consequence operator is determined by its fixed points
Theorem 1
Suppose that and are consequence operators on a set and that,for every , it happens that if and only if . Then .
Theorem 2
Suppose that is a consequence operators on a set . Define . Then, for every , there exists a such that and, for every such that ,one has .
Theorem 3
Given a set , suppose that is a subset of such that, for every , there exists a such that and, for every such that , one has . Then there exists aconsequence operator such that if and only if .