Infinite Atomized Semilattices

Atomized semilattices are subdirect product representations that facilitate theconstruction and manipulation of semilattices in practical contexts. This paper developsa careful axiomatization of atomized semilattices that extends to the infinite setting. Weextend the Full Crossing operator, which yields the quotient via principal congruences,to infinite structures. The notions of atom redundancy are generalized to the infinitecase, revealing a divergence between redundancy and weak redundancy that is absent infinite models. Finally, we prove that for every finitely generated semilattice there existsan atomization consisting exclusively of non-redundant atoms.