An Hadamard operation on rational relations

Abstract

We consider a new operation on the family of binary relations on integers called Hadamard star. View a binary relation R⊆N×N as a mapping of N into the power set of N and let R(n) denote the subset of integers m such that (n,m)∈R. Then the Hadamard star of R is the relation which assigns to each integer n the Kleene star of R(n). This is reminiscent of the Hadamard inverse of series with coefficients in a field.We characterize the rational relations whose Hadamard star is also rational and show that this property is decidable.