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# 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 $X\in K$.

## Mathematics Subject Classification

03G10*no label found*03B22

*no label found*03G25

*no label found*

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