Almost 𝜎-Dedekind complete Riesz spaces and the main inclusion theorem

Abstract

A Riesz space is <italic> almost <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Dedekind complete </italic> if it can be embedded as a super order dense Riesz subspace of a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Dedekind complete space. The location of this concept in the Main Inclusion Theorem of Luxemburg and Zaanen is investigated; it is shown that this concept is implied by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi> σ </mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Dedekind completeness and by the projection property, that it implies the property of being Archimedean, and that it is independent of the principal projection property and the property of having sufficiently many projections.