Achiral subset

If the problem were to partition a set of carbon compounds into two equivalence classes, of which one contains only chiral molecules and the other one only achiral ones, it could not be solved with the criterion of asymmetric C-atoms. In the first case, one would assign meso-forms like 9 and compounds with pseudo-asymmetric 22> C-atoms, such as 11, to the chiral equivalence class, and in the second, chiral molecules like 12 would remain in the achiral subset. However, the latter class would be devoid of chiral molecules, if the compounds under consideration have been confined to molecules with free rotation about all C—C bonds. 

Fuzzy set B is a maximal achiral subset of fuzzy set A B h achiral, B maximal achiral subset B is not necessarily unique for a given fuzzy set A. 

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... 

Proof of statement (3). Point set S represents an n-chiral simplex in consequently, any subset of k vertices of 5 defines a unique k - 1)-dimensional hyperplane that contains these k points. Take the (n - 2)-dimensional, unique hyperplane Q that contains the first n - 1, uniquely labeled vertices, and take the unique (n - l)-dimensional hyperplane A that contains the first n vertices of 5. Hyperplane is a maximum achiral subset of 5. Furthermore,... 

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... 

According to the above criteria, the boundary between the regions R and L must be just the subspace of the achiral molecules. On the other hand, the boundary between two regions of the A-space must necessarily be (n—l)-dimensional. Thus, the requirements can only be satisfied if the subset of the achiral molecules is a ( —l)-dimensional hypersurface, or a set of such surfaces. 

It is clear that, for a chiral class, the subset of the A-space corresponding to chiral molecules is -dimensional, while the subset of achiral molecules, which require the equality of two or more A s, will be of dimension less than n. 

Consider now the same arrangement of A and A embedded in En+1, by regarding E" as a subspace of En+L A two-dimensional rotation in En+1 is defined by its (n-l)-dimensional axis and by the angle a of rotation in the remaining two dimensions. [Note that in a k-dimensional space, the axis of rotation is (k-2)-dimensional.] Choose the rotation axis in En+ as the (n-l)-dimensional subset defined as the reflection hyperplane E"- of condition x i = 0 in E". With respect to this axis, a rotation of angle a = 7C in the two-dimensional plane spanned by coordinates (xi, x +i) superimposes A on A in (n+l)-dimensions. Consequently, the object A is achiral in (n+l)-dimensions (i.e., when embedded in space E"+ ). Furthermore, the superimposition of mirror images performed in En+1 is a possible motion in any Euclidean space En+k (> of which En+ is a subspace, hence A is achiral in any higher dimensions. Consequently, chirality may occur only in the lowe.st dimension where A is embeddable. Q.E.D. 

The permutational isomers under consideration correspond to the orbits of R = Dj in the set of distributions 5 e 7 of content c = (4,4). Since the skeleton is planar, P R, but the ligands are achiral, thus all isomers are achiral. Concerning the construction, we are faced with the problem of evaluating a transversal of a subset of the set of symmetry classes of distributions... 

---