In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.
In 1941, Hopf [1] first put forward the concept of both algebra structure and coalgebra structure in the study of cohomology algebra H * ( G , K ) of Lie group G. After that, more and more interesting questions about algebras and coalgebras have attracted many mathematicians to work and study on them continuously. Among those questions, it is a hot topic how to construct algebras and coalgebras on combinatorial objects.
In 2014, Vargas [2] defined a super-shuffle product ш _ and a coproduct Δ ⋄ , called cut-box coproduct by Liu and Li [3] on permutations. In 2005, Aguiar and Sottile introduced the global descents of permutations in symmetric groups [4] . On this basis, Zhao and Li derived another shuffle product and deconcatenation coproduct from the classical one on permutations. Then they proved the vector space spanned by permutations with the shuffle product that is a graded algebra and with the deconcatenation coproduct that is a graded coalgebra [5] in 2020. In the same year, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs without proof [6] . In 2023, Dong [7] proved the vector space spanned by labeled graphs with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.
In fact, matrices are related to permutations and graphs closely. A (0,1)-matrix is a matrix whose entries are all 0 or 1, also called a binary matrix. It is widely used in graph theory [8] [9] , combinatorics [10] , linear programming [11] [12] [13] and computer science [14] . In this paper, we first generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices, then we prove that the vector space with the super-shuffle product that is a graded algebra and with the cut-box coproduct that is a graded coalgebra.
This paper is organized as follows. We start by recalling some notations on (0,1)-matrices and defining the vector space M spanned by (0,1)-matrices in Section 2. In Section 3, we define the cut-box coproduct Δ on M and prove M with coproduct Δ that is a graded coalgebra. In Section 4, we define the super-shuffle product ∗ on M and prove M with product ∗ that is a graded algebra. Lastly, we summarize our main conclusions in Section 5.
2. Basic Definitions
An s × n matrix A = ( a i j ) s × n is called a (0,1)-matrix if
A = ( a 11 a 12 a 21 a 22 ⋯ a 1 n ⋯ a 2 n ⋮ ⋮ a s 1 a s 2 ⋱ ⋮ ⋯ a s n ) s × n ,
where a i j is either 0 or 1. In particular, the empty matrix is the matrix with no entries, denoted by ε .
Define
[ n ] = ( { 1 , 2 , ⋯ , n } , n > 0 , ∅ , n = 0 ,
and
[ i , j ] = ( { i , i + 1 , ⋯ , j } , i ≤ j , ∅ , i > j .
Let I = { i 1 , i 2 , ⋯ , i k } ⊆ [ s ] and J = { j 1 , j 2 , ⋯ , j q } ⊆ [ n ] , where i 1 < i 2 < ⋯ < i k ≤ s and j 1 < j 2 < ⋯ < j q ≤ n . For an s × n (0,1)-matrix A, the restriction of A on I × J is the submatrix formed by the entries, in the same relative positions, in both rows indexed by I and columns indexed by J, denoted by A I × J . In particular, if I = [ s ] and J = [ n ] , A I × J = A and if I or J is empty, A I × J = ε . For convenience, let A I denote A I × I and call A I the restriction of A on I.
Let M n = { A | A = ( a i j ) is an n × n ( 0,1 ) -matrix } and M n be the vector space spanned by M n over field K , for any non-negative integer n. For example,
In particular, M 0 = { ε } and M 0 = K M 0 . Denote
M = ∪ n = 0 ∞ M n and M = ⊕ n = 0 ∞ M n .
If A and B are both non-empty matrices, then we denote A ⋄ B = [ A O O B ] , where O’s are zero matrices. In particular, ε ⋄ A = A ⋄ ε = A for any (0,1)-matrix A.
