M obius Functions of Posets I: Introduction to Partially Ordered - PowerPoint PPT Presentation

Example: The Boolean Algebra. The Boolean algebra is B n = { S : S ⊆ { 1 , 2 , . . . , n }} partially ordered by S ≤ T if and only if S ⊆ T . { 1 , 2 } ◗ ◗ t ◗ ◗ B 3 = ◗ ◗ { 1 } { 2 } { 3 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ ∅ t

Example: The Boolean Algebra. The Boolean algebra is B n = { S : S ⊆ { 1 , 2 , . . . , n }} partially ordered by S ≤ T if and only if S ⊆ T . { 1 , 2 } { 1 , 3 } { 2 , 3 } ✑✑✑✑✑ ◗ ◗ ✑ ✑✑✑✑✑ ◗ ◗ ✑ t t t ◗ ◗ ◗ ◗ B 3 = ◗ ◗ ◗ ◗ { 1 } { 2 } { 3 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ ∅ t

Example: The Boolean Algebra. The Boolean algebra is B n = { S : S ⊆ { 1 , 2 , . . . , n }} partially ordered by S ≤ T if and only if S ⊆ T . { 1 , 2 , 3 } ✑ ◗ ✑✑✑✑✑ t ◗ ◗ ◗ ◗ ◗ { 1 , 2 } { 1 , 3 } { 2 , 3 } ✑✑✑✑✑ ◗ ◗ ✑ ◗ ✑✑✑✑✑ ◗ ✑ t t t ◗ ◗ ◗ ◗ B 3 = ◗ ◗ ◗ ◗ { 1 } { 2 } { 3 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ ∅ t

Example: The Boolean Algebra. The Boolean algebra is B n = { S : S ⊆ { 1 , 2 , . . . , n }} partially ordered by S ≤ T if and only if S ⊆ T . { 1 , 2 , 3 } ✑ ◗ ✑✑✑✑✑ t ◗ ◗ ◗ ◗ ◗ { 1 , 2 } { 1 , 3 } { 2 , 3 } ✑✑✑✑✑ ◗ ◗ ✑ ◗ ✑✑✑✑✑ ◗ ✑ t t t ◗ ◗ ◗ ◗ B 3 = ◗ ◗ ◗ ◗ { 1 } { 2 } { 3 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ ∅ t Note that B 3 looks like a cube.

Example: The Divisor Lattice.

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d .

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d . D 18 =

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d . D 18 = 1 t

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d . D 18 = 2 3 t t ❅ � ❅ � ❅ � ❅ � 1 t

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d . D 18 = 6 9 t t � ❅ � � ❅ � � ❅ � � ❅ � 2 3 t t ❅ � ❅ � ❅ � ❅ � 1 t

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d . 18 t � ❅ � ❅ D 18 = � ❅ � ❅ 6 9 t t � ❅ � � ❅ � � ❅ � � ❅ � 2 3 t t ❅ � ❅ � ❅ � ❅ � 1 t

Example: The Divisor Lattice. Given n ∈ Z > 0 the corresponding divisor lattice is D n = { d ∈ Z > 0 : d | n } partially ordered by c ≤ D n d if and only if c | d . 18 t � ❅ � ❅ D 18 = � ❅ � ❅ 6 9 t t � ❅ � � ❅ � � ❅ � � ❅ � 2 3 t t ❅ � ❅ � ❅ � ❅ � 1 t Note that D 18 looks like a rectangle.

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x .

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x .

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y .

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0.

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1.

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1.

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded:

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 ,

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n ,

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n , ˆ 0 B n = ∅ ,

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n , ˆ 0 B n = ∅ , ˆ 1 B n = { 1 , . . . , n } ,

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n , ˆ 0 B n = ∅ , ˆ 1 B n = { 1 , . . . , n } , ˆ 0 D n = 1 ,

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n , ˆ 0 B n = ∅ , ˆ 1 B n = { 1 , . . . , n } , ˆ 0 D n = 1 , ˆ 1 D n = n .

