), namely a set on which binary operations act but neither the identity nor the inversion are included, is nowadays called a magma (see e.g. Bourbaki, 1998).
1The alternative meaning of `groupoid' introduced by Hausmann & Ore (1937), namely a set on which binary operations act but neither the identity nor the inversion are included, is nowadays called a magma (see e.g. Bourbaki, 1998).
2The geometrical equivalence must be fulfilled not necessarily by the real layers but by their archetypes, i.e. the slightly idealized layers to which the real layers can be reduced by neglecting some distortions occurring in the true structure. The notion of polytypism becomes thus unequivocal only when it is used in an abstract sense to indicate a structural type with specific geometrical properties.
3The weighted reciprocal lattice is obtained by assigning to each node of the reciprocal lattice a `weight' that corresponds to F(hkl) (Shmueli, 2001).
4The translation subgroup T is a normal subgroup of G: the factor group or quotient group G/T is the set of all cosets of T in G.
5The role of extraordinary orbits was first addressed by Sándor (1968), who suggested extending the concept of `special positions' to positions having translational symmetry higher than that of the general position.
6The group isomorphism as described here is limited to crystallographic equivalence. Two space groups of type P61 and P65 are not considered isomorphic, although they are affine equivalent.
7It is emphasized that the expression `merohedral twins' often appearing in the literature is inappropriate: `merohedral' indicates the symmetry of an individual, not that of a twin (Catti & Ferraris, 1976).
8We say that the twin point group is isomorphic with a given crystallographic point group, not that it coincides with it because, although it has the same type of symmetry elements, some of these are chromatic. Only by neglecting the chromatic nature of its elements would a twin point group `coincide' with a crystallographic point group.
9A graph can be undirected (a line from point A to point B is considered to be the same as a line from point B to point A) or directed, also called a digraph (the two directions are counted as being distinct arcs or directed edges).
10An automorphism is an isomorphism (bijective mapping) from a mathematical object to itself.