Example 2. For A = ( 0 0 1 1 ) and B = ( 1 0 0 1 0 1 0 1 1 ) , we have
For A in M n , we call i a spilt of A, if A [ i ] ⋄ A [ n ] ] \ [ i ] = A , where 0 ≤ i ≤ n . By the definition, 0 and n are always splits of a (0,1)-matrix in M n when n ≥ 1 , called trivial splits. Obviously, A [ i ] × ( [ n ] \ [ i ] ) = A ( [ n ] \ [ i ] ) × [ i ] = ε when i is a trivial spilt of A; A [ i ] × ( [ n ] \ [ i ] ) = O i × ( n − i ) and A ( [ n ] \ [ i ] ) × [ i ] = O ( n − i ) × i when i is a non-trivial spilt of A. We call A indecomposible if it is non-empty and only has trivial splits.
For A in M n , n ≥ 1 , suppose that { i 0 , i 1 , ⋯ , i s } is the set of all splits of A, where 0 = i 0 < i 1 < ⋯ < i s = n . We call A [ i k − 1 + 1, i k ] an atom of A, 1 ≤ k ≤ s . Obviously, there is no non-trivial split of A [ i k − 1 + 1, i k ] for 1 ≤ k ≤ s . Let A k = A [ i k − 1 + 1 , i k ] , for 1 ≤ k ≤ s . We define the decomposition of A by A = A 1 ⋄ A 2 ⋄ ⋯ ⋄ A s .
In particular, when A is indecomposable or empty, its decomposition is itself.
2) The set of splits of [ 1 0 1 0 1 0 1 0 1 ] is { 0,3 } , so it is indecomposable. Its decomposition is itself, and so is its atom.
3. Cut-Box Coproduct and Coalgebra
In this section, we define the cut-box coproduct on the vector space M . Then we prove the space with the cut-box coproduct is a graded coalgebra.
Define the cut-box coproduct Δ on M by Δ ( A ) = ∑ j = 0 s A 1 ⋄ ⋯ ⋄ A j ⊗ A j + 1 ⋄ ⋯ ⋄ A s for non-empty matrix A in M n with decomposition A = A 1 ⋄ A 2 ⋄ ⋯ ⋄ A s , where A 1 ⋄ ⋯ ⋄ A 0 = A s + 1 ⋄ ⋯ ⋄ A s = ε . In particular, define Δ ( ε ) = ε ⊗ ε .
Define the counit ν from M to K by
ν ( A ) = ( 1, A = ε , 0, otherwise ,
for A in M.
Example 4. From Example 3 and the definition of the cut-box coproduct, we have
Proof. It is easy to verify that ν is a counit. Obviously, ( id ⊗ Δ ) ∘ Δ ( ε ) = ε ⊗ ε ⊗ ε = ( Δ ⊗ id ) ∘ Δ ( ε ) .
Suppose A in M n with n ≥ 1 and its decomposition is A = A 1 ⋄ A 2 ⋄ ⋯ ⋄ A s .
Then
( id ⊗ Δ ) ∘ Δ ( A ) = ( id ⊗ Δ ) ∘ Δ ( A 1 ⋄ A 2 ⋄ ⋯ ⋄ A s ) = ( id ⊗ Δ ) ( ∑ j = 0 s A 1 ⋄ ⋯ ⋄ A j ⊗ A j + 1 ⋄ ⋯ ⋄ A s ) = ∑ j = 0 s A 1 ⋄ ⋯ ⋄ A j ⊗ ( ∑ k = j s A j + 1 ⋄ ⋯ ⋄ A k ⊗ A k + 1 ⋄ ⋯ ⋄ A s )
= ∑ 0 ≤ j ≤ k ≤ s A 1 ⋄ ⋯ ⋄ A j ⊗ A j + 1 ⋄ ⋯ ⋄ A k ⊗ A k + 1 ⋄ ⋯ ⋄ A s = ∑ k = 0 s ( ∑ j = 0 k A 1 ⋄ ⋯ ⋄ A j ⊗ A j + 1 ⋄ ⋯ ⋄ A k ) ⊗ A k + 1 ⋄ ⋯ ⋄ A s = ( Δ ⊗ id ) ( ∑ k = 0 s A 1 ⋄ ⋯ ⋄ A k ⊗ A k + 1 ⋄ ⋯ ⋄ A s ) = ( Δ ⊗ id ) ∘ Δ ( A ) ,
where A j + 1 ⋄ ⋯ ⋄ A j = ε for 0 ≤ j ≤ s . So Δ satisfies the coassociative law.
