| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2009-09-24T11:34:30Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128078 |
| . |
| Reference:
|
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|
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|
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