The Coherent Framed Join and Biassociahedra Joint work with Samson - PowerPoint PPT Presentation

Augmented Consecutive Partitions � Let B be an ordered set � ACP B B = B

Augmented Consecutive Partitions � Let B be an ordered set � ACP B B = B � ACP B ∅ = 0 | · · · | 0 ( # empty blocks = # B + 1)

Augmented Consecutive Partitions � Let B be an ordered set � ACP B B = B � ACP B ∅ = 0 | · · · | 0 ( # empty blocks = # B + 1) � ACP { 1 , 2 , ... , 9 } { 2 , 5 , 6 , 8 } = 0 | 2 | 0 | 56 | 8 | 0

Factoring a Bipartition � Given C = B 1 | · · · | B r , for each k = 1 , 2 , . . . r : A 1 | · · · | A r

Factoring a Bipartition � Given C = B 1 | · · · | B r , for each k = 1 , 2 , . . . r : A 1 | · · · | A r � Compute a k , 1 | · · · | a k , s k : = ACP A 1 ∪···∪ A k A k b k , 1 | · · · | b k , t k : = ACP B k ∪···∪ B r B k

Factoring a Bipartition � Given C = B 1 | · · · | B r , for each k = 1 , 2 , . . . r : A 1 | · · · | A r � Compute a k , 1 | · · · | a k , s k : = ACP A 1 ∪···∪ A k A k b k , 1 | · · · | b k , t k : = ACP B k ∪···∪ B r B k � Construct the bipartition matrix   b k , 1 b k , 1 · · · a k , 1 a k , sk   . .   . . C k = . .   b k , tk b k , tk · · · a k , 1 a k , sk

Factoring a Bipartition � Given C = B 1 | · · · | B r , for each k = 1 , 2 , . . . r : A 1 | · · · | A r � Compute a k , 1 | · · · | a k , s k : = ACP A 1 ∪···∪ A k A k b k , 1 | · · · | b k , t k : = ACP B k ∪···∪ B r B k � Construct the bipartition matrix   b k , 1 b k , 1 · · · a k , 1 a k , sk   . .   . . C k = . .   b k , tk b k , tk · · · a k , 1 a k , sk � C = C 1 · · · C r

Factoring a Bipartition 56 | 7 | 8 � Example 1 | 23 | 4 1 = ACP 1 1 56 | 0 | 0 = ACP 5678 56 0 | 23 = ACP 123 23 7 | 0 = ACP 78 7 0 | 0 | 0 | 4 = ACP 1234 4 8 = ACP 8 8

Factoring a Bipartition 56 | 7 | 8 � Example 1 | 23 | 4 1 = ACP 1 1 56 | 0 | 0 = ACP 5678 56 0 | 23 = ACP 123 23 7 | 0 = ACP 78 7 0 | 0 | 0 | 4 = ACP 1234 4 8 = ACP 8 8   56 � 7 � 1 7   � 8 � 56 | 7 | 8 0 23   0 8 8 8 � 1 | 23 | 4 =   1 0 0 0 4 0 0 0 23 0 1

Graphical Representation # B + 1 B � A ← # A + 1

Graphical Representation # B + 1 B � A ← # A + 1   56 � 7 � � 56 | 7 | 8 � 1 7   � 8 � 0 23   0 8 8 8 � =   1 0 0 0 4 1 | 23 | 4 0 0 0 23 0 1 = =

Dimension of a Bipartition Matrix � A null matrix with entries of the form 0 |···| 0 0 |···| 0 has dim 0

Dimension of a Bipartition Matrix � A null matrix with entries of the form 0 |···| 0 0 |···| 0 has dim 0 � � �� B �� � : = # A + # B − 1 A

Dimension of a Bipartition Matrix � A null matrix with entries of the form 0 |···| 0 0 |···| 0 has dim 0 � � �� B �� � : = # A + # B − 1 A � | C 1 · · · C r | : = | C 1 | + · · · + | C r |

Dimension of a Bipartition Matrix � A null matrix with entries of the form 0 |···| 0 0 |···| 0 has dim 0 � � �� B �� � : = # A + # B − 1 A � | C 1 · · · C r | : = | C 1 | + · · · + | C r | � Unique factorization ⇒ Define | C | for C indecomposable