Obviously, by the definitions of Δ and ν , we have Δ ( M n ) ⊆ ⊕ M i ⊗ M n − i and ν ( M n ) = 0 for n > 0 . Hence ( M , Δ , ν ) is a graded coalgebra.
□
4. Super-Shuffle Product and Algebra
In this section, we define the super-shuffle product on the vector space M . Then we prove the space with the super-shuffle product is a graded algebra.
Define the super-shuffle product ∗ on M by
A ∗ B = ∑ C ∈ M m + n I , J : I ∪ J = [ m + n ] C I = A , C J = B C (1)
for A in M m and B in M n , where C traverses all matrices in M m + n with the restriction on I is A, on J is B, on I × J and J × I are arbitrary (0,1)-matrices. Obviously, the product ∗ is commutative and ε ∗ A = A ∗ ε = A , for any A in M. Define the unit μ from K to M by μ ( 1 ) = ε .
Example 5. For A = [ 1 1 1 0 ] , B = [ 1 ] we have
Here, we color the entries of C in A ∗ B restricted to A red and to B blue, respectively. Although [ 1 0 1 0 1 1 1 1 0 ] and [ 1 0 1 0 1 1 1 1 0 ] are same matrices, we consider they are different. Then each term in A ∗ B is unique.
Let W = { i 1 , i 2 , ⋯ , i n } be a set of positive integers where i 1 < i 2 < ⋯ < i n . Define a mapping st W from W to [ | W | ] by st W ( i a ) = a for 1 ≤ a ≤ n , and call it the standardization of W [6] . For a subset T of W, denote st W ( T ) = { st W ( i ) | i ∈ T } . Obviously, st W is a 1-1 mapping from the set of subsets of W to the set of subsets of [ | W | ] . Therefore, for any subset H of [ | W | ] , there must exist a unique subset P of W such that H = st W ( P ) .
Remark 1. ( [15] ) Let W be a set of positive integers and P be a subset of W. Then there exists a unique subset H in [ | W | ] such that
st P ( i ) = st H ( st W ( i ) ) ,
for any i in P. Actually, H = st W ( P ) .
Example 6. For W = { 3,5,7,8,9 } and P = { 3,7,8 } , st P ( 3 ) = 1 , st P ( 7 ) = 2 , st P ( 8 ) = 3 , st W ( 3 ) = 1 , st W ( 7 ) = 3 and st W ( 8 ) = 4 . Take H = st W ( P ) = st W ( { 3,7,8 } ) = { 1,3,4 } . Furthermore, st H ( st W ( 3 ) ) = 1 , st H ( st W ( 7 ) ) = 2 , st H ( st W ( 8 ) ) = 3 .
Next, in order to prove ( M , ∗ , μ ) is a graded algebra, we give one lemma.
Lemma 2. Assume A = ( a i j ) n × n is a (0,1)-matrix, W ⊆ [ n ] and H ⊆ [ | W | ] . Then there exists a subset P of W such that A P = ( A W ) H .
Proof. By the definition of st W , there must exist a subset P of W such that st W ( P ) = H . Next, we prove A P = ( A W ) H .
Let A P be B = ( b i j ) , A W be C = ( c i j ) and ( A W ) H be D = ( d i j ) . Obviously, B and D are both (0,1)-matrices. We just need to show that b i j = d i j for each i , j in [ | P | ] . For b i j in, there must exist i ′ and j ′ in P such that st P ( i ′ ) = i , st P ( j ′ ) = j and. Since P is a subset of W, there must exist i ′ ′ and j ′ ′ in [ | W | ] such that st W ( i ′ ) = i ′ ′ , st W ( j ′ ) = j ′ ′ and a i ′ j ′ = c i ′ ′ j ′ ′ . On the other hand, we have st W ( P ) = H and st W is a 1-1 mapping from the set of subsets of W to the set of subsets of [ | W | ] , therefore i ′ ′ and j ′ ′ are in H. Then there must exist i ′ ′ ′ and in [ | H | ] such that st H ( i ′ ′ ) = i ′ ′ ′ , st H ( j ′ ′ ) = j ′ ′ ′ and c i ′ ′ j ′ ′ = d i ′ ′ ′ j ′ ′ ′ . Hence, b i j = d i ′ ′ ′ j ′ ′ ′ . By Remark 1, we have i = st P ( i ′ ) = st H ( st W ( i ′ ) ) = st H ( i ′ ′ ) = i ′ ′ ′ and j = st P ( j ′ ) = st H ( st W ( j ′ ) ) = st H ( j ′ ′ ) = j ′ ′ ′ .
