Lattice-ordered algebras that are subdirect products of valuation domains

Abstract

An <italic>f</italic> -ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) <italic>A</italic> is called an <italic>SV-ring</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A slash upper P"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A/P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a valuation domain for every prime ideal <italic>P</italic> of <italic>A</italic> . If <italic>M</italic> is a maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -ideal of <italic>A</italic> , then the <italic>rank</italic> of <italic>A</italic> at <italic>M</italic> is the number of minimal prime ideals of <italic>A</italic> contained in <italic>M , rank</italic> of <italic>A</italic> is the sup of the ranks of <italic>A</italic> at each of its maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -ideals. If the latter is a positive integer, then <italic>A</italic> is said to have <italic>finite rank</italic> , and if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A = C(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the ring of all real-valued continuous functions on a Tychonoff space, the <italic>rank</italic> of <italic>X</italic> is defined to be the rank of the <italic>f</italic> -ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <italic>X</italic> is called an <italic>SV-space</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <italic>ST</italic> -ring. <italic>X</italic> has finite rank <italic>k</italic> iff <italic>k</italic> is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general <italic>f</italic> -rings these two concepts are unrelated, but if <italic>A</italic> is uniformly complete (in particular, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper C left-parenthesis upper X right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A = C(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ) then if <italic>A</italic> is an <italic>SV</italic> -ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space <italic>X</italic> of finite rank that fails to be an <italic>SV</italic> -space.