Dimension of a Bipartition Matrix � A null matrix with entries of the form 0 |···| 0 0 |···| 0 has dim 0 � � �� B �� � : = # A + # B − 1 A � | C 1 · · · C r | : = | C 1 | + · · · + | C r | � Unique factorization ⇒ Define | C | for C indecomposable � β ij � � Let C = be a q × p indecomposable bipartition matrix α ij over { a j , b i }

Dimension of a Bipartition Matrix β ij α ij = 0 |···| 0 for all ( i , j ) , let C i ∗ denote the i th row of C � If α ij

Dimension of a Bipartition Matrix β ij α ij = 0 |···| 0 for all ( i , j ) , let C i ∗ denote the i th row of C � If α ij � Let ( λ i 1 · · · λ i p ) be the λ matrix associated with C i ∗

Dimension of a Bipartition Matrix β ij α ij = 0 |···| 0 for all ( i , j ) , let C i ∗ denote the i th row of C � If α ij � Let ( λ i 1 · · · λ i p ) be the λ matrix associated with C i ∗ � Define � � | · · · | � � A 1 | · · · | A n � A � 1 | · · · | A � A 1 ∪ A � A n ∪ A � n : = 1 n

Dimension of a Bipartition Matrix β ij α ij = 0 |···| 0 for all ( i , j ) , let C i ∗ denote the i th row of C � If α ij � Let ( λ i 1 · · · λ i p ) be the λ matrix associated with C i ∗ � Define � � | · · · | � � A 1 | · · · | A n � A � 1 | · · · | A � A 1 ∪ A � A n ∪ A � n : = 1 n � Form partitions ∧ 1 ( α i 1 ) � · · · � µ λ i α i : = µ λ i p ( α ip ))

Dimension of a Bipartition Matrix β ij α ij = 0 |···| 0 for all ( i , j ) , let C i ∗ denote the i th row of C � If α ij � Let ( λ i 1 · · · λ i p ) be the λ matrix associated with C i ∗ � Define � � | · · · | � � A 1 | · · · | A n � A � 1 | · · · | A � A 1 ∪ A � A n ∪ A � n : = 1 n � Form partitions ∧ 1 ( α i 1 ) � · · · � µ λ i α i : = µ λ i p ( α ip )) ∧ � Define | C | : = | α i | ∑ 1 ≤ i ≤ q

Dimension of a Bipartition Matrix � �� 0 �� � = | 13 | = 1 0 � Example 1 3

Dimension of a Bipartition Matrix � �� 0 �� � = | 13 | = 1 0 � Example 1 3 � | C | is not necessarily the sum of the dim’s of its entries

Dimension of a Bipartition Matrix � �� 0 �� � = | 13 | = 1 0 � Example 1 3 � | C | is not necessarily the sum of the dim’s of its entries ∨ β ij � If c ij = 0 |···| 0 for all ( i , j ) , form partitions β j in each column

Dimension of a Bipartition Matrix � �� 0 �� � = | 13 | = 1 0 � Example 1 3 � | C | is not necessarily the sum of the dim’s of its entries ∨ β ij � If c ij = 0 |···| 0 for all ( i , j ) , form partitions β j in each column ∨ � Define | C | = ∑ | β j | 1 ≤ j ≤ p

Dimension of a Bipartition Matrix � �� 0 �� � = | 13 | = 1 0 � Example 1 3 � | C | is not necessarily the sum of the dim’s of its entries ∨ β ij � If c ij = 0 |···| 0 for all ( i , j ) , form partitions β j in each column ∨ � Define | C | = ∑ | β j | 1 ≤ j ≤ p � Otherwise...

