Dimension in algebraic frames

Martínez, Jorge

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 437-474.

Résumé

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 $p_0$, if such a maximum exists, and $\mathrm{\infty }$ 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.

Détail

Classification : 06D22, 06F15, 06F25
Mots-clés : algebraic frame; dimension; $d$-elements; $z$-elements; lattice-ordered group; $f$-ring