Dimension in algebraic frames

Jorge Martinez

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 2, page 437-474
  • ISSN: 0011-4642

Abstract

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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 < ⋯ < p n , if such a maximum exists, and ∞ 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 d L and z L of d -elements and z -elements, respectively. The more concrete illustrations regarding the frame convex ℓ -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 ℓ -dimension then A must be hyperarchimedean. The d -dimension of an ℓ -group is invariant under formation of direct products, whereas ℓ -dimension is not. r -dimension of a commutative semiprime f -ring is either 0 or infinite, but this fails if nilpotent elements are present. s p -dimension coincides with classical Krull dimension in commutative semiprime f -rings with bounded inversion.

How to cite

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Martinez, Jorge. "Dimension in algebraic frames." Czechoslovak Mathematical Journal 56.2 (2006): 437-474. <http://eudml.org/doc/31040>.

@article{Martinez2006,
abstract = {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.},
author = {Martinez, Jorge},
journal = {Czechoslovak Mathematical Journal},
keywords = {algebraic frame; dimension; $d$-elements; $z$-elements; lattice-ordered group; $f$-ring; algebraic frame; dimension; -elements; -elements; lattice-ordered group; -ring},
language = {eng},
number = {2},
pages = {437-474},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Dimension in algebraic frames},
url = {http://eudml.org/doc/31040},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Martinez, Jorge
TI - Dimension in algebraic frames
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 437
EP - 474
AB - 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.
LA - eng
KW - algebraic frame; dimension; $d$-elements; $z$-elements; lattice-ordered group; $f$-ring; algebraic frame; dimension; -elements; -elements; lattice-ordered group; -ring
UR - http://eudml.org/doc/31040
ER -

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