283 research outputs found
Joint Laver diamonds and grounded forcing axioms
I explore two separate topics: the concept of jointness for set-theoretic
guessing principles, and the notion of grounded forcing axioms. A family of
guessing sequences is said to be joint if all of its members can guess any
given family of targets independently and simultaneously. I primarily
investigate jointness in the case of various kinds of Laver diamonds. In the
case of measurable cardinals I show that, while the assertions that there are
joint families of Laver diamonds of a given length get strictly stronger with
increasing length, they are all equiconsistent. This is contrasted with the
case of partially strong cardinals, where we can derive additional consistency
strength, and ordinary diamond sequences, where large joint families exist
whenever even one diamond sequence does. Grounded forcing axioms modify the
usual forcing axioms by restricting the posets considered to a suitable ground
model. I focus on the grounded Martin's axiom which states that Martin's axioms
holds for posets coming from some ccc ground model. I examine the new axiom's
effects on the cardinal characteristics of the continuum and show that it is
quite a bit more robust under mild forcing than Martin's axiom itself.Comment: This is my PhD dissertatio
Goldblatt-Thomason Theorems for Modal Intuitionistic Logics
We prove Goldblatt-Thomason theorems for frames and models of a wide variety
of modal intuitionistic logics, including ones studied by Wolter and
Zakharyaschev, Goldblatt, Fischer Servi, and Plotkin and Sterling. We use the
framework of dialgebraic logic to describe most of these logics and derive
results in a uniform way
Extension of sectional pseudocomplementation in posets
Sectional pseudocomplementation (sp-complementation) on a poset is a partial
operation which associates with every pair of elements, where , the pseudocomplement of in the upper section . Any total
extension of is said to be an extended sp-complementation and is
considered as an implication-like operation. Extended sp-complementations have
already be studied on semilattices and lattices. We describe several naturally
arising classes of general posets with extended sp-complementation, present
respective elementary properties of this operation, demonstrate that two other
known attempts to isolate particular such classes are in fact not quite
correct, and suggest suitable improvements.Comment: pdfLaTeX, 28 pages, contains LaTeX figures and tables. V2:
Proposition 3.6 corrected, the final part of Section 3.3 reorganized, Remark
3 edited, Theorem 7.6 and Corollary 7.7 strengthened, proof of Theorem 7.10
edite
Every topos has an optimal noetherian form
The search, of almost a century long, for a unified axiomatic framework for
establishing homomorphism theorems of classical algebra (such as Noether
isomorphism theorems and homological diagram lemmas) has led to the notion of a
`noetherian form', which is a generalization of an abelian category suitable to
encompass categories of non-abelian algebraic structures (such as non-abelian
groups, or rings with identity, or cocommutative Hopf algebras over any field,
and many others). In this paper, we show that, surprisingly, even the category
of sets, and more generally, any topos, fits under the framework of a
noetherian form. Moreover, we give an intrinsic characterization of such
noetherian form and show that it is very closely related to the known
noetherian form of a semi-abelian category. In fact, we show that for a pointed
category having finite products and sums, the existence of the type of
noetherian form that any topos possesses is equivalent to the category being
semi-abelian (this result is unexpected since only trivial toposes can be
semi-abelian). We also show that these noetherian forms are optimal, in a
suitable sense.Comment: 66 pages, submitted for publicatio
Contact semilattices
We devise exact conditions under which a join semilattice with a weak contact
relation can be semilattice embedded into a Boolean algebra with an overlap
contact relation, equivalently, into a distributive lattice with additive
contact relation. A similar characterization is proved with respect to Boolean
algebras and distributive lattices with weak contact, not necessarily additive,
nor overlap.Comment: v3: noticed that former Condition (D2-) is pleonastic; added two new
equivalent conditions in Theorem 3.2. We realized all this after the paper
has been published: variations with respect to the published version are
printed in a blue character. v2: solved a problem left open in v1; added a
counterexample; a few fixe
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