Thus, for each i , j in [ | P | ] , b i j = d i j , i.e., A P = ( A W ) H . □
Theorem 3. ( M , ∗ , μ ) is a graded algebra.
Proof. It is easy to verify that μ is a unit. For A in M h , B in M k and C in M l , we have
A ∗ B = ∑ X ∈ M h + k H , K : H ∪ K = [ h + k ] X H = A , X K = B X .
Then for any term Y in ( A ∗ B ) ∗ C , there exist two disjoint subsets W and L of [ h + k + l ] with | W | = h + k and | L | = l such that Y W is a term in A ∗ B and Y L = C . It means
( A ∗ B ) ∗ C = ∑ X ∈ M h + k H , K : H ∪ K = [ h + k ] X H = A , X K = B ∑ Y ∈ M h + k + l W , L : W ∪ L = [ h + k + l ] Y W = X , Y L = C Y . (2)
For a fixed W in [ h + k + l ] with cardinality h + k , there exist two disjoint subsets H and K of [ h + k ] with | H | = h and | K | = k such that
( Y W ) H = A and ( Y W ) K = B .
Since H is a subset of [ | W | ] = [ h + k ] , due to the Lemma 2, there exists a subset P of W corresponding to H with | P | = h such that H = st W ( P ) and
Y P = ( Y W ) H = A .
Similarly, there exists a subset Q of W with | Q | = k corresponding to K such that K = st W ( Q ) and
Y Q = ( Y W ) K = B .
In (2), for a fixed subset W in [ h + k + l ] with cardinality h + k , H traverses all subsets with cardinality h in [ h + k ] , since Y W traverses all terms in A ∗ B . Meanwhile, P traverses all subsets with cardinality h in W. Therefore, P traverses all subsets with cardinality h in [ h + k + l ] when W traverses all subsets with cardinality h + k in [ h + k + l ] . Similarly, Q traverses all subsets with candinality k in [ h + k + l ] when W traverses all subsets with cardinality h + k in [ h + k + l ] from W = P ∪ Q . Thus (2) can be rewritten as
( A ∗ B ) ∗ C = ∑ Y ∈ M h + k + l P , Q , L : P ∪ Q ∪ L = [ h + k + l ] Y P = A , Y Q = B , Y L = C Y . (3)
Similarly, A ∗ ( B ∗ C ) can be rewritten as (3). Hence, ∗ satisfies the associative law and ( M , ∗ , μ ) is an algebra.
By the definitions of the product ∗ and μ , we have M r ∗ M s ⊆ M r + s and μ ( K ) ⊆ M 0 . So ( M , ∗ , μ ) is a graded algebra. □
5. Conclusion and Suggestion
Let M be the vector space spanned by (0,1)-matrices. Firstly, we introduce splits and the decomposition of a (0,1)-matrix. Then we define the cut-box coproduct Δ and the super-shuffle product ∗ on M . We prove the cut-box coproduct Δ satisfies coassociativity and the super-shuffle product ∗ satisfies associativity, i.e., ( M , Δ , ν ) is a graded coalgebra and ( M , ∗ , μ ) is a graded algebra.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11701339 and 12071265).
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Song, S.F. and Li, H.L. (2023) Super-Shuffle Product and Cut-Box Coproduct on (0,1)-Matrices. Open Journal of Applied Sciences, 13, 1326-1335. https://doi.org/10.4236/ojapps.2023.138105
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