superfluity of the third defining property for finite consequence operator

In this entry, we demonstrate the claim made in section 1 of thehttp://planetmath.org/node/8646parent entry that the defining conditions forfinitary consequence operator given there are redundant becauseone 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 $C(X)\\subseteq C(Y)$ .

单词	SuperfluityOfTheThirdDefiningPropertyForFiniteConsequenceOperator
释义	superfluity of the third defining property for finite consequence operator In this entry, we demonstrate the claim made in section 1 of thehttp://planetmath.org/node/8646parent entry that the defining conditions forfinitary consequence operator given there are redundant becauseone 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 $C(X)\\subseteq C(Y)$ .
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