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We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation T=Ta+Tc$T = T_a + T_c$, where Ta$T_a$ is the sum of an absolutely order summable family of disjointness preserving operators and Tc$T_c$ is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators. [ABSTRACT FROM AUTHOR]

In this note we prove that positive multi-valued operators on Riesz spaces which are everywhere defined are always continuous. Further, we prove that the sum of two positive relations having the same null part is itself positive. This result allows us to study the domination problem in the class of positive relations. [ABSTRACT FROM AUTHOR]

Let U and V be two Archimedean Riesz spaces. An operator S : U → V is said to be unbounded order continuous (uo-continuous), if rα uo → 0 in U implies Srα uo → 0 in V . In this paper, we give some properties of the uo-continuous dual Uuo~ of U. We show that a nonzero linear functional f on U is uo-continuous if and only if f is a linear combination of finitely many order continuous lattice homomorphisms. The result allows us to characterize the uo-continuous dual Uuo~. In general, by giving an example that the uo-continuous dual U~ uo is not a band in U~, we obtain the conditions for the uo-continuous dual of a Banach lattice U to be a band in U~. Then, we examine the properties of uo-continuous operators. We show that S is an order continuous operator if and only if S is an unbounded order continuous operator when S is a lattice homomorphism between two Riesz spaces U and V . Finally, we proved that if an order bounded operator S : U → V between Archimedean Riesz space U and atomic Dedekind complete Riesz space V is uo-continuous, then /S/ is uo-continuous. [ABSTRACT FROM AUTHOR]

In this paper we introduce two new classes of operators which are called strongly order continuous and order-to-unbounded order continuous operators and we study their lattice properties. With under some conditions, we show that order continuity is equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all strongly continuous linear functionals on a Riesz space E is a band of E~. [ABSTRACT FROM AUTHOR]