I am given (possibly total) order $\phi$ (represented as a directed acyclic graph), how can I construct an identity element $E_I$ for the order (i.e. $\phi$ $\otimes$ $E_I$=$\phi$)? let's assume $\otimes$ to be the product order ($(a_1,b_1)\succ (a_2,b_2)$ iff $a_1\succ a_2$ and $b_1\succ b_2$ ).
2026-04-07 20:50:36.1775595036
identity element for partial orders
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If I understand your question correctly, I have the same problem as @dfeuer, that it isn't entirely well-defined. What I think you're trying to do is the following:
We take all partially ordered sets and mod out by the relation $\sim$ that is given by $A\sim B$ whenever $A$ and $B$ are order-isomorphic. Next we want to consider this as an algebraic structure and see that it forms a monoid. First of all, the definition of product you mention neatly descends to one on isomorphism classes, so there's no problem there. Also, associativity doesn't seem to give much of a problem.
As to the identity element: notice that all ordered singleton sets are order-isomorphic, the only possible order being $a\succ a$ if the set is $\{a\}$. Therefore, there is only one isomorphism class corresponding to these sets. If we take this class as $E_I$, it is the identity element. Why? Just choose a representative singleton set and see what happens. In the product order you mention, the condition $b_1\succ b_2$ is automatically fulfilled, since $b_1=b_2$ and you get $(a_1,b_1)\succ(a_2,b_2)\iff a_1\succ a_2$. The product therefore is order-isomorphic to your original order.
If this wasn't your question and you meant something different, please explain what it is you do want.