Notes on \(R1\)-ideals in partial abelian monoids (Q1866818)

A cancellative positive partial abelian monoid (CPAM) is an algebra \(\mathcal P =(P;\oplus ,0)\) of type \((2,0)\) which is a partial commutative monoid satisfying the left cancellation law and \(a\oplus b=0\) implies \(a=0\). The author generalizes effect algebras or \(D\)-posets. Several concepts of ideals are introduced. Especially, an \(R1\)-ideal \(I\) is defined in such way that, for each CPAM \(\mathcal P\), \(\mathcal P /I\) is also a CPAM. The author studies congruences on CPAM associated with \(R1\)-ideals and the lattice of all \(R1\)-ideals on a CPAM. An \(R1\)-ideal satisfying one more condition is called a Riesz ideal, the lattice of all Riesz ideals is a sublattice of the lattice of all ideals as shown by the author.