Projective geometry can be equivalent to a hierarchical network {lattice theory}|. Lattice theory is similar to fiber-bundle theory and similar to set theory. Hierarchical networks {lattice network} have highest node and lowest nodes. Two nodes can connect through intermediate-level nodes. Two nodes can have no connections. Projective geometry uses complex continuous functions. Lattice networks use real discrete values at lattice nodes, so calculations are simpler.
Lattice-network operations are commutative and associative, and can be distributive or not distributive.
quantum mechanics
Lattice theory is like quantum mechanics. Both are discontinuous, have intermediate states between states/nodes, and have different paths from one state/node to another state/node.
types
Node subsets can have least upper bounds and greatest lower bounds (complete lattice). Lattices can be graphs, polyhedra, or simplexes. Lattices can be quasi-ordered lattices, oriented graphs, or semilattices. Lattices can have independent branches (modular orthocomplemented lattice). Higher-dimension lattices can have vector-space factors (one-dimensional subspace), finite Abelian-group factors {cyclic component}, or combinatorial topologies.
Physical Sciences>Physics>Quantum Mechanics>Theory
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Date Modified: 2022.0224