A mode (idempotent and entropic algebra) is a Lallement sum of its cancellative submodes over a normal band if it has a congruence with a normal band quotient and cancellative congruence classes. We show that such a sum embeds as a subreduct into a semimodule over a certain ring, and discuss some consequences of this fact. The result generalizes a similar earlier result of the authors proved in the case when the normal band is a semilattice.
Let $P_k$ be a path on $k$ vertices. In an earlier paper we have proved that each polyhedral map $G$ on any compact $2$-manifold $\mathbb{M}$ with Euler characteristic $\chi (\mathbb{M})\le 0$ contains a path $P_k$ such that each vertex of this path has, in $G$, degree $\le k\left\lfloor \frac{5+\sqrt{49-24 \chi (\mathbb{M})}}{2}\right\rfloor $. Moreover, this bound is attained for $k=1$ or $k\ge 2$, $k$ even. In this paper we prove that for each odd $k\ge \frac{4}{3} \left\lfloor \frac{5+\sqrt{49-24\chi (\mathbb{M})}}{2}\right\rfloor +1$, this bound is the best possible on infinitely many compact $2$-manifolds, but on infinitely many other compact $2$-manifolds the upper bound can be lowered to $\left\lfloor (k-\frac{1}{3})\frac{5+\sqrt{49-24\chi (\mathbb{M})}}{2}\right\rfloor $.