Priority queue for partially ordered priorities with infima
Although the exact problem posed in the original question does seem to be difficult (and I would be interested in a solution to that problem, especially the infima finding part). I just wanted to note that if the partially ordered set indeed consists of vectors using a product order, and if it is sufficient to just have the guarantee that the priority queue returns the values in an order that is "compatible" with the partial order (that is, smaller elements are always returned before larger elements), then there is a fairly easy way to do it.

The idea is essentially to find a topological ordering of the partially ordered set. That is, a total order '$\le_T$' such that $\mathbf{a}\le\mathbf{b}\implies \mathbf{a}\le_T\mathbf{b}$. For vectors using a product order, this is fairly easy: just use a lexicographical order '$\le_S$', where the first "component" is the sum of all the components used for the product order (the rest of the components are essentially arbitrary, so you could also stick to a weak order). We can then see that 
$$\mathbf{a}&lt;\mathbf{b}\implies \forall_i(a_i\le b_i)\text{ and }\exists_i(a_i&lt;b_i)\implies(\sum_i a_i)&lt;(\sum_i b_i)\implies\mathbf{a}\le_S\mathbf{b}$$
and
$$\mathbf{a}=\mathbf{b}\implies \forall_i(a_i=b_i)\implies(\sum_i a_i)=(\sum_i b_i)\implies\mathbf{a}\le_S\mathbf{b},$$
and thus that $\mathbf{a}\le\mathbf{b}\implies\mathbf{a}\le_S\mathbf{b}$. We can thus use this order with a priority queue and be sure that smaller elements (in the product order) will always be extracted before larger elements.