Decomposition of strongly separative monoids

Strong separativity is a weak form of cancellativity for commutative monoids. This notion can be naturally extended to po+-monoids, that is, commutative monoids endowed with a positive, compatible preordering. Every strongly separative po+-monoid can be embedded, with respect to the preordering, into a direct product image, where the Gaʹs are partially preordered abelian groups, and the imageʹs are special sorts of lexicographical powers of the positive reals. As a corollary, we prove that the universal theory of strongly separative po+-monoids is decidable. Hence, the word problem in finitely presented strongly separative po+-monoids is uniformly solvable.