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n , ˆ 0 B n = ∅ , ˆ 1 B n = { 1 , . . . , n } , ˆ 0 D n = 1 , ˆ 1 D n = n . If x ≤ y in P then the corresponding closed interval is [ x , y ] = { z : x ≤ z ≤ y } .

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: 0 C n = 0 , ˆ ˆ 1 C n = n , ˆ 0 B n = ∅ , ˆ 1 B n = { 1 , . . . , n } , ˆ 0 D n = 1 , ˆ 1 D n = n . If x ≤ y in P then the corresponding closed interval is [ x , y ] = { z : x ≤ z ≤ y } . Open and half-open intervals are defined analogously.

In a poset P , a minimal element is x ∈ P such that there is no y ∈ P with y < x . A maximal element is x ∈ P such that there is no y ∈ P with y > x . y x t t ❅ ❅ Example. The poset on the left has w ❅ t minimal elements u and v , ❅ u v t t and maximal elements x and y . A poset has a zero if it has a unique minimal element, ˆ 0. A poset has a one if it has a unique maximal element, ˆ 1. A poset if bounded if it has both a ˆ 0 and a ˆ 1. Example. Our three fundamental examples are bounded: ˆ 0 C n = 0 , ˆ 1 C n = n , ˆ 0 B n = ∅ , ˆ 1 B n = { 1 , . . . , n } , ˆ 0 D n = 1 , ˆ 1 D n = n . If x ≤ y in P then the corresponding closed interval is [ x , y ] = { z : x ≤ z ≤ y } . Open and half-open intervals are defined analogously. Note that [ x , y ] is a poset in its own right and it has a zero and a one: ˆ ˆ 0 [ x , y ] = x , 1 [ x , y ] = y .

Example: The Chain. In C 9 we have the interval [ 4 , 7 ] :

Example: The Chain. In C 9 we have the interval [ 4 , 7 ] : 7 s 6 s 5 s 4 s

Example: The Chain. In C 9 we have the interval [ 4 , 7 ] : 7 s 6 s 5 s 4 s This interval looks like C 3 .

Example: The Boolean Algebra. In B 7 we have the interval [ { 3 } , { 2 , 3 , 5 , 6 } ] :

Example: The Boolean Algebra. In B 7 we have the interval [ { 3 } , { 2 , 3 , 5 , 6 } ] : t { 3 }

Example: The Boolean Algebra. In B 7 we have the interval [ { 3 } , { 2 , 3 , 5 , 6 } ] : { 2 , 3 } { 3 , 5 } { 3 , 6 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ t { 3 }

Example: The Boolean Algebra. In B 7 we have the interval [ { 3 } , { 2 , 3 , 5 , 6 } ] : { 2 , 3 , 5 } { 2 , 3 , 6 } { 3 , 5 , 6 } ◗ ✑✑✑✑✑ ◗ ✑ ✑✑✑✑✑ ◗ ◗ ✑ t t t ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ { 2 , 3 } { 3 , 5 } { 3 , 6 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ t { 3 }

Example: The Boolean Algebra. In B 7 we have the interval [ { 3 } , { 2 , 3 , 5 , 6 } ] : { 2 , 3 , 5 , 6 } ✑✑✑✑✑ ✑ ◗ ◗ t ◗ ◗ ◗ ◗ { 2 , 3 , 5 } { 2 , 3 , 6 } { 3 , 5 , 6 } ✑✑✑✑✑ ◗ ◗ ✑ ✑✑✑✑✑ ◗ ◗ ✑ t t t ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ { 2 , 3 } { 3 , 5 } { 3 , 6 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ t { 3 }

Example: The Boolean Algebra. In B 7 we have the interval [ { 3 } , { 2 , 3 , 5 , 6 } ] : { 2 , 3 , 5 , 6 } ✑✑✑✑✑ ✑ ◗ ◗ t ◗ ◗ ◗ ◗ { 2 , 3 , 5 } { 2 , 3 , 6 } { 3 , 5 , 6 } ✑✑✑✑✑ ◗ ◗ ✑ ✑✑✑✑✑ ◗ ◗ ✑ t t t ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ { 2 , 3 } { 3 , 5 } { 3 , 6 } ◗ ◗ ✑✑✑✑✑ ✑ t t t ◗ ◗ ◗ ◗ t { 3 } Note that this interval looks like B 3 .

