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The definition of a lattice and its superlattice is given algebraically. A coincidence site lattice (CSL) is defined as an intersection lattice of any two crystal lattices, and a complete pattern-shift lattice (DSCL) as the set theoretically smallest lattice containing both crystal lattices as superlattices. In the case where the two lattices are related by a non-singular matrix (having non-zero determinant), the so-called 0-lattice may be generated from the two crystal lattices. Any translation of the 0-lattice by all the vectors of one of the crystal lattices forms a lattice, i.e. a reduced 0-lattice. As a result of the theory of groups and numbers, the reduced 0-lattice (abbreviated to ROL) is homomorphic to the DSCL. It is shown that the factor group of all cosets of lattice 1 in the DSCL (in the ROL) is isomorphic with the factor group of all cosets of the CSL in lattice 2 (in the 0-lattice). The volume of a unit cell is derived for all the lattices generated by the two crystal lattices. Secondly, the reciprocal of a lattice is introduced and the reciprocity between the CSL and the DSCL determined by the reciprocals of the two crystal lattices is shown as a special case of a theorem mentioned about modules over a ring. Finally a complete diagram of relationships between b-lattices and 0-lattices for direct lattices and reciprocal lattices is given.
