# operations on consequence operators

Let $L$ be a set and let $\mathcal{C}$ be the set of all consequence operators on $S$. Then we may define a binary relation $\leq\,\subset\mathcal{C}\times\mathcal{C}$ and binary operations $\wedge,\vee,\veebar\colon\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ as follows:

###### Definition 1

For $C_{1},C_{2}\in\mathcal{C}$, we have $C_{1}\leq C_{2}$ when, for all $X\subseteq L$, we have $C_{1}(X)\subseteq C_{2}(X)$

###### Definition 2

For $C_{1},C_{2}\in\mathcal{C}$, we have $(C_{1}\wedge C_{2})(X)=C_{1}(X)\cap C_{2}(X)$ for all $X\subseteq L$.

###### Definition 3

For $C_{1},C_{2}\in\mathcal{C}$, we have $(C_{1}\vee C_{2})(X)=C_{1}(X)\cup C_{2}(X)$ for all $X\subseteq L$.

###### Definition 4

For $C_{1},C_{2}\in\mathcal{C}$, we have $(C_{1}\veebar C_{2})(X)=\cap\{Y\mid X\subseteq Y\subseteq L\land C_{1}(Y)=C_{2% }(Y)=Y\}$ for all $X\subseteq L$.

Title operations on consequence operators OperationsOnConsequenceOperators 2013-03-22 16:29:38 2013-03-22 16:29:38 rspuzio (6075) rspuzio (6075) 6 rspuzio (6075) Definition msc 03G10 msc 03B22 msc 03G25