Boolean-Monotone

Let $I = (S,\prec)$ be a partially ordered set. Let $\mathcal{A}_i,~i\in I$ be a family of complex algebras indexed by $I$

We say that the algebras $\mathcal{A}_i,~i\in I$ are bm independent if the following holds:

• For all $\xi \prec \eta \succ \rho$ and $a_{\xi} \in \mathcal{A}_{\xi}, a_{\eta} \in \mathcal{A}_{\eta}, a_{\rho} \in \mathcal{A}_{\rho}$, $\varphi(a_{\xi}a_{\eta}a_{\rho})=\varphi(a_{\eta})a_{\xi}a_{\rho}$,
• For $n\geq1$, $1 \leq k\leq l \leq n$ and elements of $I$:

$$\xi_1 \succ \xi_2 \cdots \succ \xi_k \not\sim\cdots\not\sim\xi_l \prec \xi_{l+1} \prec\cdots\prec\xi_n,$$ it holds that, with $a_{\xi_i} \in \mathcal{A}_{\xi_i}$, $1\leq i \leq n$: $$\varphi(a_{\xi_1}\cdots a_{\xi_n})=\varphi(a_{\xi_1})\cdots\varphi(a_{\xi_n}).$$