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 $C\mathrm{:}\mathrm{P}\mathit{}\mathrm{(}L\mathrm{)}\mathrm{\to}\mathrm{P}\mathit{}\mathrm{(}L\mathrm{)}$ 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 $C(X)=\bigcup _{Y\in \mathcal{F}(X)}C(Y)$.
Then $C$ also satisfies the following property: For all $X\mathrm{,}Y\mathrm{\subseteq}L$, if $X\mathrm{\subseteq}Y$, then $C\mathit{}\mathrm{(}X\mathrm{)}\mathrm{\subseteq}C\mathit{}\mathrm{(}Y\mathrm{)}$.
Title  superfluity of the third defining property for finite consequence operator 

Canonical name  SuperfluityOfTheThirdDefiningPropertyForFiniteConsequenceOperator 
Date of creation  20130322 16:30:13 
Last modified on  20130322 16:30:13 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  5 
Author  rspuzio (6075) 
Entry type  Theorem^{} 
Classification  msc 03G25 
Classification  msc 03G10 
Classification  msc 03B22 