Equivalences Induced by $\left(\mathrm{n}, 1\right),\ \left(\mathrm{n},\mathrm{n}-1\right)$ and $\left(\mathrm{n},\mathrm{k}\right)$ - Equivalences (1 < k < n -1)

In this paper we generate equivalences induced by
$\left(\mathrm{n},1\right),\ \left(\mathrm{n},\mathrm{n}-1\right)$
and
$\left(\mathrm{n},\mathrm{k}\right)$
-equivalences (1 < k < n -1). The connections between such equivalences are proved. Moreover, properties of
$\left(\mathrm{n},\mathrm{m}\right)$
- relations are investigated, and especially properties of relations induced from other relations, namely
$\left(\mathrm{n},\mathrm{m}\right)$
- equivalence relations. Couple of examples are given for generalized equivalences.