Log in

Relevant bibliographies by topics / Inductive sets

Contents

Journal articles

Academic literature on the topic 'Inductive sets'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles

Select a source type:

Consult the lists of relevant articles, books, theses, conference reports, and other scholarly sources on the topic 'Inductive sets.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Journal articles on the topic "Inductive sets"

1

Lubarsky, Robert S. "μ-definable sets of integers". Journal of Symbolic Logic 58, № 1 (1993): 291–313. http://dx.doi.org/10.2307/2275338.

Full text

Abstract:

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

APA, Harvard, Vancouver, ISO, and other styles

2

Bosch, Carlos, and Jan Kučera. "Closed bounded sets in inductive limits of $\Cal K$-spaces." Czechoslovak Mathematical Journal 43, no. 2 (1993): 221–23. http://dx.doi.org/10.21136/cmj.1993.128401.

Full text

APA, Harvard, Vancouver, ISO, and other styles

3

Bell, John L. "Finite sets and frege structures." Journal of Symbolic Logic 64, no. 4 (1999): 1552–56. http://dx.doi.org/10.2307/2586795.

Full text

Abstract:

Call a family of subsets of a set E inductive if and is closed under unions with disjoint singletons, that is, ifA Frege structure is a pair (E, ν) with ν a map to E whose domain dom(ν) is an inductive family of subsets of E such thatIn [2] it is shown in a constructive setting that each Frege structure determines a subset which is the domain of a model of Peano's axioms. In this note we establish, within the same constructive setting, three facts. First, we show that the least inductive family of subsets of a set E is precisely the family of decidable Kuratowski finite subsets of E. Secondly, we establish that the procedure presented in [2] can be reversed, that is, any set containing the domain of a model of Peano's axioms determines a map which turns the set into a minimal Frege structure: here by a minimal Frege structure is meant one in which dom(ν) is the least inductive family of subsets of E. And finally, we show that the procedures leading from minimal Frege structures to models of Peano's axioms and vice-versa are mutually inverse. It follows that the postulation of a (minimal) Frege structure is constructively equivalent to the postulation of a model of Peano's axioms.All arguments will be formulated within constructive (intuitionistic) set theory.

APA, Harvard, Vancouver, ISO, and other styles

4

CURI, GIOVANNI. "ABSTRACT INDUCTIVE AND CO-INDUCTIVE DEFINITIONS." Journal of Symbolic Logic 83, no. 2 (2018): 598–616. http://dx.doi.org/10.1017/jsl.2018.13.

Full text

Abstract:

AbstractIn [G. Curi, On Tarski’s fixed point theorem. Proc. Amer. Math. Soc., 143 (2015), pp. 4439–4455], a notion of abstract inductive definition is formulated to extend Aczel’s theory of inductive definitions to the setting of complete lattices. In this article, after discussing a further extension of the theory to structures of much larger size than complete lattices, as the class of all sets or the class of ordinals, a similar generalization is carried out for the theory of co-inductive definitions on a set. As a corollary, a constructive version of the general form of Tarski’s fixed point theorem is derived.

APA, Harvard, Vancouver, ISO, and other styles

5

ARCISZEWSKI, TOMASZ, and WOJCIECH ZIARKO. "Inductive Learning in Civil Engineering: Rough Sets Approach." Computer-Aided Civil and Infrastructure Engineering 5, no. 1 (2008): 19–28. http://dx.doi.org/10.1111/j.1467-8667.1990.tb00038.x.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Dybjer, Peter. "A general formulation of simultaneous inductive-recursive definitions in type theory." Journal of Symbolic Logic 65, no. 2 (2000): 525–49. http://dx.doi.org/10.2307/2586554.

Full text

Abstract:

AbstractThe first example of a simultaneous inductive-recursive definition in intuitionistic type theory is Martin-Löfs universe à la Tarski. A set U0of codes for small sets is generated inductively at the same time as a function T0, which maps a code to the corresponding small set, is defined by recursion on the way the elements of U0are generated.In this paper we argue that there is an underlyinggeneralnotion of simultaneous inductive-recursive definition which is implicit in Martin-Löf's intuitionistic type theory. We extend previously given schematic formulations of inductive definitions in type theory to encompass a general notion of simultaneous induction-recursion. This enables us to give a unified treatment of several interesting constructions including various universe constructions by Palmgren, Griffor, Rathjen, and Setzer and a constructive version of Aczel's Frege structures. Consistency of a restricted version of the extension is shown by constructing a realisability model in the style of Allen.

APA, Harvard, Vancouver, ISO, and other styles

7

Zanger, Daniel Z. "Talagrand’s inductive method and isoperimetric inequalities involving random sets." Statistics & Probability Letters 78, no. 7 (2008): 861–68. http://dx.doi.org/10.1016/j.spl.2007.09.012.

Full text

APA, Harvard, Vancouver, ISO, and other styles

8

Korobeĭnik, Yu F. "INDUCTIVE AND PROJECTIVE TOPOLOGIES. SUFFICIENT SETS AND REPRESENTING SYSTEMS." Mathematics of the USSR-Izvestiya 28, no. 3 (1987): 529–54. http://dx.doi.org/10.1070/im1987v028n03abeh000896.

Full text

APA, Harvard, Vancouver, ISO, and other styles

9

Alexandru, Andrei, and Gabriel Ciobanu. "Fuzzy Results for Finitely Supported Structures." Mathematics 9, no. 14 (2021): 1651. http://dx.doi.org/10.3390/math9141651.

Full text

Abstract:

We present a survey of some results published recently by the authors regarding the fuzzy aspects of finitely supported structures. Considering the notion of finite support, we introduce a new degree of membership association between a crisp set and a finitely supported function modelling a degree of membership for each element in the crisp set. We define and study the notions of invariant set, invariant complete lattices, invariant monoids and invariant strong inductive sets. The finitely supported (fuzzy) subgroups of an invariant group, as well as the L-fuzzy sets on an invariant set (with L being an invariant complete lattice) form invariant complete lattices. We present some fixed point results (particularly some extensions of the classical Tarski theorem, Bourbaki–Witt theorem or Tarski–Kantorovitch theorem) for finitely supported self-functions defined on invariant complete lattices and on invariant strong inductive sets; these results also provide new finiteness properties of infinite fuzzy sets. We show that apparently, large sets do not contain uniformly supported, infinite subsets, and so they are invariant strong inductive sets satisfying finiteness and fixed-point properties.

APA, Harvard, Vancouver, ISO, and other styles

10

LUMSDAINE, PETER LEFANU, and MICHAEL SHULMAN. "Semantics of higher inductive types." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 1 (2019): 159–208. http://dx.doi.org/10.1017/s030500411900015x.

Full text

Abstract:

AbstractHigher inductive typesare a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings.We introduce the notion ofcell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category hasweakly stable typal initial algebrasfor any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations.Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

APA, Harvard, Vancouver, ISO, and other styles

More sources

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!