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# consequence operator determined by a class of subsets

###### Theorem 1.

Let $L$ be a set and let $K$ be a subset of $\mathcal{P}(L)$. The the mapping $C\colon\mathcal{P}(L)\to\mathcal{P}(L)$ defined as $C(X)=\cap\{Y\in K\mid X\subseteq Y\}$ is a consequence operator.

###### Proof.

We need to check that $C$ satisfies the defining properties.

*Property 1:*
Since every element of the set $\{Y\in K\mid X\subseteq Y\}$
contains $X$, we have $X\subseteq C(X)$.

*Property 2:*
For every element $Y$ of $K$ such that $X\subseteq Y$, it also is the case that
$C(X)\subseteq Y$ because an intersection of a family of sets is a subset of
any member of the family. In other words (or rather, symbols),

$\{Y\in K\mid X\subseteq Y\}\subseteq\{Y\in K\mid C(X)\subseteq Y\},$ |

hence $C(C(X))\subseteq C(X)$. By the first property proven above, $C(X)\subseteq C(C(X))$ so $C(C(X))=C(X)$. Thus, $C\circ C=C$.

*Property 3:*
Let $X$ and $Y$ be two subsets of $L$ such that $X\subseteq Y$. Then if,
for some other subset $Z$ of $L$, we have $Y\subset Z$, it follows that
$X\subset Z$. Hence,

$\{Z\in K\mid Y\subseteq Z\}\subseteq\{Z\in K\mid X\subseteq Z\},$ |

so $C(X)\subseteq C(Y)$.

∎

## Mathematics Subject Classification

03G25*no label found*03G10

*no label found*03B22

*no label found*

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