Conventions for Bipartition Matrices � Deleting or inserting empty blocks in an entry of a bipartition matrix may preserve or change dimension

Conventions for Bipartition Matrices � Deleting or inserting empty blocks in an entry of a bipartition matrix may preserve or change dimension � Discard bipartition matrices whose dimension increases when empty blocks are inserted

Conventions for Bipartition Matrices � Deleting or inserting empty blocks in an entry of a bipartition matrix may preserve or change dimension � Discard bipartition matrices whose dimension increases when empty blocks are inserted � Example Discard the 1-dim’l indecomposable matrix � 0 | 1 � 0 | 1 1 C = 1 | 0 1 | 0 1 Inserting empty blocks in the third entry transforms C into the 3-dim’l decomposable � 0 � � 0 | 1 � 0 0 � 1 � 0 | 1 0 | 1 1 1 0 1 1 1 1 = . 0 0 0 0 0 0 0 1 1 | 0 1 | 0 0 | 1 1 1 0

Conventions for Bipartition Matrices � Equate bipartition matrices of the same dimension that differ only in the number of empty blocks in their entries

Conventions for Bipartition Matrices � Equate bipartition matrices of the same dimension that differ only in the number of empty blocks in their entries     0 0 0 0 1 3 1 3   =   � Example 0 | 0 | 0 0 | 0 | 0 0 | 0 0 | 0 0 | 1 | 0 0 | 0 | 3 1 | 0 0 | 3

Conventions for Bipartition Matrices � Equate bipartition matrices of the same dimension that differ only in the number of empty blocks in their entries     0 0 0 0 1 3 1 3   =   � Example 0 | 0 | 0 0 | 0 | 0 0 | 0 0 | 0 0 | 1 | 0 0 | 0 | 3 1 | 0 0 | 3 � Only preserve empty blocks necessary to preserve dimension

Conventions for Bipartition Matrices � Equate bipartition matrices of the same dimension that differ only in the number of empty blocks in their entries     0 0 0 0 1 3 1 3   =   � Example 0 | 0 | 0 0 | 0 | 0 0 | 0 0 | 0 0 | 1 | 0 0 | 0 | 3 1 | 0 0 | 3 � Only preserve empty blocks necessary to preserve dimension � Example Preserve all empty blocks in   0 0 1 3   C = 0 | 0 0 | 0 1 | 0 0 | 3 Removing empty blocks in the second row increases dimension

Coherence � β ij � � Definition A q × p indecomposable bipartition matrix α ij over { a j , b i } is

Coherence � β ij � � Definition A q × p indecomposable bipartition matrix α ij over { a j , b i } is � column coherent if ∧ ∧ α 1 ) � ∆ ( q − 1 ) ( P # ( a 1 ∪···∪ a p ) ) π ( α q ) × · · · × π (

Coherence � β ij � � Definition A q × p indecomposable bipartition matrix α ij over { a j , b i } is � column coherent if ∧ ∧ α 1 ) � ∆ ( q − 1 ) ( P # ( a 1 ∪···∪ a p ) ) π ( α q ) × · · · × π ( � row coherent if ∨ ∨ β p ) � ∆ ( p − 1 ) ( P # ( b 1 ∪···∪ b q ) ) π ( β 1 ) × · · · × π (

Coherence � β ij � � Definition A q × p indecomposable bipartition matrix α ij over { a j , b i } is � column coherent if ∧ ∧ α 1 ) � ∆ ( q − 1 ) ( P # ( a 1 ∪···∪ a p ) ) π ( α q ) × · · · × π ( � row coherent if ∨ ∨ β p ) � ∆ ( p − 1 ) ( P # ( b 1 ∪···∪ b q ) ) π ( β 1 ) × · · · × π ( � coherent if column and row coherent

Coherent Framed Elements � Given a ( m ) and b ( n ) of orders m and n , and r ≥ 1 , let β α ∈ P � r ( a ( m )) × P � r ( b ( n ))

Coherent Framed Elements � Given a ( m ) and b ( n ) of orders m and n , and r ≥ 1 , let β α ∈ P � r ( a ( m )) × P � r ( b ( n )) � If r = 1 or mn = 0 , the set of coherent framed elements �� β �� α � c β : = α

Coherent Framed Elements � Given a ( m ) and b ( n ) of orders m and n , and r ≥ 1 , let β α ∈ P � r ( a ( m )) × P � r ( b ( n )) � If r = 1 or mn = 0 , the set of coherent framed elements �� β �� α � c β : = α � Otherwise, assume inductively that the set of coherent framed elements α � � c β � has been defined for all β � α � ∈ P � ( a ( s )) × P � ( b ( t )) such that ( s , t ) ≤ ( m , n ) and s + t < m + n

