consequence operator is determined by its fixed points

Theorem 1

Suppose that $C_{1}$ and $C_{2}$ are consequence operators on a set $L$ and that,for every $X\\subseteq L$ , it happens that $C_{1}(X)=X$ if and only if $C_{2}(X)=X$ . Then $C_{1}=C_{2}$ .

Theorem 2

Suppose that $C$ is a consequence operators on a set $L$ . Define $K=\\{X\\subseteq L\\mid C(X)=X\\}$ . Then, for every $X\\in L$ , there exists a $Y\\in K$ such that $X\\subseteq Y$ and, for every $Z\\in K$ such that $X\\subseteq Z$ ,one has $Y\\subseteq Z$ .

Theorem 3

Given a set $L$ , suppose that $K$ is a subset of $L$ such that, for every $X\\in L$ , there exists a $Y\\in K$ such that $X\\subseteq Y$ and, for every $Z\\in K$ such that $X\\subseteq Z$ , one has $Y\\subseteq Z$ . Then there exists aconsequence operator $C\\colon\\mathcal{P}(L)\\to\\mathcal{P}(L)$ such that $C(X)=X$ if and only if $X\\in K$ .

单词	ConsequenceOperatorIsDeterminedByItsFixedPoints
释义	consequence operator is determined by its fixed points Theorem 1 Suppose that $C_{1}$ and $C_{2}$ are consequence operators on a set $L$ and that,for every $X\\subseteq L$ , it happens that $C_{1}(X)=X$ if and only if $C_{2}(X)=X$ . Then $C_{1}=C_{2}$ . Theorem 2 Suppose that $C$ is a consequence operators on a set $L$ . Define $K=\\{X\\subseteq L\\mid C(X)=X\\}$ . Then, for every $X\\in L$ , there exists a $Y\\in K$ such that $X\\subseteq Y$ and, for every $Z\\in K$ such that $X\\subseteq Z$ ,one has $Y\\subseteq Z$ . Theorem 3 Given a set $L$ , suppose that $K$ is a subset of $L$ such that, for every $X\\in L$ , there exists a $Y\\in K$ such that $X\\subseteq Y$ and, for every $Z\\in K$ such that $X\\subseteq Z$ , one has $Y\\subseteq Z$ . Then there exists aconsequence operator $C\\colon\\mathcal{P}(L)\\to\\mathcal{P}(L)$ such that $C(X)=X$ if and only if $X\\in K$ .
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