Abstract
The paper contains a systematic study of the lateral partial order sqsubseteq in a Riesz space (the relation x sqsubseteq y means that x is a fragment of y) with applications to nonlinear analysis of Riesz spaces. We introduce and study lateral fields, lateral ideals, lateral bands and consistent subsets and show the importance of these notions to the theory of orthogonally additive operators, like ideals and bands are important for linear operators. We prove the existence of a lateral band projection, provide an elegant formula for it and prove some properties of this orthogonally additive operator. One of our main results (Theorem 7.5) asserts that, if D is a lateral field in a Riesz space E with the intersection property, X a vector space and T_0:Drightarrow X an orthogonally additive operator, then there exists an orthogonally additive extension T:Erightarrow X of T_0. The intersection property of E means that every two-point subset of E has an infimum with respect to the lateral order. In particular, the principal projection property implies the intersection property.
Highlights
The main idea of the paper is to show the importance of the so called lateral partial order on a Riesz space for analysis of Riesz spaces, especially for the study of orthogonally additive operators
Among well known results on the subject, it is worth mentioning that the set Fe of all fragments of an element e of a Riesz space E is a Boolean algebra with respect to the lateral order with zero 0 and unit e
We show that the kernel of a positive orthogonally additive operator is a lateral ideal, not every lateral ideal equals the kernel of some orthogonally additive operator
Summary
The main idea of the paper is to show the importance of the so called lateral partial order on a Riesz space for analysis of Riesz spaces, especially for the study of orthogonally additive operators. The new notion proposed in [10] and developed in the present paper naturally generalizes the lateral convergence from laterally increasing nets to arbitrary laterally bounded nets Due to this idea, the main result of [10] asserts that the lateral continuity of an orthogonally additive operator T at zero implies the lateral continuity of T at any point, which is even impossible to formulate for the old notion because there is no nontrivial laterally increasing net which laterally converges to zero. The main result of the section Theorem 5.4 asserts that a Riesz space E is laterally complete if and only if every consistent set A ⊂ E has a lateral supremum. The main result of the section (Theorem 7.5) asserts that, if D is a lateral field in a Riesz space E with the intersection property, X a vector space and T0 : D → X an orthogonally additive operator there exists an orthogonally additive extension T : E → X of T0
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