Realizations of semilocal ℓ-groups over k[x 1,x 2,…x n

Abstract

Doering and Lequain in 1999 introduced a weak approximation theorem for dependent valuation rings and they proved that every finitely generated lattice-ordered group can be realized as the group of divisibility of a semilocal Bézout overring of a polynomial ring over a field k in infinitely many variables, where each of the valuation rings appearing in the finite intersection has residue field k. Moreover, they proved that every semilocal lattice-ordered group admits a lexico-cardinal decomposition form. In this work, we focus on realizing the semilocal -group over a polynomial ring in finitely many variables. We prove that every semilocal lattice-ordered group having a finite rational rank can be realized as the group of divisibility of a Bézout overring of up to lexico-cardinal decomposition, where k is a field and are indeterminates over k and n depends on the group. As a corollary, we prove that every semilocal -group either finitely generated or divisible with finite rational rank is realizable over where each of the valuation rings appearing in the finite intersection has residue field k.