On a combinatorial theorem and its application to nonnegative matrices. (Russian. English summary) Zbl 0082.24402

Pták, Vlastimil

Czech. Math. J. 8(83), 487-495 (1958).

Let \(P(N)\) be the set consisting of all subsets of \(N=\{1,2,\ldots,n\}\) where \(n\) is a natural number, \(F=\{f\}\) the set of all mappings defined on \(P(N)\) with ranges in \(P(N)\) and such that \(f(A\cup B)=f(A)\cup f(B)\). A mapping \(f\) which does not map \(N\) in the empty set \(\emptyset\) is said to be irreducible if \(f(A)\subseteq A\) implies \(A=\emptyset\) or \(A=N\). Irreducible mappings are characterized and successfully applied on a problem of irreducibility of a square nonnegative matrix of the order \(n\).
This is a joint review for the article under review and Zbl 0082.24403.

Reviewer: S. Kurepa

MSC:

15A99	Basic linear algebra
05A05	Permutations, words, matrices

Citations:

Zbl 0082.24403