A New Characterization of C-Complete Riesz Spaces

Abstract In the present paper, we expose a deep analogy of some recently discovered objects of the Riesz space theory with classical notions of Analysis such as order ideals, bands, order projections and properties like Dedekind completeness, principal projection property. A Riesz space E is said to be C-complete if every nonempty subset of fragments of an element $$e \in E^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>E</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:math> has a supremum. One of our results asserts that, a Riesz space E is C-complete if and only if every lateral band of E is a projection lateral band. Another result characterizes projective lateral bands in an arbitrary Riesz space and provides an explicit formula for the corresponding lateral band projection, which was previously known for C-complete Riesz spaces. Our characterization of a projective lateral band consists of two conditions, one of which is actually a property of the space, which we call relative C-completeness, and the other one is a property of bands, which we call relative lateral bands. A similar characterization is obtained for the relative C-completeness. We describe the lateral disjoint complement $$A^\dag $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>†</mml:mo> </mml:msup> </mml:math> to any subset A of E , show that $$A^{\dag \dag }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>†</mml:mo> <mml:mo>†</mml:mo> </mml:mrow> </mml:msup> </mml:math> equals the relative lateral closure of the lateral ideal generated by A and prove that $$A^\dag = A^{\dag \dag \dag }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>†</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>†</mml:mo> <mml:mo>†</mml:mo> <mml:mo>†</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Then we obtain some corollaries on the extension of orthogonally additive operators. We also provide many examples confirming the sharpness of our results.