The main aim of this paper is to obtain compositive cone factorizations from non-compositive ones by itereration. This is possible if and only if certain colimits of (possibly large) chains exist. In particular, we show that (strong-epi, mono) factorizations of cones exist if and only if joint coequalizers and colimits of chains of regular epimorphisms exist.

We study free sequences and related notions on Boolean algebras. A free sequence on a BA $A$ is a sequence $\langle a_{\xi }:\xi <\alpha \rangle$ of elements of $A$, with $\alpha$ an ordinal, such that for all $F,G\in {\left[\alpha \right]}^{<\omega }$ with $F<G$ we have ${\prod }_{\xi \in F}{a}_{\xi }·{\prod }_{\xi \in G}-a_{\xi }\ne 0$. A free sequence of length $\alpha$ exists iff the Stone space $Ult\left(A\right)$ has a free sequence of length $\alpha$ in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function $${𝔣}_{sp}\left(A\right)=\left\{\right|\alpha |:A\text{has}\text{an}\text{infinite}\text{maximal}\text{free}\text{sequence}\text{of}\text{length}\alpha \}$$ and the associated min-max function $$𝔣\left(A\right)=min\left({𝔣}_{sp}\left(A\right)\right).$$ Among the results...