Hierarchies of Boolean algebras

1970 ◽  
Vol 35 (3) ◽  
pp. 365-374 ◽  
Author(s):  
Lawrence Feiner
Keyword(s):  
Number Theory ◽  
Standard Model ◽  
Boolean Algebra ◽  
Boolean Algebras ◽  
Natural Numbers ◽  
Recursive Structure ◽  
The Standard Model ◽  
Free Generators ◽  
Basic Facts

A denumerable structure is said to be recursive iff its universe is a recursive subset of the natural numbers and its relations and operations are recursive. For example, the standard model of number theory is recursive. A structure is said to be recursively presentable iff it is isomorphic to a recursive structure. For example, a Boolean algebra generated by ℵ0 free generators is easily seen to be recursively presentable. (For basic facts concerning Boolean algebras, the reader is referred to R. Sikorski [9] and A. Tarski and A. Mostowski [10].)

scholarly journals CHOICE-FREE STONE DUALITY

2019 ◽  
Vol 85 (1) ◽  
pp. 109-148
Author(s):  
NICK BEZHANISHVILI ◽  
WESLEY H. HOLLIDAY
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Stone Space ◽  
Spectral Space ◽  
Topological Representation ◽  
Stone Duality ◽  
Closed Sets ◽  
Spectral Spaces ◽  
Basic Facts ◽  
Regular Open

AbstractThe standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. This connection makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales. Unlike Stone’s representation of Boolean algebras via Stone spaces, our choice-free topological representation of Boolean algebras does not show that every Boolean algebra can be represented as a field of sets; but like Stone’s representation, it provides the benefit of a topological perspective on Boolean algebras, only now without choice. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.

Theories without countable models

1972 ◽  
Vol 37 (3) ◽  
pp. 562-568
Author(s):  
Andreas Blass
Keyword(s):  
Standard Model ◽  
Countable Model ◽  
Compactness Theorem ◽  
Complete Theory ◽  
Natural Numbers ◽  
Axiom Schema ◽  
First Order ◽  
Order Language ◽  
The Standard Model ◽  
Skolem Theorem

Consider the Löwenheim-Skolem theorem in the form: If a theory in a countable first-order language has a model, then it has a countable model. As is well known, this theorem becomes false if one omits the hypothesis that the language be countable, for one then has the following trivial counterexample.Example 1. Let the language have uncountably many constants, and let the theory say that they are unequal.To motivate some of our future definitions and to introduce some notation, we present another, less trivial, counterexample.Example 2. Let L0 be the language whose n-place predicate (resp. function) symbols are all the n-place predicates (resp. functions) on the set ω of natural numbers. Let be the standard model for L0; we use the usual notation Th() for its complete theory. Add to L0 a new constant e, and add to Th() an axiom schema saying that e is infinite. By the compactness theorem, the resulting theory T has models. However, none of its models are countable. Although this fact is well known, we sketch a proof in order to refer to it later.By [5, p. 81], there is a family {Aα ∣ < α < c} of infinite subsets of ω, the intersection of any two of which is finite.

Recursive Boolean algebras with recursive atoms

1981 ◽  
Vol 46 (3) ◽  
pp. 595-616 ◽  
Author(s):  
Jeffrey B. Remmel
Keyword(s):  
Boolean Algebra ◽  
Boolean Algebras ◽  
Isomorphism Type ◽  
Natural Numbers ◽  
Recursive Isomorphism ◽  
The Ideal

A Boolean algebra (henceforth abbreviated B.A.) is said to be recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Let denote the set of atoms of and denote the ideal generated by the atoms of . Given recursive B.A.s and , we write ≈ if is isomorphic to and ≈r if is recursively isomorphic to , i.e., if there is a partial recursive isomorphism from onto .Recursive B.A.s have been studied by several authors including Ershov [2], Fiener [3], [4], Goncharov [5], [6], [7], LaRoche [8], Nurtazin [7], and the author [10], [11]. This paper continues a study of the recursion theoretic relationships among , , and the recursive isomorphism type of a recursive B.A. we started in [11]. We refer the reader to [11] for any unexplained notation and definitions. In [11], we were mainly concerned with the possible recursion theoretic properties of the set of atoms in recursive B.A.s. We found that even if we insist that be recursive, there is considerable freedom for the properties of . For example, we showed that if is a recursive B.A. such that is recursive and is infinite, then (i) there exists a recursive B.A. such that and both and are recursive and (ii) for any nonzero r.e. degree δ, there exist recursive B.A.s , , … such that for each i, is of degree δ, is recursive, is immune if i is even and is not immune if i is odd, and no two B.A.s in the sequence are recursively isomorphic.

