# 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 a consequence operator $C\colon\mathcal{P}(L)\to\mathcal{P}(L)$ such that $C(X)=X$ if and only if $X\in K$.

Title consequence operator is determined by its fixed points ConsequenceOperatorIsDeterminedByItsFixedPoints 2013-03-22 16:29:48 2013-03-22 16:29:48 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Theorem msc 03G10 msc 03B22 msc 03G25