# superfluity of the third defining property for finite consequence operator

In this entry, we demonstrate the claim made in section 1 of the http://planetmath.org/node/8646parent 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. 1.

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

2. 2.

$C\circ C=C$

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

Title superfluity of the third defining property for finite consequence operator SuperfluityOfTheThirdDefiningPropertyForFiniteConsequenceOperator 2013-03-22 16:30:13 2013-03-22 16:30:13 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Theorem  msc 03G25 msc 03G10 msc 03B22