A network is an acyclic directed graph in which there are sources that have some messages and want to transmit to receivers through the combination of messages in intermediate nodes. The goal is to find a collection of functions that allow to combine messages in order to satisfy the demand of the receivers. In this paper, we study the fractional solvability problem of a network using an extension of the solvability problem in closure operators given by Gadouleau in 2013. We define linear programming problems via the desired extension in order to study capacities; using some information inequalities and characteristic-dependent linear rank inequalities, we obtain upper bounds on linear capacity of some networks and closure operators over some fields.

A linear rank inequality is a linear inequality that holds by dimensions of vector spaces over any finite field. A characteristic-dependent linear rank inequality is also a linear inequality that involves dimensions of vector spaces but this holds over finite fields of determined characteristics, and does not in general hold over other characteristics. In this paper, using as guide binary matrices whose ranks depend on the finite field where they are defined, we show a theorem which explicitly produces characteristic-dependent linear rank inequalities; this theorem generalizes results previously obtained in the literature.

A network is an acyclic directed graph in which there are sources that have some messages and want to transmit to receivers through the combination of messages in intermediate nodes. The goal is to find a collection of functions that allow to combine messages in order to satisfy the demand of the receivers. In this paper, we study the fractional solvability problem of a network using an extension of the solvability problem in closure operators given by Gadouleau in 2013. We define linear programming problems via the desired extension in order to study capacities; using some information inequalities and characteristic-dependent linear rank inequalities, we obtain upper bounds on linear capacity of some networks and closure operators over some fields.