NDLI: Codimension and pseudometric in co-Heyting algebras

Content Provider	Springer Nature Link
Author	Darnière, Luck Junker, Markus
Copyright Year	2011
Abstract	In this paper, we introduce a notion of dimension and codimension for every element of a bounded distributive lattice L. These notions prove to have a good behavior when L is a co-Heyting algebra. In this case the codimension gives rise to a pseudometric on L which satisfies the ultrametric triangle inequality. We prove that the Hausdorff completion of L with respect to this pseudometric is precisely the projective limit of all its finite dimensional quotients. This completion has some familiar metric properties, such as the convergence of every monotonic sequence in a compact subset. It coincides with the profinite completion of L if and only if it is compact or equivalently if every finite dimensional quotient of L is finite. In this case we say that L is precompact. If L is precompact and Hausdorff, it inherits many of the remarkable properties of its completion, specially those regarding the join/meet irreducible elements. Since every finitely presented co-Heyting algebra is precompact Hausdorff, all the results we prove on the algebraic structure of the latter apply in particular to the former. As an application, we obtain the existence for every positive integers n, d of a term t n, d such that in every co-Heyting algebra generated by an n-tuple a, t n, d(a) is precisely the maximal element of codimension d.
Ending Page	282
Page Count	32
Starting Page	251
File Format	PDF
ISSN	00025240
e-ISSN	14208911
Journal	algebra universalis
Issue Number	3
Volume Number	64
Language	English
Publisher	SP Birkhäuser Verlag Basel
Publisher Date	2011-01-20
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Complete lattices, completions slices general topology Topological lattices, order topologies Lattices and duality Lattice theory Algebra Heyting algebras dimension duality codimension
Content Type	Text
Resource Type	Article
Subject	Logic Algebra and Number Theory

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