Function minimum associated to a congruence on integral \(n\)-tuple space (Q1895808)

Let \(R\) be any congruence on the monoid \(N^k\), \(N\) being the set of natural numbers and consider a fixed well-ordering of \(N^k\). Define a map \(\mu:N^k/R\to N^k\) by associating with every class \(\text{mod }R\) the element of that class which is minimal according to the chosen order. It is shown that the complement of the image of \(\mu\) is an ideal and \(\mu\) is completely determined by its values at the set of classes \(\text{mod }R\) containing the generators of that ideal.