7,367 research outputs found
A Proof of Tarski’s Fixed Point Theorem by Application of Galois Connections
Two examples of Galois connections and their dual forms are considered. One
of them is applied to formulate a criterion when a given subset of a complete lattice forms
a complete lattice. The second, closely related to the first, is used to prove in a short way
the Knaster-Tarski’s fixed point theore
Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements
We study Archimedean atomic lattice effect algebras whose set of sharp
elements is a complete lattice. We show properties of centers, compatibility
centers and central atoms of such lattice effect algebras. Moreover, we prove
that if such effect algebra is separable and modular then there exists a
faithful state on . Further, if an atomic lattice effect algebra is densely
embeddable into a complete lattice effect algebra and the
compatiblity center of is not a Boolean algebra then there exists an
-continuous subadditive state on
Dedekind sigma-complete l-groups and Riesz spaces as varieties
We prove that the category of Dedekind -complete Riesz spaces is an
infinitary variety, and we provide an explicit equational axiomatization. In
fact, we show that finitely many axioms suffice over the usual equational
axiomatization of Riesz spaces. Our main result is that , regarded
as a Dedekind -complete Riesz space, generates this category as a
quasi-variety, and therefore as a variety. Analogous results are established
for the categories of (i) Dedekind -complete Riesz spaces with a weak
order unit, (ii) Dedekind -complete lattice-ordered groups, and (iii)
Dedekind -complete lattice-ordered groups with a weak order unit.Comment: 15 page
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