how to combine different partial orders (Poset) Given two posets $\prec_A$ and $\prec_B$ where $A\neq B$ and $A\cap B\neq \emptyset$, is there any way to combine them while preserving the exact information they exhibit - namely dominance relation ( $x\prec y$) and incomparability.
 (i.e. if $x$ and $y$ are incomparable in $\prec_A$ then they are incomparable in $\prec_{AB}$)? 
 A: If $A\cap B=\varnothing$ you can define an order $<$ on $A\cup B$ by $x<y$ if, and only if:


*

*$x,y\in A$ and $x\prec_A y$.

*$x,y\in B$ and $x\prec_B y$.


This will preserve comparability. A diagram for this order can be obtained by placing the diagram for $A$ on the same plane as $B$.

Again assuming $A\cap B=\varnothing$, you can define an order $\ll$ on $A\cup B$ by $x\ll y$ if, and only if:


*

*$x,y\in A$ and $x\prec_A y$

*$x,y\in B$ and $x\prec_B y$.

*$x\in A$ and $y\in B$


This will also preserve comparability. A diagram for this order can be obtained by placing the diagram for $A$ below the diagram for $B$ and connecting $A$'s maximal elements to $B$'s minimal elements if they exist.
A: Example where the combination of 2 Posets is not a Poset:

*

*<A={{1},{2,3},{4,5}}=SA and <B={{2,3},{5,6},{7}}=SB

*<(SA||SB)={{1},{2,3},{2,3},{4,5},{5,6},{7}} is not equal to <(∪)

where :

*

*∪ ={1,2,3,4,5,6,7}

*SA||SB being the concatenation of SA with SB

*SA||SB={{1},{2,3},{4,5},{2,3},{5,6},{7}}

*<(SA||SB) being the sorted concatenation of SA and SB

