Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf{c}\left(H\right)$ of $H$ is the smallest integer $N$ with the following property: for each $a\in H$ and each two factorizations $z,z^{\prime }$ of $a$, there exist factorizations $z=z_0,...,z_k=z^{\prime }$ of $a$ such that, for each $i\in [1,k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition...

For an order embedding $G\stackrel{h}{\to }\to \Gamma$ of a partly ordered group $G$ into an $l$-group $\Gamma$ a topology ${𝒯}_{\widehat{W}}$ is introduced on $\Gamma$ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h\left(G\right)$, $h\left(X_t\right)$ and $(h\left(X_t\right)\setminus \{h\left(g_1\right),\cdots ,h\left(g_n\right)\})$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,{𝒯}_{\widehat{W}})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.