Gelfand theorem implies Stone representation theorem of Boolean rings.
Geometric and higher order logic in terms of abstract Stone duality.
Large Subalgebras of a Boolean Algebra
Maximal free sequences in a Boolean algebra
We study free sequences and related notions on Boolean algebras. A free sequence on a BA is a sequence of elements of , with an ordinal, such that for all with we have . A free sequence of length exists iff the Stone space has a free sequence of length in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function and the associated min-max function Among the results...
Measurable Refinement Monoids and Applications to Distributive Semilattices, Heyting Algebras, and Stone Spaces.
Nonisomorphic thin-tall superatomic Boolean algebras
Nonsupercompactness and the reduced measure algebra
On preimages of ultrafilters in ZF
We show that given infinite sets and a function which is onto and -to-one for some , the preimage of any ultrafilter of under extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms and a finite-to-one onto function such that for each free ultrafilter of its preimage under does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact pseudometric...
On Priestley duals of products
On the representation of orthocomplemented posets
Partially additive states on orthomodular posets
We fix a Boolean subalgebra B of an orthomodular poset P and study the mappings s:P → [0,1] which respect the ordering and the orthocomplementation in P and which are additive on B. We call such functions B-states on P. We first show that every P possesses "enough" two-valued B-states. This improves the main result in [13], where B is the centre of P. Moreover, it allows us to construct a closure-space representation of orthomodular lattices. We do this in the third section. This result may also...
Pointwise periodic groups
Quasi locally connected toposes.
Reflexive algebras and sigma algebras.
Remarks on the Stone Spaces of the Integers and the Reals without AC
In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.
Representation and duality for Hilbert algebras
In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are...
Representation of certain classes of distributive lattices by sections of sheaves.
Representation of Hilbert algebras and implicative semilattices
In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring.
Representation Theorem for Minimal -Algebras
Sequential convergences on Boolean algebras defined by systems of maximal filters
We study sequential convergences defined on a Boolean algebra by systems of maximal filters. We describe the order properties of the system of all such convergences. We introduce the category of 2-generated convergence Boolean algebras and generalize the construction of Novák sequential envelope to such algebras.