DECOMPOSITION OF STRONGLY SEPARATIVE MONOIDS

Abstract. Strong separativity is a weak form of cancellativity for commuta-tive monoids. This notion can be naturally extended to po+-monoids, that is, commutative monoids endowed with a positive, compatible preordering. Ev-ery strongly separative po+-monoid can be embedded, with respect to the pre-ordering, into a direct product a∈A(G a ∪R(a)), where the Ga-s are partially preordered abelian groups, and the R(a)-s 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. 1