operations on consequence operators

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

Definition 1

For $C_{\mathrm{1}}\mathrm{,}{C}_{\mathrm{2}}\mathrm{\in }\mathrm{C}$, we have $C_{\mathrm{1}}\mathrm{\le }{C}_{\mathrm{2}}$ when, for all $X\mathrm{\subseteq }L$, we have $C_{\mathrm{1}}\mathit{}\mathrm{(}X\mathrm{)}\mathrm{\subseteq }{C}_{\mathrm{2}}\mathit{}\mathrm{(}X\mathrm{)}$

Definition 2

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

Definition 3

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

Definition 4

For $C_{\mathrm{1}}\mathrm{,}{C}_{\mathrm{2}}\mathrm{\in }\mathrm{C}$, we have $\mathrm{(}{C}_{\mathrm{1}}\mathrm{⊻}{C}_{\mathrm{2}}\mathrm{)}\mathit{}\mathrm{(}X\mathrm{)}\mathrm{=}\mathrm{\cap }\mathrm{\{}Y\mathrm{\mid }X\mathrm{\subseteq }Y\mathrm{\subseteq }L\mathrm{\wedge }{C}_{\mathrm{1}}\mathit{}\mathrm{(}Y\mathrm{)}\mathrm{=}{C}_{\mathrm{2}}\mathit{}\mathrm{(}Y\mathrm{)}\mathrm{=}Y\mathrm{\}}$ for all $X\mathrm{\subseteq }L$.

Title	operations on consequence operators
Canonical name	OperationsOnConsequenceOperators
Date of creation	2013-03-22 16:29:38
Last modified on	2013-03-22 16:29:38
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 03G10
Classification	msc 03B22
Classification	msc 03G25