2003/10/28 by Friedrich Wehrung · 1 citation
Computer Science · Mathematics · Decision Sciences · #Advanced Algebra and Logic #Rings, Modules, and Algebras #Fuzzy and Soft Set Theory #Mathematics #Semilattice #Quotient #Finitely-generated abelian group #Idempotence #Order (exchange) #Rank (graph theory) #Combinatorics #Pure mathematics #Ring (chemistry) #Lattice (music) #Discrete mathematics
paper · pdf · doi:10.1090/s0002-9947-03-03369-5
openalex publication_date 2003/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/22
We find a distributive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis logical-or comma 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ∨ </mml:mo> <mml:mo>,</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\vee ,0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -semilattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript omega 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Sω 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of size <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal alef 1"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal"> ℵ </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">ℵ 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: [—] There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript omega 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Sω 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . [—] There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript omega 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Sω 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These results are established by constructing an infinitary statement, denoted here by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper U normal upper R normal upper P Subscript normal s normal r"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">U</mml:mi> <mml:mi mathvariant="normal">R</mml:mi> <mml:msub> <mml:mi mathvariant="normal">P</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm URPsr</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript omega 1"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Sω 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .