Endomorphisms of the Separated Product of Lattices
To describe the evolution of separated entities remaining separated, we proposeto study endomorphisms (join-preserving maps, sending atoms to atoms) of theseparated product of cao lattices (complete, atomistic orthocomplementedlattices). Morphisms have been used successfully to describe the evolution ofentities, and the separated product is a model for the property lattice of separatedsystems; its set of atoms is the Cartesian product of each atom space. Let L bethe separated product of two cao lattices having the covering property and f anendomorphism of L. We prove that the center F(L) of L is the power set ofΩ1 × Ω2 where Ω i is the atom space ofF(L i ) (Theorem 1), f preserves irreduciblecomponents (Theorem 2), and if L is irreducible there exist two endomorphismsf1 and f2 and a permutation σsuch that the restriction of f to atoms is given byf(p1, p2) = (f1(pσ(1)), f2(pσ(2)))(Theorem 3). For generalizations of these resultsto separated products of families of cao lattices, we develop new general argumentsinvolving a topology we define on the set of atoms of a cao lattice.
KeywordsField Theory Elementary Particle Quantum Field Theory Property Lattice Separate Entity
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