Partial symmetry

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The symmetry operations of a space group are isometries operating on the whole crystal pattern and are also called total operations or global operations. More generally, the crystal space can be divided in N components S₁ to S_N, and a coincidence operation φ(S_i)→S_j can act on just the i-th component S_i to bring it to coincide with the j-th component S_j. Such an operation is not one of the operations of the space group of the crystal because it is not a coincidence operation of the whole crystal space; it is not even defined, in general, for any component k different from i. It is called a partial operation: from the mathematical viewpoint, partial operations are space-groupoid operations.

When i = j, i.e. when the operation is φ(S_i)→S_i and brings a component to coincide with itself, the partial operation is of special type and is called local. A local operation is in fact a symmetry operation, which is defined only on a part of the crystal space: local operations may constitute a subperiodic group.