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r � Compute a 1 | · · · | a p : = ACP A 1 ∪···∪ A k A k

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r � Compute a 1 | · · · | a p : = ACP A 1 ∪···∪ A k A k � Compute b 1 | · · · | b q : = ACP B k ∪···∪ B r B k

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r � Compute a 1 | · · · | a p : = ACP A 1 ∪···∪ A k A k � Compute b 1 | · · · | b q : = ACP B k ∪···∪ B r B k � � � � β � � Choose R ∈ N q × p and indecomposable i over a j , b i α � j w.r.t. R

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r � Compute a 1 | · · · | a p : = ACP A 1 ∪···∪ A k A k � Compute b 1 | · · · | b q : = ACP B k ∪···∪ B r B k � � � � β � � Choose R ∈ N q × p and indecomposable i over a j , b i α � j w.r.t. R j � c β � � Choose c k ij ∈ α � i

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r � Compute a 1 | · · · | a p : = ACP A 1 ∪···∪ A k A k � Compute b 1 | · · · | b q : = ACP B k ∪···∪ B r B k � � � � β � � Choose R ∈ N q × p and indecomposable i over a j , b i α � j w.r.t. R j � c β � � Choose c k ij ∈ α � i � � c k � Form the coherent framed matrix C k = ij

Coherent Framed Matrices � Given B 1 | · · · | B r = β α , for k = 1 , 2 , . . . , r : A 1 | · · · | A r � Compute a 1 | · · · | a p : = ACP A 1 ∪···∪ A k A k � Compute b 1 | · · · | b q : = ACP B k ∪···∪ B r B k � � � � β � � Choose R ∈ N q × p and indecomposable i over a j , b i α � j w.r.t. R j � c β � � Choose c k ij ∈ α � i � � c k � Form the coherent framed matrix C k = ij � The set of coherent framed elements α � c β : = { C 1 · · · C r } , where C i ranges over all possible coherent framed matrices and the product is formal juxtaposition

The Coherent Framed Join of Ordered Sets � Definition The coherent framed join of a ( m ) and b ( n ) is the set � a ( m ) � pp b ( n ) : = α � c β β α ∈ P � r ( a ( m )) × P � r ( b ( n )) r ≥ 1

The Coherent Framed Join of Ordered Sets � Definition The coherent framed join of a ( m ) and b ( n ) is the set � a ( m ) � pp b ( n ) : = α � c β β α ∈ P � r ( a ( m )) × P � r ( b ( n )) r ≥ 1 � Example � � 0 � � 1 �� � � 1 � � 0 1 � pp 1 = 0 | 1 1 | 0 1 1 1 | 0 = 0 | 1 = 1 , 1 , 0 0 0 0 1 1

The Coherent Framed Join of Ordered Sets � Definition The coherent framed join of a ( m ) and b ( n ) is the set � a ( m ) � pp b ( n ) : = α � c β β α ∈ P � r ( a ( m )) × P � r ( b ( n )) r ≥ 1 � Example � � 0 � � 1 �� � � 1 � � 0 1 � pp 1 = 0 | 1 1 | 0 1 1 1 | 0 = 0 | 1 = 1 , 1 , 0 0 0 0 1 1 � PP 2 , 2 = KK 2 , 2 ↔ 1 � pp 1 0 | 1 1 | 0 1 1 | 0 1 0 | 1

The Coherent Framed Join of Ordered Sets � Definition The coherent framed join of a ( m ) and b ( n ) is the set � a ( m ) � pp b ( n ) : = α � c β β α ∈ P � r ( a ( m )) × P � r ( b ( n )) r ≥ 1 � Example � � 0 � � 1 �� � � 1 � � 0 1 � pp 1 = 0 | 1 1 | 0 1 1 1 | 0 = 0 | 1 = 1 , 1 , 0 0 0 0 1 1 � PP 2 , 2 = KK 2 , 2 ↔ 1 � pp 1 0 | 1 1 | 0 1 1 | 0 1 0 | 1 � The Hopf relation holds up to homotopy

For 12 � pp 1, let W be the set obtained by � Example 1 inserting empty blocks into 12 in all possible ways that preserve coherence

