Decompositions of elements into intersections of primal elements and into intersections
of p-components are studied in certain lattice-ordered commutative semigroups, by
making use of the new development in commutative ideal theory without finiteness conditions,
due to Fuchs-Heinzer-Olberding . Several results concerning ideals can be phrased
as theorems in ‘abstract ideal theory’.
The intersections we consider are in general not irredundant, and the associated prime
elements are not unique. However, one can establish a canonical intersection that is often
irredundant with uniquely determined associated primes.