Abstract
It is shown that in a meet-continuous lattice L endowed with a multiplicative auxiliary order ≺ the family of all members of L which satisfy the axiom of approximation, i.e. α = \(\bigvee\){β ∈ L : β ≺ α}, is closed under finite infs and arbitrary sups. This is a key ingredient of a meet-continuous lattice proof that both regularity and complete regularity of many valued topology have subbasic characterizations. As a consequence, the frame law can now be eliminated from some fundamental results on completely regular L-valued topological spaces (e.g., this is the case in regard to the Tychonoff embedding theorem).
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The grant MTM2006-14925-C02-02 from the Ministry of Education and Science of Spain and FEDER is gratefully acknowledged by the second named author.
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Höhle, U., Kubiak, T. Approximating Orders in Meet-Continuous Lattices and Regularity Axioms in Many Valued Topology. Order 25, 9–17 (2008). https://doi.org/10.1007/s11083-007-9074-0
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DOI: https://doi.org/10.1007/s11083-007-9074-0
Keywords
- Meet-continuous lattices
- Multiplicative auxiliary order
- L-valued topology
- Regularity
- Complete regularity
- L-cube