For 12 � pp 1, let W be the set obtained by � Example 1 inserting empty blocks into 12 in all possible ways that preserve coherence � 12 , 0 | 1 1 | 2 , 0 | 1 2 | 1 , 1 | 0 1 | 2 , 1 | 0 2 | 1 , 1 | 0 1 � W = 0 | 12 , � 0 | 0 | 1 1 | 2 | 0 , 0 | 0 | 1 2 | 1 | 0 , 0 | 1 | 0 1 | 0 | 2 , 0 | 1 | 0 2 | 0 | 1 , 1 | 0 | 0 0 | 1 | 2 , 1 | 0 | 0 0 | 2 | 1

For 12 � pp 1, let W be the set obtained by � Example 1 inserting empty blocks into 12 in all possible ways that preserve coherence � 12 , 0 | 1 1 | 2 , 0 | 1 2 | 1 , 1 | 0 1 | 2 , 1 | 0 2 | 1 , 1 | 0 1 � W = 0 | 12 , � 0 | 0 | 1 1 | 2 | 0 , 0 | 0 | 1 2 | 1 | 0 , 0 | 1 | 0 1 | 0 | 2 , 0 | 1 | 0 2 | 0 | 1 , 1 | 0 | 0 0 | 1 | 2 , 1 | 0 | 0 0 | 2 | 1 � 0 � � 1 � 0 | 1 � Note that 1 1 12 | 0 = 12 is incoherent because 0 0 0 0 12 ∧ ∧ α 1 ) = 12 × 12 �� ∆ ( 1 ) ( P 2 ) π ( α 2 ) × π (

For 12 � pp 1, let W be the set obtained by � Example 1 inserting empty blocks into 12 in all possible ways that preserve coherence � 12 , 0 | 1 1 | 2 , 0 | 1 2 | 1 , 1 | 0 1 | 2 , 1 | 0 2 | 1 , 1 | 0 1 � W = 0 | 12 , � 0 | 0 | 1 1 | 2 | 0 , 0 | 0 | 1 2 | 1 | 0 , 0 | 1 | 0 1 | 0 | 2 , 0 | 1 | 0 2 | 0 | 1 , 1 | 0 | 0 0 | 1 | 2 , 1 | 0 | 0 0 | 2 | 1 � 0 � � 1 � 0 | 1 � Note that 1 1 12 | 0 = 12 is incoherent because 0 0 0 0 12 ∧ ∧ α 1 ) = 12 × 12 �� ∆ ( 1 ) ( P 2 ) π ( α 2 ) × π ( � Replace entries in all possible ways to obtain coherence �� 0 | 0 � � 1 � � � 1 � 0 | 0 � � 1 �� � � 0 12 | 0 � c 0 | 1 = 1 1 1 1 1 1 12 2 | 1 2 | 1 , , 0 | 0 0 0 0 0 0 0 0 0 | 0 0 0 0 12 1 | 2 1 | 2

For 12 � pp 1, let W be the set obtained by � Example 1 inserting empty blocks into 12 in all possible ways that preserve coherence � 12 , 0 | 1 1 | 2 , 0 | 1 2 | 1 , 1 | 0 1 | 2 , 1 | 0 2 | 1 , 1 | 0 1 � W = 0 | 12 , � 0 | 0 | 1 1 | 2 | 0 , 0 | 0 | 1 2 | 1 | 0 , 0 | 1 | 0 1 | 0 | 2 , 0 | 1 | 0 2 | 0 | 1 , 1 | 0 | 0 0 | 1 | 2 , 1 | 0 | 0 0 | 2 | 1 � 0 � � 1 � 0 | 1 � Note that 1 1 12 | 0 = 12 is incoherent because 0 0 0 0 12 ∧ ∧ α 1 ) = 12 × 12 �� ∆ ( 1 ) ( P 2 ) π ( α 2 ) × π ( � Replace entries in all possible ways to obtain coherence �� 0 | 0 � � 1 � � � 1 � 0 | 0 � � 1 �� � � 0 12 | 0 � c 0 | 1 = 1 1 1 1 1 1 12 2 | 1 2 | 1 , , 0 | 0 0 0 0 0 0 0 0 0 | 0 0 0 0 12 1 | 2 1 | 2 � 12 � pp 1 = W ∪ ( 12 | 0 � c 0 | 1 )

