Information theory structure definition

From a information theory point of view, structure is a property of a system S with an original description X that allows to encode it in a more simplified description, namely Y. Therefore, Y is a new description of X that exploits its structure.

If given the description Y we have some function that maps to description X then there is no loss of information in this mapping and the information of X is totally encoded in Y. Then, we can also say that Y is a better description for the original system than X.

In information theory, the measure that relates two description of the same system is the mutual information:

$\$I(X|Y)=H(X)-H(X|Y)\$$

However this kind of reduction without information loss is not always possible and usually when we get the simplifiest version of X description exploiting its structure we loss information in such mapping.