Maximal achiral subset

Fuzzy set B is a maximal achiral subset of fuzzy set A B h achiral, B [Pg.161]

Here fuzzy set is a maximal achiral subset, fuzzy set 5 is a maximal mass achiral subset, fuzzy set C is a minimal achiral fuzzy superset, and fuzzy set C is a minimal mass achiral superset of fuzzy set A. 

Fuzzy chirality measures can also be defined in terms of a maximal achiral subset B, maximal mass achiral subset B, minimal achiral subset C, and minimal mass achiral subset C, of fuzzy set A, discussed in Section IV. If fuzzy set D denotes any one of these fuzzy sets, D e B, B ,C,C , then a fuzzy chirality measure A ts.o ) of fuzzy set A is provided by... 

For more general objects, several chirality measures have been proposed based on the concepts of maximal achiral subsets and minimum achiral supersets [240]. A maximal achiral subset of an object is a subset that cannot be increased within the object without becoming chiral, and a minimal achiral superset of an object cannot be decreased while containing the object and staying achiral. Note that for some objects neither the maximal achiral subset nor the minimal achiral superset is necessarily unique, and their collection gives a fairly detailed chirality characterization [240], for example, by measuring the deviation of their volumes from that of the original object and from one another. 

The actual determination of a set M for some chiral set T and the calculation of the volume v(T) are usually rather difficult problems (see some relevant comments in references [51-53,58,240,242]), and the same applies for superset N. However, within a RBSM framework, the analogous chirality measures given in terms of a discretization procedure using polycubes (or lattice animals in 2D) [240] do not require the explicit determination of a maximal volume (area) achiral subset M and the calculation of its exact volume (or area) v(M). 

If M(x, Mr, Not, and Nr are maximal volume achiral subset, maximal volume R-subset, minimal volume achiral superset, and minimal volume R-superset of a set T, respectively, then... 

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