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Additive maps preserving the scrambling index are bijective. (English) Zbl 1413.05038

Let \(\mathbb B=\{ 0,1\}\) be the (commutative) Boolean semiring with \(a+b=1\) except for \(a=b=0\), \(a\cdot b=0\) except for \(a=b=1\) and the usual distributive law. Let \(M_n (\mathbb B)\) be the semiring of \(n\times n\) matrices over \(\mathbb B\). To each \(A\in M_n(\mathbb B)\) associate the digraph \(G_A\) with adjacency matrix \(A\). The digraph \(G_A\) (and by extension \(A\)) is said to have scrambing index \(k(A)\) if \(k(A)\) is the least positive integer \(k\) such that for each pair \(u\), \(v\) of vertices in \(G_A\) there exists a vertex \(w\) with directed paths of length \(k\) from \(u\) to \(w\) and from \(v\) to \(w\); if no such \(k\) exists then we put \(k(A)=0\) (see M. Akelbek and S. Kirkland [Linear Algebra Appl. 430, No. 4, 1111–1130 (2009; Zbl 1167.05030)]). In particular, if \(A\) is a primitive matrix and so for some positive integer \(t\) the matrix \(A^t\) has all entries \(1\), then \(k(A)\) satisfies \(0<k(A)\leq t\).
Now assume that \(n\geq 3\). A mapping \(T:M_n(\mathbb B)\rightarrow M_n(\mathbb B)\) is called additive if \(T(A+B)=T(A)+T(B)\) for all \(A\), \(B\). Using earlier results, the authors investigate conditions under which an additive mapping \(T\) preserves the scrambling index: \(k(T(A))=k(A)\) for all \(A\). The main result of the paper is that if \(T\) is additive then the following are equivalent (see Corollary 2.23): (1) \(k(T(A))=k(A)\) for all \(A\) with \(k(A)\neq 0\); (2) \(T\) is bijective; (3) there exists a permutation matrix \(P\) such that \(T(A)=P^T AP\) for all \(A\); and (4) \(T\) preserves the scrambing index for all \(A\). The final section of the paper gives a partial extension of this result to additive mappings of \(M_n (\mathcal S)\) over more general semirings \(\mathcal S\).

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
15A86 Linear preserver problems
15B34 Boolean and Hadamard matrices
16Y60 Semirings
05C20 Directed graphs (digraphs), tournaments

Citations:

Zbl 1167.05030
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