Example: The Divisor Lattice. In D 80 we have the interval [ 2 , 40 ] :

Example: The Divisor Lattice. In D 80 we have the interval [ 2 , 40 ] : 2 t

Example: The Divisor Lattice. In D 80 we have the interval [ 2 , 40 ] : 10 4 t t ❅ � ❅ � ❅ � ❅ � 2 t

Example: The Divisor Lattice. In D 80 we have the interval [ 2 , 40 ] : 20 8 t t � ❅ � � ❅ � � ❅ � � ❅ � 10 4 t t ❅ � ❅ � ❅ � ❅ � 2 t

Example: The Divisor Lattice. In D 80 we have the interval [ 2 , 40 ] : 40 t � ❅ � ❅ � ❅ � ❅ 20 8 t t � ❅ � � ❅ � � ❅ � � ❅ � 10 4 t t ❅ � ❅ � ❅ � ❅ � 2 t

Example: The Divisor Lattice. In D 80 we have the interval [ 2 , 40 ] : 40 t � ❅ � ❅ � ❅ � ❅ 20 8 t t � ❅ � � ❅ � � ❅ � � ❅ � 10 4 t t ❅ � ❅ � ❅ � ❅ � 2 t Note that this interval looks like D 18 .

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y ,

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y .

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that 1. x ∨ y ≥ x and x ∨ y ≥ y ,

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that 1. x ∨ y ≥ x and x ∨ y ≥ y , 2. if z ≥ x and z ≥ y then z ≥ x ∧ y .

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that 1. x ∨ y ≥ x and x ∨ y ≥ y , 2. if z ≥ x and z ≥ y then z ≥ x ∧ y . We say P is a lattice if every x , y ∈ P have both a meet and a join.

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that 1. x ∨ y ≥ x and x ∨ y ≥ y , 2. if z ≥ x and z ≥ y then z ≥ x ∧ y . We say P is a lattice if every x , y ∈ P have both a meet and a join. Example. 1. C n is a lattice with i ∧ j = min { i , j } and i ∨ j = max { i , j } .

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that 1. x ∨ y ≥ x and x ∨ y ≥ y , 2. if z ≥ x and z ≥ y then z ≥ x ∧ y . We say P is a lattice if every x , y ∈ P have both a meet and a join. Example. 1. C n is a lattice with i ∧ j = min { i , j } and i ∨ j = max { i , j } . 2. B n is a lattice with S ∧ T = S ∩ T and S ∨ T = S ∪ T .

If P is a poset then x , y ∈ P have a greatest lower bound or meet if there is an element x ∧ y in P such that 1. x ∧ y ≤ x and x ∧ y ≤ y , 2. if z ≤ x and z ≤ y then z ≤ x ∧ y . Also x , y ∈ P have a least upper bound or join if there is an element x ∨ y in P such that 1. x ∨ y ≥ x and x ∨ y ≥ y , 2. if z ≥ x and z ≥ y then z ≥ x ∧ y . We say P is a lattice if every x , y ∈ P have both a meet and a join. Example. 1. C n is a lattice with i ∧ j = min { i , j } and i ∨ j = max { i , j } . 2. B n is a lattice with S ∧ T = S ∩ T and S ∨ T = S ∪ T . 3. D n is a lattice with c ∧ d = gcd { c , d } and c ∨ d = lcm { c , d } .

Outline Motivating Examples Poset Basics Isomorphism and Products

For posets P and Q , an order preserving map is f : P → Q with x ≤ P y = ⇒ f ( x ) ≤ Q f ( y ) .

For posets P and Q , an order preserving map is f : P → Q with x ≤ P y = ⇒ f ( x ) ≤ Q f ( y ) . An isomorphism is a bijection f : P → Q such that both f and f − 1 are order preserving. In this case P and Q are isomorphic , written P ∼ = Q .