General Extension Research Articles

Quotients of Boolean algebras and regular subalgebras

Let \({\mathbb{B}}\) and \({\mathbb{C}}\) be Boolean algebras and \({e: \mathbb{B}\rightarrow \mathbb{C}}\) an embedding. We examine the hierarchy of ideals on \({\mathbb{C}}\) for which \({ \bar{e}: \mathbb{B}\rightarrow \mathbb{C} / \fancyscript{I}}\) is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between \({\fancyscript{P}(\omega)/{{\rm fin}}}\) in the ground model and in its extension. If M is an extension of V containing a new subset of ω, then in M there is an almost disjoint refinement of the family ([ω]ω)V. Moreover, there is, in M, exactly one ideal \({\fancyscript{I}}\) on ω such that \({(\fancyscript{P}(\omega)/{{\rm fin}})^V}\) is a dense subalgebra of \({(\fancyscript{P}(\omega)/\fancyscript{I})^M}\) if and only if M does not contain an independent (splitting) real. We show that for a generic extension V[G], the canonical embedding $$\fancyscript{P}^V(\omega){/}{\rm fin}\hookrightarrow \fancyscript{P}(\omega){/}(U(Os)(\mathbb{B}))^G$$ is a regular one, where \({U(Os)(\mathbb{B})}\) is the Urysohn closure of the zero-convergence structure on \({\mathbb{B}}\).

Presenting degenerate Ringel–Hall algebras of cyclic quivers

Using the generators labelled by simple and sincere semisimple modules for the Ringel–Hall algebra H q ( n ) of a cyclic quiver Δ ( n ) , we give a presentation for the degenerate algebra H 0 ( n ) . This is achieved by establishing a presentation for the generic extension monoid algebra of Δ ( n ) . As an application, we show that both the degenerate Ringel–Hall algebra and the degenerate quantum affine s l n admit multiplicative bases.

Universally measurable sets in generic extensions

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets.

Kripke models for subtheories of CZF

In this paper a method to construct Kripke models for subtheories of constructive set theory is introduced that uses constructions from classical model theory such as constructible sets and generic extensions. Under the main construction all axioms except the collection axioms can be shown to hold in the constructed Kripke model. It is shown that by carefully choosing the classical models various instances of the collection axioms, such as exponentiation, can be forced to hold as well. The paper does not contain any deep results. It consists of first observations on the subject, and is meant to introduce some notions that could serve as a foundation for further research.

Incompatible Ω-Complete Theories

AbstractIn 1985 the second author showed that if there is a proper class of measurable Woodin cardinals andandare generic extensions ofVsatisfying CH thenandagree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axiomsAsuch that relative to large cardinal axioms ZFC +Ais Ω-complete for all of third-order arithmetic. Going further, for each specifiable segmentVλof the universe of sets (for example, one might takeVλto be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axiomsAsuch that relative to large cardinal axioms ZFC +Ais Ω-complete for the theory ofVλ. If such theories exist, extend one another, and are unique in the sense that any other such theoryBwith the same level of Ω-completeness asAis actually Ω-equivalent toAover ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.

An architecture for generic extensions

We examine what is necessary to allow generic libraries to be used naturally in a multi-language, potentially distributed environment. Language-neutral library interfaces usually do not support the full range of programming idioms that are available when a library is used natively. We investigate how to structure the language bindings of the neutral interface to achieve a better expressibility and code re-use. We furthermore address how language-neutral interfaces can be extended with import bindings to recover the desired programming idioms. We also address the question of how these extensions can be organized to minimize the performance overhead that arises from using objects in manners not anticipated by the original library designers. Our approach is to treat a library as a software component and to view the problem as one of component extension. We use C ++ as an example of a mature language, with libraries using a variety of patterns, and use the Standard Template Library as an example of a complex library for which efficiency is important. By viewing the library extension problem as one of component organization, we enhance software composibility, hierarchy maintenance and architecture independence.

Generic extensions and generic polynomials for semidirect products

This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group A ⋊ G over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension. Generic polynomials for several small groups are constructed.

Efficient and Type-Safe Generic Data Storage

In this paper we present an elegant method for sequentializing arbitrary data using the generic language extension of the functional programming language Clean. We show how the proposed operations can be used to store values of any concrete data type in several kinds of IO containers (such as files or arrays of characters), and how to manipulate stored data efficiently. Moreover, by extending stored data with encoded type information, data manipulation will be type-safe. Defining these operations generically has the advantage that specific instances for user defined data types can be generated fully automatically. Compared to traditional sequentialization methods (or to common data manipulation, using relational data bases) our operations are an order of magnitude faster.

