Related papers: Vector lattices admitting a positively homogeneous…

We develop a functional calculus on Archimedean vector lattices for semicontinuous positively homogeneous real-valued functions defined on $\R^n$ which are bounded on the unit sphere. It is further shown that this semicontinuous Archimedean…

We characterize the Archimedean lattice-ordered algebras with identity that admit a polynomial growth continuous function calculus. More precisely, for an $n$-tuple $\mathbf{x}=(x_1,\dots,x_n)$ in an Archimedean lattice-ordered algebra $X$…

We study completions of Archimedean vector lattices relative to any nonempty set of positively-homogeneous functions on finite-dimensional real vector spaces. Examples of such completions include square mean closed and geometric closed…

We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of…

We prove that the category of unital hyperarchimedean vector lattices is equivalent to the category of Boolean algebras. The key result needed to establish the equivalence is that, via the Yosida representation, such a vector lattice is…

We show that a free vector lattice over a real vector space $V$ can be realised canonically as a vector lattice of real-valued positively homogeneous functions on any linear subspace of its dual space that separates the points of $V$. This…

Given an Archimedean vector lattice $E$, we present one elementary property of $E$ which is equivalent to the entire traditional list of axioms which makes $E$ a $\Phi$-algebra. We call a vector lattice with this property ``square closed".…

We define a free uniformly complete vector lattice over a set of generators and give its concrete representation as a space of continuous positively homogeneous functions.

We show that for an ideal $H$ in an Archimedean vector lattice $F$ the following conditions are equivalent: $\bullet$ $H$ is a projection band; $\bullet$ Any collection of mutually disjoint vectors in $H$, which is order bounded in $F$, is…

We show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence and…

In this paper, we study Artinian and Noetherian properties in vector lattices and provide a concrete representation of these spaces. Furthermore, we describe for which Archimedean uniformly complete vector lattices every decreasing sequence…

Let $S$ be a non-empty, closed subspace of a locally compact group $G$ that is a subsemigroup of $G$. Suppose that $X, Y$, and $Z$ are Banach lattices that are vector sublattices of the order dual $\mathrm{C}_{\mathrm{c}}(S,\mathbb R)^\sim$…

In the paper, we revisit several approaches to the concept of uniform completion $X^{\mathrm{ru}}$ of a vector lattice $X$. We show that many of these approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly…

Let $X$ be an Archimedean vector lattice. We investigate subalgebras of $\mathscr{L}(X)$ consisting of regular operators that contain all rank-one operators of the form $a \otimes \varphi_b$, where $a$ and $b$ are atoms of $X$ and…

A synaptic algebra $A$ is a generalization of the self-adjoint part of a von Neumann algebra. We study a linear subspace $V$ of $A$ in regard to the question of when $V$ is a vector lattice. Our main theorem states that if $V$ contains the…

We start a systematic analysis of the first-order model theory of free lattices. Firstly, we prove that the free lattices of finite rank are not positively indistinguishable, as there is a positive $\exists \forall$-sentence true in…

A celebrated theorem of Hadwiger states that the Euler-Poincar\'e characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty…

We study D\'iaz-Dineen's problem for regular homogeneous vector-valued polynomials. In particular, we prove that if $E^*$ and $F^*$ are lattice isomorphic with at least one having order continuous norm, then $\mathcal{P}^r(^n E; G^*)$ and…

We consider all compatible topologies of an arbitrary finite-dimensional vector space over a non-trivial valuation field whose metric completion is a locally compact space. We construct the canonical lattice isomorphism between the lattice…

Let $L$ be a (non necessarily unital) truncated vector lattice of real-valued functions on a nonempty set $X$. A nonzero linear functional $\psi$ on $L$ is called a truncation homomorphism if it preserves truncation, i.e.,% \[ \psi\left(…