A Generalization of the Jaffard-Ohm-Kaplansky Theorem
Algebraic frame, Quantale, Prüfer domain, Lattice-ordered group
The well-known Jaffard–Ohm–Kaplansky Theorem states that every abelian ℓ-group can be realized as the group of divisibility of a commutative Bézout domain. To date there is no realization (except in certain circumstances) of an arbitrary, not necessarily abelian, ℓ-group as the group of divisibility of an integral domain. We show that using filters on lattices we can construct a nice quantal frame whose “group of divisibility” is the given ℓ-group. We then show that our construction when applied to an abelian ℓ-group gives rise to the lattice of ideals of any Prüfer domain assured by the Jaffard–Ohm–Kaplansky Theorem. Thus, we are assured of the appropriate generalization of the Jaffard–Ohm–Kaplansky Theorem
Iberkleid, Wolf and McGovern, Warren William, "A Generalization of the Jaffard-Ohm-Kaplansky Theorem" (2009). Mathematics Faculty Articles. 21.