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Homesuperfluity of the third defining property for finite consequence operator

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# superfluity of the third defining property for finite consequence operator

In this entry, we demonstrate the claim made in section 1 of the parent entry that the defining conditions for finitary consequence operator given there are redundant because one of them may be derived from the other two.

###### Theorem.

Let $L$ be a set. Suppose that a mapping $C\colon\mathcal{P}(L)\to\mathcal{P}(L)$ satisfies the following three properties:

1. For all $X\subseteq L$, it happens that $X\subseteq C(X)$.

2. $C\circ C=C$

3. For all $X\in L$, it happens that $C(X)=\bigcup\limits_{{Y\in\mathcal{F}(X)}}C(Y)$.

Then $C$ also satisfies the following property: For all $X,Y\subseteq L$, if $X\subseteq Y$, then $C(X)\subseteq C(Y)$.

Type of Math Object:

Theorem

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Reference

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## Mathematics Subject Classification

03G25*no label found*03G10

*no label found*03B22

*no label found*

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