-** A *poset* $({\mathcal{P}},\leq)$ is a partially ordered set. This means that $\leq$ is a binary relation on a nonempty set ${\mathcal{P}}$, satisfying $$\begin{array}{lcll}
\mbox{for any } {\mathcal{O}}\in{\mathcal{P}}& \Rightarrow & {\mathcal{O}}\leq {\mathcal{O}}& reflexive\\
{\mathcal{O}}_1\leq{\mathcal{O}}_2 \mbox{ and } {\mathcal{O}}_2\leq{\mathcal{O}}_1 & \Rightarrow & 
 {\mathcal{O}}_1={\mathcal{O}}_2 & antisymmetric\\
{\mathcal{O}}_1\leq{\mathcal{O}}_2 \mbox{ and } {\mathcal{O}}_2\leq {\mathcal{O}}_3 & \Rightarrow & 
{\mathcal{O}}_1\leq {\mathcal{O}}_3 & transitive
\end{array}$$ A poset is said to be *directed* if for any pair ${\mathcal{O}}_1,{\mathcal{O}}_2\in{\mathcal{P}}$ there exists ${\mathcal{O}}_3\in{\mathcal{P}}$ such that ${\mathcal{O}}_1,{\mathcal{O}}_2\leq {\mathcal{O}}_3$. For our aim, important examples of posets are provided by the standard simplices. A *standard* $n-simplex$ is defined as $$\Delta_n {\equiv}\big\{ ({\lambda}_0,\ldots,{\lambda}_n)\in \mathbb{R}^{n+1} \ 
  | \ {\lambda}_0+\cdots +{\lambda}_n=1, \ \ {\lambda}_i\in[0,1]\big\}.$$ It is clear that $\Delta_0$ is a point, $\Delta_1$ is a closed interval etc... The *inclusion maps* $d^n_i$ between standard simplices are maps $d^n_i:\Delta_{n-1}\longrightarrow \Delta_n$ defined as $$d^n_i({\lambda}_0,\ldots,{\lambda}_{n-1})= 
({\lambda}_0,{\lambda}_1,\ldots,{\lambda}_{i-1}, 0, {\lambda}_{i},\ldots{\lambda}_{n-1}),$$ for $n\geq 1$ and $0\leq i\leq n$. Now, note that a standard n-simplex $\Delta_n$ can be regarded as a partially ordered set with respect to the inclusion of its subsimplices. A *singular n-simplex* of a poset ${\mathcal{P}}$ is an order preserving map $f:\Delta_n\longrightarrow {\mathcal{P}}$. We denote by $\Sigma_n({\mathcal{P}})$ the collection of singular n-simplices of ${\mathcal{P}}$ and by ${\Sigma}_*({\mathcal{P}})$ the collection of 