Let V = {a1,a2 ,...,an} be a finite set with n ≥ 2 and Pn（V） the set of all primitive binary relations on V. For Q E Pn（V）, denote by G（Q） the directed graph corresponding to Q. For positive integer d ≤ n, let Pn（V, d） = {Q ： Q ∈ Pn（V） and G（Q） contains exactly d loops}. In this paper, it is proved that the set of common consequent indices of binary relations in Pn （V, d） is {1, 2,..., n -[d/2] }. Furthermore, the minimal extremal binary relations are described.
We discuss a family of restricted m-ary overpartition functions bm,j （n）, which is the number of m-ary overpartitions of n with at most i ＋ j copies of the non-overlined part m^i allowed, and obtain a family of congruences for bm,lm-1 （n）.
A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.
