Dimension in algebraic frames |
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Authors: | Jorge Martínez |
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Institution: | (1) Department of Mathematics, University of Florida, P. O. Box 118105, Gainesville, FL 32611-8105, USA |
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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 < ... < 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 dL and zL 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. sp-dimension coincides with classical Krull dimension in commutative semiprime f-rings with bounded inversion. |
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Keywords: | algebraic frame dimension d-elements z-elements lattice-ordered group f-ring |
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