In the paper we prove that every orthosymmetric lattice bilinear map on the cartesian product of a vector lattice with itself can be extended to an orthosymmetric lattice bilinear map on the cartesian product of the Dedekind completion with itself. The main tool used in our proof is the technique associated with extension to a vector subspace generated by adjoining one element. As an application, we prove that if $(A,*)$ is a commutative $d$-algebra and $A^{𝔡}$ its Dedekind completion, then, $A^{𝔡}$ can be equipped...

In the paper we study the existence of nonzero positive invariant elements for positive operators in Riesz spaces. The class of Riesz spaces for which the results are valid is large enough to contain all the Banach lattices with order continuous norms. All the results obtained in earlier works deal with positive operators in KB-spaces and in many of them the approach is based upon the use of Banach limits. The methods created for KB-spaces cannot be extended to our more general setting; that is...

Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\widehat{\tau })$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau$ has the Fatou property, then the order intervals of $G$ are $\tau$-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\widehat{\tau })$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on...

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 $TA\to A$ a Riesz homomorphism such that $T^2\left(f\right)=T\left(fT\left(e\right)\right)$ for all $f\in A$. Then every Riesz homomorphism extension $\tilde{T}$ of $T$ from the Dedekind completion $A^{\delta }$ of $A$ into itself satisfies ${\tilde{T}}^2\left(f\right)=\tilde{T}\left(fT\left(e\right)\right)$ for all $f\in A^{\delta }$. 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...

In this paper we investigate conditions for a system of sequences of elements of a vector lattice; analogous conditions for systems of sequences of reals were studied by D. E. Peek.

Two problems posed by Choquet and Foias are solved:(i) Let $T$ be a positive linear operator on the space $C\left(X\right)$ of continuous real-valued functions on a compact Hausdorff space $X$. It is shown that if $n^{-1}{\sum }_{r=0}^{n-1}{T}^{r}1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$.(ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A\left(K\right)$ of continuous affine real-valued functions on $K$, such that$$\inf \left\{\right(T^{n}1\left)\right(x):n\ge \}\<1$$for each $x$ in ${\partial }_{\ell }K$, but $\parallel T^{n}1\parallel$ does not converge to 0.