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...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.展开更多
基金Foundation item： the Natural Science Foundation of Jiangsu Province （No. BK2007030） the Natural Science Foundation of Education Committee of Jiangsu Province （No. 07KJD110207）.
文摘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.