scholarly journals A Lattice-Based Identity-Based Proxy Blind Signature Scheme in the Standard Model

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Lili Zhang ◽  
Yanqin Ma
Keyword(s):  
Number Theory ◽  
Standard Model ◽  
Blind Signature ◽  
Signature Scheme ◽  
Signature Schemes ◽  
Proxy Blind Signature ◽  
The Standard Model ◽  
Identity Based ◽  
Lattice Based Cryptography ◽  
Solution Problem

A proxy blind signature scheme is a special form of blind signature which allowed a designated person called proxy signer to sign on behalf of original signers without knowing the content of the message. It combines the advantages of proxy signature and blind signature. Up to date, most proxy blind signature schemes rely on hard number theory problems, discrete logarithm, and bilinear pairings. Unfortunately, the above underlying number theory problems will be solvable in the postquantum era. Lattice-based cryptography is enjoying great interest these days, due to implementation simplicity and provable security reductions. Moreover, lattice-based cryptography is believed to be hard even for quantum computers. In this paper, we present a new identity-based proxy blind signature scheme from lattices without random oracles. The new scheme is proven to be strongly unforgeable under the standard hardness assumption of the short integer solution problem (SIS) and the inhomogeneous small integer solution problem (ISIS). Furthermore, the secret key size and the signature length of our scheme are invariant and much shorter than those of the previous lattice-based proxy blind signature schemes. To the best of our knowledge, our construction is the first short lattice-based identity-based proxy blind signature scheme in the standard model.

Nonisomorphism of lattices of recursively enumerable sets

1993 ◽  
Vol 58 (4) ◽  
pp. 1177-1188 ◽  
Author(s):  
John Todd Hammond
Keyword(s):  
Boolean Algebra ◽  
Isomorphism Class ◽  
Boolean Algebras ◽  
Turing Degree ◽  
Symmetric Difference ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Definition Of

Let ω be the set of natural numbers, let be the lattice of recursively enumerable subsets of ω, and let A be the lattice of subsets of ω which are recursively enumerable in A. If U, V ⊆ ω, put U =* V if the symmetric difference of U and V is finite.A natural and interesting question is then to discover what the relation is between the Turing degree of A and the isomorphism class of A. The first result of this form was by Lachlan, who proved [6] that there is a set A ⊆ ω such that A ≇ . He did this by finding a set A ⊆ ω and a set C ϵ A such that the structure ({W ϵ A∣W ⊇ C},∪,∩)/=* is a Boolean algebra and is not isomorphic to the structure ({W ϵ ∣W ⊇ D},∪,∩)/=* for any D ϵ . There is a nonrecursive ordinal which is recursive in the set A which he constructs, so his set A is not (see, for example, Shoenfield [11] for a definition of what it means for a set A ⊆ ω to be ). Feiner then improved this result substantially by proving [1] that for any B ⊆ ω, B′ ≇ B, where B′ is the Turing jump of B. To do this, he showed that for each X ⊆= ω there is a Boolean algebra which is but not and then applied a theorem of Lachlan [6] (definitions of and Boolean algebras will be given in §2). Feiner's result is of particular interest for the case B = ⊘, for it shows that the set A of Lachlan can actually be chosen to be arithmetical (in fact, ⊘′), answering a question that Lachlan posed in his paper. Little else has been known.

The consistency of number theory via Herbrand's theorem

1973 ◽  
Vol 38 (1) ◽  
pp. 29-58 ◽  
Author(s):  
T. M. Scanlon
Keyword(s):  
Number Theory ◽  
Standard Model ◽  
Fundamental Theorem ◽  
Elementary Number Theory ◽  
Consistency Proof ◽  
Crucial Point ◽  
Quantification Theory ◽  
Elementary Number ◽  
Herbrand’S Theorem ◽  
The Standard Model