A GENERIC MULTIPLICATION IN QUANTIZED SCHUR ALGEBRAS

We define a generic multiplication in quantized Schur algebras and thus obtain a new algebra structure in the Schur algebras. We prove that via a modified version of the map from quantum groups to quantized Schur algebras, defined in (A. A. Beilinson, G. Lusztig and R. MacPherson, A geometric setting for the quantum deformation of GL(n), Duke Math. J. 61 (1990), 655-677), a subalgebra of this new algebra is a quotient of the monoid algebra in Hall algebras studied in (M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0, Represent. Theory 5 (2001), 147-163). We also prove that the subalgebra of the new algebra gives a geometric realization of a positive part of 0-Schur algebras, defined in (S. Donkin, The q-Schur Algebra, London Mathematical Society Lecture Note Series 253. Cambridge University Press, Cambridge, 1998, x + 179. ISBN: 0-521-64558-1.). Consequently, we obtain a multiplicative basis for the positive part of 0-Schur algebras.

Ultrafilters with property $( {s})$

A set X ⊆ 2 ! has property (s) (Marczewski (Szpilrajn)) iff for every perfect set P ⊆ 2 ! there exists a perfect set Q ⊆ P such that Q ⊆ X or Q∩X = ∅. Suppose U is a nonprincipal ultrafilter on !. It is not difficult to see that ifU is preserved by Sacks forcing, i.e., it generates an ultrafilter in the generic extension after forcing with the partial order of perfect sets, then U has property (s) in the ground model. It is known that selective ultrafilters or even P-points are preserved by Sacks forcing. On the other hand (answering a question raised by Hrusak) we show that assuming CH (or more generally MA) there exists an ultrafilter U with property (s) such that U does not generate an ultrafilter in any extension which adds a new subset of !.

Revisiting the global electroweak fit of the Standard Model and beyond with Gfitter

The global fit of the Standard Model to electroweak precision data, routinely performed by the LEP electroweak working group and others, demonstrated impressively the predictive power of electroweak unification and quantum loop corrections. We have revisited this fit in view of (i) the development of the new generic fitting package, Gfitter, allowing for flexible and efficient model testing in high-energy physics, (ii) the insertion of constraints from direct Higgs searches at LEP and the Tevatron, and (iii) a more thorough statistical interpretation of the results. Gfitter is a modular fitting toolkit, which features predictive theoretical models as independent plug-ins, and a statistical analysis of the fit results using toy Monte Carlo techniques. The state-of-the-art electroweak Standard Model is fully implemented, as well as generic extensions to it. Theoretical uncertainties are explicitly included in the fit through scale parameters varying within given error ranges. This paper introduces the Gfitter project, and presents state-of-the-art results for the global electroweak fit in the Standard Model (SM), and for a model with an extended Higgs sector (2HDM). Numerical and graphical results for fits with and without including the constraints from the direct Higgs searches at LEP and Tevatron are given. Perspectives for future colliders are analysed and discussed. In the SM fit including the direct Higgs searches, we find M H =116.4 −1.3 +18.3 GeV, and the 2σ and 3σ allowed regions [114,145] GeV and [[113,168] and [180,225]] GeV, respectively. For the strong coupling strength at fourth perturbative order we obtain α S (M 2 )=0.1193 −0.0027 +0.0028 (exp )±0.0001 (theo). Finally, for the mass of the top quark, excluding the direct measurements, we find m t =178.2 −4.2 +9.8 GeV. In the 2HDM we exclude a charged-Higgs mass below 240 GeV at 95% confidence level. This limit increases towards larger tan β, e.g., $M_{H^{\pm}}<780 \mbox{GeV}$ is excluded for tan β=70.

A game on Boolean algebras describing the collapse of the continuum

The game G ls is played on a complete Boolean algebra B in ω -many moves. At the beginning White chooses a non-zero element p of B and, in the n th move, White chooses a positive p n < p and Black responds by choosing an i n ∈ { 0 , 1 } . White wins the play iff lim sup p n i n = 0 . It is shown that White has a winning strategy in this game iff forcing by B collapses the continuum to ω in some generic extension. On the other hand, if a complete Boolean algebra B carries a strictly positive Maharam submeasure or contains a countable dense subset, then Black has a winning strategy in the game G ls played on B . A Suslin algebra on which the game is undetermined is constructed and the game G ls is compared with the well-known cut-and-choose games G c&c , G fin ( λ ) and G ω ( λ ) introduced by Jech.

