Up to now, the only strategy being tested is ``take the pair with smallest least common multiple of the leading monomials'', however, the interface for choosing a different strategy is implemented. A possible (i.e. reasonable) strategy to be tested is ``take the pair with smallest leading monomial of the S-polynomial''.
For the variant of the Gröbner bases algorithm, in which pairs for which a criterion can be applied are deleted immediately, we also have to fix an ordering on the pairs with equal ``lcm''. This ordering is called is_labeled_pa_greater and is a refinement of is_pa_greater in the way that pairs with equal ``lcm'' are ordered by a lexical ordering on the pairs of labels (of the polynomials in the pair).