Cancellation in primely generated refinement monoids.
A commutative monoid \(M\) is a refinement groupoid if the equation \(a_1+a_2=b_1+b_2\) implies the existence of \(c_{11},c_{12},c_{21},c_{22}\in M\) such that \(a_1=c_{11}+c_{12}\), \(a_2=c_{21}+c_{22}\), \(b_1=c_{11}+c_{21}\), \(b_2=c_{12}+c_{22}\). Further, \(M\) is: cancellative if \(a+c=b+c\) implies \(a=b\), separative if \(2a=a+b=2b\) implies \(a=b\) and strongly separative if \(2a=a+b\) implies \(a=b\). It is proved that any primely generated refinement monoid \(M\) has separative cancellation, and even strong separative cancellation provided \(M\) has no nonzero idempotents. In addition, \(M\) is a semilattice with respect to the order \(\leq\) defined by \(a\leq b\) iff \(a+c=b\) for some \(c\in M\). It is shown that any finitely generated refinement monoid is primely generated (i.e. every element is a sum of prime elements of \(M\)). A number of questions answered in the paper arises in module theory and they are reformulated and generalized for refinement groupoids.
scientific article; zbMATH DE number 2158749
scientific article