Non-gaussian $r-$congruences

This paper is an invitation to the study and use of general theory of non-gaussian $r-$congruences in the theory of numbers. In this work we classify the two kinds of $r-$congruences that exist (namely the trivial and non-trivial types) and establish their foundational properties as well as their algebra. Among other results, we show that the consideration of the non-trivial $r-$ congruences entails the elucidation of a distinguished cyclic subgroup of the permutation group, while the trivial $r-$congruences (chief among which is the well-known gaussian congruence) leads to the trivial subgroup. The order of this cyclic group is also computed.