Series and Toeplitz matrices (a global implicit approach)

In this paper we deal with properties of series and Toeplitz matrices from set-theoretic point of view. This paper is motivated by results of N. N. Kholshchevnikova, V. I. Malykhin and T. Salat. We consider cardinal characteristics of series and matrices (which usually lie between N-1 and 2(N0)). So far this is no more interesting under CK, hence we reformulate all this in the language of Galois-Tukey connections between binary relations characterizing properties of series, sequences and matrices. To avoid further trivialities and to preserve forcing properties we consider Borel connections. We survey some of our former results in a new setting of Galois-Tukey connections and give also some new results and formulate several problems.