Stacking mice

AbstractWe show that either of the following hypotheses imply that there is an inner model with a proper class of strong cardinals and a proper class of Woodin cardinals. 1) There is a countably closed cardinal κ ≥ ℵ such that □κ and □(κ) fail. 2) There is a cardinal κ such that κ is weakly compact in the generic extension by Col(κ, κ+). Of special interest is 1) with κ = ℵ3 since it follows from PFA by theorems of Todorcevic and Velickovic. Our main new technical result, which is due to the first author, is a weak covering theorem for the model obtained by stacking mice over Kc∥κ.

An upper cardinal bound on absolute E-rings

We show that for every abelian group A of cardinality ≥ κ(ω) there exists a generic extension of the universe, where A is countable with 2 N 0 injective endomorphisms. As an immediate consequence of this result there are no absolute E-rings of cardinality > κ(ω). This paper does not require any specific prior knowledge of forcing or model theory and can be considered accessible also for graduate students.

The strength of choiceless patterns of singular and weakly compact cardinals

We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that AD , the Axiom of Determinacy, holds in the L ( R ) of a generic extension of HOD : (a) ZF + every uncountable cardinal is singular, and (b) ZF + every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.

Supersymmetric electroweak symmetry breaking

In the minimal supersymmetric standard model (the MSSM), the electroweak symmetry is restored as supersymmetry-breaking terms are turned off. We describe a generic extension of the MSSM where the electroweak symmetry is broken in the supersymmetric limit. We call this limit the ``sEWSB'' phase, short for supersymmetric electroweak symmetry breaking. We define this phase in an effective field theory that only contains the MSSM degrees of freedom. The sEWSB vacua naturally have an inverted scalar spectrum, where the heaviest $CP$-even Higgs state has standard model-like couplings to the massive vector bosons; experimental constraints in the scalar Higgs sector are more easily satisfied than in the MSSM.

A model for a very good scale and a bad scale

AbstractGiven a supercompact cardinalκand a regular cardinalλ&lt;κ, we describe a type of forcing such that in the generic extension the cofinality ofκisλ, there is a very good scale atκ, a bad scale atκ, and SCH atκfails. When creating our model we have great freedom in assigning the value of 2κ, and so we can make SCH hold or fail arbitrarily badly.

Order-isomorphic [formula omitted]-orderings in Cohen extensions

In this paper we prove that, in the Cohen extension (adding ℵ 2 -generic reals) of a model M of ZFC+CH containing a simplified ( ω 1 , 1 ) -morass, η 1 -orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η 1 -orderings can be extended to an order-isomorphism between the η 1 -orderings in the Cohen extension of M . We use the simplified ( ω 1 , 1 ) -morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding ℵ 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω 1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η 1 -orderings are order-isomorphic.

Modules with absolute endomorphism rings

Eklof and Shelah [8] call an abelian group absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. More generally, we say that an R-module is absolutely rigid if its endomorphism ring is just the ring of scalar multiplications by elements of R in every generic extension of the universe. In [8] it is proved that there do not exist absolutely rigid abelian groups of size ≥ κ(ω), where κ(ω) is the first ω-Erdős cardinal (for its definition see the introduction). A similar result holds for rigid systems of abelian groups. On the other hand, recently Gobel and Shelah [15] proved that for modules of size < κ(ω) this phenomenon disappears. Their result on Rω-modules (i.e. on R-modules with countably many distinguished submodules) that establishes the existence of ‘well-behaving’ fully rigid systems of abelian groups of large sizes < κ(ω) will be extended here to a large class of R-modules by proving the existence of modules of any sizes < κ(ω) with endomorphism rings which are absolute. In order to cover rings as general as possible, we utilize a method developed by Brenner, Butler and Corner (see [2, 3, 5]) to reduce the number of distinguished submodules required in the construction from ℵ0 to five.

Comparing libraries for generic programming in haskell

Datatype-generic programming is defining functions that depend on the structure, or "shape", of datatypes. It has been around for more than 10 years, and a lot of progress has been made, in particular in the lazy functional programming language Haskell. There are morethan 10 proposals for generic programming libraries orlanguage extensions for Haskell. To compare and characterise the many generic programming libraries in atyped functional language, we introduce a set of criteria and develop a generic programming benchmark: a set of characteristic examples testing various facets of datatype-generic programming. We have implemented the benchmark for nine existing Haskell generic programming libraries and present the evaluation of the libraries. The comparison is useful for reaching a common standard for generic programming, but also for a programmer who has to choose a particular approach for datatype-generic programming.