The Differential � Let m = { 1 , 2 , . . . , m } ; let ρ ∈ m � pp n

The Differential � Let m = { 1 , 2 , . . . , m } ; let ρ ∈ m � pp n � For top dim’l ρ = n m define � n � = { codim 1 elements of m � pp n } ˜ ∂ m

The Differential � Let m = { 1 , 2 , . . . , m } ; let ρ ∈ m � pp n � For top dim’l ρ = n m define � n � = { codim 1 elements of m � pp n } ˜ ∂ m � Example � � 0 | 0 �� 1 � �� 1 �� � 0 ˜ 0 | 1 1 | 2 , 0 | 1 2 | 1 , 1 | 0 1 | 2 , 1 | 0 2 | 1 , 1 | 0 ∂ ( 1 1 1 1 1 12 ) = 0 | 12 , 2 | 1 , 12 0 0 0 0 | 0 0 0 0 0 12 1 | 2

The Differential � For lower dim’l cells insert empty blocks and subdivide in all possible ways that preserve coherence

The Differential � For lower dim’l cells insert empty blocks and subdivide in all possible ways that preserve coherence � � 0 | 1 = 0 | 1 | 0 1 | 0 | 2 ∪ 0 | 0 | 1 � ˜ ∂ 1 | 2 1 | 2 | 0

The Differential � For lower dim’l cells insert empty blocks and subdivide in all possible ways that preserve coherence � � 0 | 1 = 0 | 1 | 0 1 | 0 | 2 ∪ 0 | 0 | 1 � ˜ ∂ 1 | 2 1 | 2 | 0 � � � ˜ 1 | 0 = 1 | 0 | 0 0 | 1 | 2 ∪ 1 | 0 | 0 ∂ 0 | 12 0 | 2 | 1

The Differential � For lower dim’l cells insert empty blocks and subdivide in all possible ways that preserve coherence � � 0 | 1 = 0 | 1 | 0 1 | 0 | 2 ∪ 0 | 0 | 1 � ˜ ∂ 1 | 2 1 | 2 | 0 � � � ˜ 1 | 0 = 1 | 0 | 0 0 | 1 | 2 ∪ 1 | 0 | 0 ∂ 0 | 12 0 | 2 | 1 �� 0 | 0 �� 1 �� � 0 | 0 �� 1 � 0 | 0 �� 1 � ∪ � � ˜ 1 1 1 1 1 1 2 | 1 2 | 1 2 | 1 ∂ = 0 0 0 0 0 | 0 0 0 0 0 | 0 0 0 0 12 1 | 2 2 | 1

PP(2,3) = KK(2,3)

PP(2,3) = KK(2,3)

The Reduced Coherent Framed Join of Ordered Sets � Define an equivalence relation ∼ on a ( m ) � pp b ( n ) :

The Reduced Coherent Framed Join of Ordered Sets � Define an equivalence relation ∼ on a ( m ) � pp b ( n ) : � � � � ∼ C � = � C = c � iff c ij and c � c ij ij differ only in the ij number or placement of empty blocks 0 0

The Reduced Coherent Framed Join of Ordered Sets � Define an equivalence relation ∼ on a ( m ) � pp b ( n ) : � � � � ∼ C � = � C = c � iff c ij and c � c ij ij differ only in the ij number or placement of empty blocks 0 0 � � � � � 0 � = 0 | 0 0 | 0 0 | 0 0 | 0 � Example 0 = 1 3 1 | 0 0 | 3 0 | 1 3 | 0

The Reduced Coherent Framed Join of Ordered Sets � Define an equivalence relation ∼ on a ( m ) � pp b ( n ) : � � � � ∼ C � = � C = c � iff c ij and c � c ij ij differ only in the ij number or placement of empty blocks 0 0 � � � � � 0 � = 0 | 0 0 | 0 0 | 0 0 | 0 � Example 0 = 1 3 1 | 0 0 | 3 0 | 1 3 | 0 � Definition The reduced coherent framed join of a ( m ) and b ( n ) is the set a ( m ) � kk b ( n ) = a ( m ) � pp b ( n ) / ∼