In an algebraic frame $L$ the dimension, $dim\left(L\right)$, 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...

This paper continues the investigation into Krull-style dimensions in algebraic frames. Let $L$ be an algebraic frame. $dim\left(L\right)$ is the supremum of the lengths $k$ of sequences $p_0<p_1<\cdots <p_k$ of (proper) prime elements of $L$. Recently, Th. Coquand, H. Lombardi and M.-F. Roy have formulated a characterization which describes the dimension of $L$ in terms of the dimensions of certain boundary quotients of $L$. This paper gives a purely frame-theoretic proof of this result, at once generalizing it to frames which are not necessarily...