2 research outputs found
A Johnson-Kist type representation for truncated vector lattices
We introduce the notion of (maximal) multi-truncations on a vector lattice as
a generalization of the notion of truncations, an object of recent origin. We
obtain a Johnson-Kist type representation of vector lattices with maximal
multi-truncations as vector lattices of almost-finite extended-real continuous
functions. The spectrum that allow such a representation is a particular set of
prime ideals equipped with the hull-kernel topology. Various representations
from the existing literature will appear as special cases of our general
result
Structural aspects of truncated archimedean vector lattices: simple elements, good sequences
The truncation operation facilitates the articulation and analysis of several
aspects of the structure of archimedean vector lattices; we investigate two
such aspects in this article. We refer to archimedean vector lattices equipped
with a truncation as \emph{truncs}. In the first part of the article we review
the basic definitions, state the (pointed) Yosida Representation Theorem for
truncs, and then prove a representation theorem which subsumes and extends the
(pointfree) Madden Representation Theorem. The proof has the virtue of being
much shorter than the one in the literature, but the real novelty of the
theorem lies in the fact that the topological data dual to a given trunc is
a (localic) compactification, i.e., a dense pointed frame surjection out of a compact regular pointed frame . The representation is an
amalgam of the Yosida and Madden representations; the compact frame is
sufficient to describe the behavior of the bounded part of in the
sense that separates the points of the compact Hausdorff
pointed space dual to , while the frame is just sufficient to
capture the behavior of the unbounded part of in . The
truncation operation lends itself to identifying those elements of a trunc
which behave like characteristic functions, and in the second part of the
article we characterize in several ways those truncs composed of linear
combinations of such elements. Along the way, we show that the category of such
truncs is equivalent to the category of pointed Boolean spaces, and to the
category of generalized Boolean algebras. The short third part contains a
characterization of the kernels of truncation homomorphisms in terms of
pointwise closure. In it we correct an error in the literature