In this paper we define the derivative and the Denjoy integral of mappings from a vector lattice to a complete vector lattice and show the fundamental theorem of calculus.
In a previous paper we defined a Denjoy integral for mappings from a vector lattice to a complete vector lattice. In this paper we define a Henstock-Kurzweil integral for mappings from a vector lattice to a complete vector lattice and consider the relation between these two integrals.
The main topic of the first section of this paper is the following theorem: let A be an Archimedean f-algebra with unit element e, and T : A → A a Riesz homomorphism such that T 2 (f) = T(fT(e)) for all f ∈ A. Then every Riesz homomorphism extension Te of T from the Dedekind completion A δ of A into itself satisfies Te2 (f) = Te(fT(e)) for all f ∈ A δ . In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application of the above result is a new approach to the Dedekind completion of commutative d-algebras.
It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator $b\colon E\times E\rightarrow F$ where $E$ and $F$ are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost $f$-algebras.
The extension of a lattice ordered group $A$ by a generalized Boolean algebra $B$ will be denoted by $A_B$. In this paper we apply subdirect decompositions of $A_B$ for dealing with a question proposed by Conrad and Darnel. Further, in the case when $A$ is linearly ordered we investigate (i) the completely subdirect decompositions of $A_B$ and those of $B$, and (ii) the values of elements of $A_B$ and the radical $R(A_B)$.