That elementary number theory is consistent and can be given a metamathematical consistency proof has been well known since Gentzen's 1936 paper, and a number of different proofs of this result have since been offered. What is presented here is essentially a simplified and generalized version of the proof given by Ackermann in 1940 [1]; but the proof given here applies to systems formalized in standard quantification theory rather than in Hilbert's ε-calculus, and is based upon the analysis of quantificational reasoning given by Herbrand's Fundamental Theorem. Dreben and Denton sketch such a proof in [2], but at a crucial point they follow Ackermann in tying the strategy of their proof too closely to the standard model. This makes the proof more complex than it need be and restricts its application to systems with induction on the standard well-ordering. The present proof is both simpler and more general in that it applies to systems ZR of number theory with induction on arbitrary recursive well-orderings R. This generalization was first obtained by Tait in [11] using functionals of lowest type. Some technical devices employed below are similar to ones used by Tait, but while the proof-theoretic methods employed here are naturally characterizable in terms of functionals of lowest type the present proof avoids the introduction of such functionals into the languages studied.

R-maximal Boolean algebras

1979 ◽  
Vol 44 (4) ◽  
pp. 533-548 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Boolean Algebra ◽  
Vector Space ◽  
Boolean Algebras ◽  
Natural Numbers ◽  
Recursively Enumerable ◽  
Similarities And Differences

Metakides and Nerode in [2] suggested the study of what they termed the lattice of recursively enumerable substructures of a recursively presented model. For example, Metakides and Nerode in [3] introduced the lattice of of recusively enumerable subspaces, , of a recursively presented vector space V∞. The similarities and differences between and ℰ, the lattice of recursively enumerable subsets of the natural numbers N as defined in [9], have been studied by Metakides and Nerode, Kalantari, Remmel, Retzlaff, and Shore. In [6], we studied some similarities and differences between ℰ and the lattice of recursively enumerable sub-algebras of a weakly recursively presented Boolean algebra and this paper continues that study. A weakly recursively presented Boolean algebra (W.R.P.B.A.), , consists of a recursive subset of N, ∣∣, called the field of , and operations (meet), (join), and (complement) which are partial recursive and under which becomes a Boolean algebra. We shall write and for the zero and unit of . If S is a subset of , we let (S)* denote the subalgebra generated by S. Given sub-algebras B and C of , we let B + C denote (B ⋃ C)*. A subalgebra B of is recursively enumerable (recursive) if {x ∈ ∣∣ x ∈ B} is a recursively enumerable (recursive) subset of ∣∣. The set of all recursively enumerable subalgebras of , , forms a lattice under the operations of intersection and sum (+).

Indirect measurements in micro-space with the EM

1995 ◽  
Vol 53 ◽  
pp. 598-599
Author(s):  
Sterling P. Newberry
Keyword(s):  
Electron Microscope ◽  
Standard Model ◽  
Particle Physics ◽  
Information Storage ◽  
Storage Space ◽  
Indirect Measurements ◽  
Storage And Retrieval ◽  
Adequate Information ◽  
Compelling Reason ◽  
The Standard Model

At the 1958 meeting of our society, then known as EMSA, the author introduced the concept of microspace and suggested its use to provide adequate information storage space and the use of electron microscope techniques to provide storage and retrieval access. At this current meeting of MSA, he wishes to suggest an additional use of the power of the electron microscope.The author has been contemplating this new use for some time and would have suggested it in the EMSA fiftieth year commemorative volume, but for page limitations. There is compelling reason to put forth this suggestion today because problems have arisen in the “Standard Model” of particle physics and funds are being greatly reduced just as we need higher energy machines to resolve these problems. Therefore, any techniques which complement or augment what we can accomplish during this austerity period with the machines at hand is worth exploring.

An Interview with the Physicist that Her Head in the Universe and Both Feet on the Earth

2019 ◽  
Author(s):  
Adib Rifqi Setiawan
Keyword(s):  
Standard Model ◽  
Particle Physics ◽  
Space Time ◽  
Additional Dimension ◽  
The Earth ◽  
The Standard Model ◽  
The Universe

Put simply, Lisa Randall’s job is to figure out how the universe works, and what it’s made of. Her contributions to theoretical particle physics include two models of space-time that bear her name. The first Randall–Sundrum model addressed a problem with the Standard Model of the universe, and the second concerned the possibility of a warped additional dimension of space. In this work, we caught up with Randall to talk about why she chose a career in physics, where she finds inspiration, and what advice she’d offer budding physicists. This article has been edited for clarity. My favourite quote in this interview is, “Figure out what you enjoy, what your talents are, and what you’re most curious to learn about.” If you insterest in her work, you can contact her on Twitter @lirarandall.

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