Graphene's hexagonal lattice can be regarded as two interleaving triangular lattices. Examples of relational algebras include semilattices, lattices, and Boolean algebras. The corresponding reciprocal lattices have the same symmetry as the direct lattice. The triclinic lattice is the least symmetric of the 14 three-dimensional Bravais lattices. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. Infinite Diophantine equations are related to theta functions and infinite dimensional lattices. The fourteen three-dimensional lattices, classified by lattice system, are shown above. Heyting algebras are an example of distributive lattices where some members might be lacking complements. In two dimensions there are two orthorhombic Bravais lattices&colon; primitive rectangular and centered rectangular. The three dimensions of space afford 14 distinct Bravais lattices describing the translational symmetry. Even positive definite unimodular lattices exist only in dimensions divisible by 8. Generalized Boolean lattices are exactly the ideals of Boolean lattices. They have a cubic MgAgAs-type structure formed by three interpenetrating face-centered-cubic (fcc) lattices. This has been observed both in photo-refractive, photonic lattices, as well as photo-reactive systems. Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.