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International Journal of Theoretical Physics

, Volume 39, Issue 11, pp 2559–2581 | Cite as

Endomorphisms of the Separated Product of Lattices

  • Boris Ischi
Article

Abstract

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.

Keywords

Field Theory Elementary Particle Quantum Field Theory Property Lattice Separate Entity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Boris Ischi

There are no affiliations available

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