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Title: Dimension in algebraic frames (English)
Author: Martínez, Jorge
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 56
Issue: 2
Year: 2006
Pages: 437-474
Summary lang: English
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Category: math
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Summary: In an algebraic frame $L$ the dimension, $\dim (L)$, is defined, as in classical ideal theory, to be the maximum of the lengths $n$ of chains of primes $p_0<p_1<\cdots <p_n$, if such a maximum exists, and $\infty $ otherwise. A notion of “dominance” is then defined among the compact elements of $L$, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of $L$, including the frames $dL$ and $zL$ of $d$-elements and $z$-elements, respectively. The more concrete illustrations regarding the frame convex $\ell $-subgroups of a lattice-ordered group and its various natural frame quotients occupy the second half of this exposition. For example, it is shown that if $A$ is a commutative semiprime $f$-ring with finite $\ell $-dimension then $A$ must be hyperarchimedean. The $d$-dimension of an $\ell $-group is invariant under formation of direct products, whereas $\ell $-dimension is not. $r$-dimension of a commutative semiprime $f$-ring is either 0 or infinite, but this fails if nilpotent elements are present. $sp$-dimension coincides with classical Krull dimension in commutative semiprime $f$-rings with bounded inversion. (English)
Keyword: algebraic frame
Keyword: dimension
Keyword: $d$-elements
Keyword: $z$-elements
Keyword: lattice-ordered group
Keyword: $f$-ring
MSC: 06D22
MSC: 06F15
MSC: 06F25
idZBL: Zbl 1164.06311
idMR: MR2291748
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Date available: 2009-09-24T11:34:30Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/128078
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