Relativization and extension of solutions of irreflexive relations
Moses Richardson · Pacific Journal of Mathematics · 1955
Introduction.Let >-be an irreflexive binary relation defined over a domain 2) of elements α, 6, c, .We represent the system (5), >-) by an oriented graph G by regarding the elements of 3 as vertices of G and inserting an arc ab of the graph, oriented from a to b, if and only if a >b.The sentence "α >b"is read "α dominates 6".A set V of vertices is termed internally satisfactory 1 if and only if x G V and γ E V implies x ^j-y.A set V of vertices is termed externally satisfactory if and only if γ E 5) -F implies that there exists an % E F such that % >-y.A set F of vertices is termed a solution of G, or of (3, >~), if and only if it is both internally and externally satisfactory.In [4], various sufficient conditions for the existence of solutions were established.By a subsystem Oo,/*") of the system ( §>,>-) is meant a system where 5) 0 C 5) and the relation >-for the subsystem is merely the restriction of the relation >-for the supersystem (5), >-).Let Go be the graph of the subsystem (^o> >"") an d l et ^o he a solution ofIn this paper, some sufficient conditions for the existence of relativizations and extensions of solutions are presented.More elegant and more effective extension theorems, especially with a view toward possible applications to the theory of ra-person games, remain to be desired.It is hoped that the present paper may serve to stimulate interest in this apparently difficult problem. A theorem on relativization.If H is a subgraph of the graph G, then the graph obtained by adding to H all the arcs of G which join pairs of vertices of H will be termed the juncture of H (relative to G) and will be denoted by //.In [2], internally satisfactory is called satisfactory with respect to non-domination, and in [4] it is called ^/--satisfactory.