On the Set of Common Consequent Indices of a Class of Binary Relations

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.

Let V={a_1,a_2,...,a_n} be a finite set with n≥2 and P_n(V)the set of all primitive binary relations on V.For Q∈P_n(V),denote by G(Q)the directed graph corresponding to Q. For positive integer d≤n,let P_n(V,d)={Q:Q∈P_n(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 P_n(V,d) is {1,2,...,n-「d/2」}.Furthermore,the minimal extremal binary relations are described.