### 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.

*the Hasse diagram for R.*

### 7.3: Partial orders: Hasse Diagrams

Example (7.37)

Consider the digraph for the partial order in Example 7.35.Fig.

*the Hasse diagram for R.*

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,a*6=*x* ⇒*x*/Ra.

*An element y* ∈*A is called a minimal element of A if whenever*
*b* ∈*A,b*6=*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,a*6=*x* ⇒*x*/Ra.

*An element y* ∈*A is called a minimal element of A if whenever*
*b* ∈*A,b*6=*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,a*6=*x* ⇒*x*/Ra.

*An element y* ∈*A is called a minimal element of A if whenever*
*b* ∈*A,b*6=*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)

*R)is a poset, then an element x* ∈*A is called a maximal*
*element of A if for all a*∈*A,a*6=*x* ⇒*x*/Ra.

*An element y* ∈*A is called a minimal element of A if whenever*
*b* ∈*A,b*6=*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)

*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)

*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)

*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).*