Ry′′for all other upper bounds y′′of B.
Theorem (7.4)
If the poset(A,R)has a greatest(least) element, then that element is unique.
Definition (7.19)
Let(A,R)be a poset with B⊆A.An element x ∈A is called a lower bound of B if xRb for all b∈B.Likewise, an element y ∈A is called an upper bound of B if bRy for all b ∈B.
An element x′ ∈A is called a greatest lower bound (glb) of B if it is a lower bound of B and if for all other lower bounds x′′of B we have x′′Rx′.Similarly, y′∈A is a least upper bound (lub) of B if it is an upper bound of B and if y′Ry′′for all other upper bounds y′′of B.
Theorem (7.4)
If the poset(A,R)has a greatest(least) element, then that element is unique.
Definition (7.19)
Let(A,R)be a poset with B⊆A.An element x ∈A is called a lower bound of B if xRb for all b∈B.Likewise, an element y ∈A is called an upper bound of B if bRy for all b ∈B.
An element x′ ∈A is called a greatest lower bound (glb) of B if it is a lower bound of B and if for all other lower bounds x′′of B we have x′′Rx′.Similarly, y′ ∈A is a least upper bound (lub) of B if it is an upper bound of B and if y′Ry′′for all other upper bounds y′′of B.
Theorem (7.4)
If the poset(A,R)has a greatest(least) element, then that element is unique.
Definition (7.19)
Let(A,R)be a poset with B⊆A.An element x ∈A is called a lower bound of B if xRb for all b∈B.Likewise, an element y ∈A is called an upper bound of B if bRy for all b ∈B.
An element x′ ∈A is called a greatest lower bound (glb) of B if it is a lower bound of B and if for all other lower bounds x′′of B we have x′′Rx′.Similarly, y′∈A is a least upper bound (lub) of B if it is an upper bound of B and if y′Ry′′for all other upper bounds y′′of B.
Theorem (7.5)
If(A,R)is a poset and B ⊆A,then B has at most one lub(glb).
