E-Resolution: Extension of Resolution to Include the Equality Relation

1983 ◽  
pp. 273-280
Author(s):  
J. B. Morris
Keyword(s):  
Equality Relation

Commutative regular rings and Boolean-valued fields

1984 ◽  
Vol 49 (1) ◽  
pp. 281-297 ◽  
Author(s):  
Kay Smith
Keyword(s):  
Set Theory ◽  
Regular Ring ◽  
Galois Theory ◽  
Algebraic Closure ◽  
Original Proof ◽  
Valued Fields ◽  
Boolean Space ◽  
Equality Relation ◽  
The One ◽  
Regular Rings

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring (the original proof is due to Carson [2]).Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. (Lemma 4.2 shows that their equality relation is the same as the one used in this paper.) To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space (see, for example, [9] and [8]). These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory.

scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

General models and extensionality

1972 ◽  
Vol 37 (2) ◽  
pp. 395-397 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Axiom Schema ◽  
Nonstandard Model ◽  
Equality Relation ◽  
Definition Of

It is well known that equality is definable in type theory. Thus, in the language of [2], the equality relation between elements of type α is definable as , i.e., xα = yα iff every set which contains xα also contains yα. However, in a nonstandard model of type theory, the sets may be so sparse that the wff above does not denote the true equality relation. We shall use this observation to construct a general model in the sense of [2] in which the Axiom of Extensionality is not valid. Thus Theorem 2 of [2] is technically incorrect. However, it is easy to remedy the situation by slightly modifying the definition of general model.Our construction will show that the Axiom Schema of Extensionality is independent even if one takes as an axiom schema.We shall assume familiarity with, and use the notation of, [2] and §§2–3 of [1].

R. e. presented linear orders

1983 ◽  
Vol 48 (2) ◽  
pp. 369-376 ◽  
Author(s):  
Dev Kumar Roy
Keyword(s):  
Linear Order ◽  
Order Relation ◽  
Recursion Theory ◽  
Linear Orders ◽  
Preliminary Results ◽  
Negative Results ◽  
Recursively Enumerable ◽  
Equality Relation ◽  
The Law

This paper looks at linear orders in the following way. A preordering is given, which is linear and recursively enumerable. By performing the natural identification, one obtains a linear order for which equality is not necessarily recursive. A format similar to Metakides and Nerode's [3] is used to study these linear orders. In effective studies of linear orders thus far, the law of antisymmetry (x ≦ y ∧ y ≦ x ⇒ y) has been assumed, so that if the order relation x ≦ y is r.e. then x < y is also r.e. Here the assumption is dropped, so that x < y may not be r.e. and the equality relation may not be recursive; the possibility that equality is not recursive leads to new twists which sometimes lead to negative results.Reported here are some interesting preliminary results with simple proofs, which are obtained if one looks at these objects with a view to doing recursion theory in the style of Metakides and Nerode. (This style, set in [3], is seen in many subsequent papers by Metakides and Nerode, Kalantari, Remmel, Retzlaff, Shore, and others, e.g. [1], [4], [6], [7], [8], [11]. In a sequel, further investigations will be reported which look at r.e. presented linear orders in this fashion and in the context of Rosenstein's comprehensive work [10].Obviously, only countable linear orders are under consideration here. For recursion-theoretic notation and terminology see Rogers [9].

scholarly journals Resource convertibility and ordered commutative monoids

2015 ◽  
Vol 27 (6) ◽  
pp. 850-938 ◽  
Author(s):  
TOBIAS FRITZ
Keyword(s):  
Information Theory ◽  
Communication Channel ◽  
Linear Logic ◽  
Commutative Monoids ◽  
Science And Engineering ◽  
Von Neumann ◽  
Torsion Freeness ◽  
Equality Relation ◽  
Image Position ◽  
Reaction Equations

Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general mathematical toolbox for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as\mathrm{2H_2 + O_2} \lra \mathrm{2H_2O,}are concerned both with a convertibility relation ‘→’ and a combination operation ‘+.’ We study ordered commutative monoids from an algebraic and functional-analytic perspective and derive a wealth of results which should have applications to concrete resource theories, such as a formula for rates of conversion. As a running example showing that ordered commutative monoids are also of purely mathematical interest without the resource-theoretic interpretation, we exemplify our results with the ordered commutative monoid of graphs.While closely related to both Girard's linear logic and to Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on the foundations of thermodynamics.Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

A coherence theorem for Martin-Löf's type theory

1998 ◽  
Vol 8 (4) ◽  
pp. 413-436 ◽  
Author(s):  
MICHAEL HEDBERG
Keyword(s):  
Large Class ◽  
Equivalence Relation ◽  
Type Theory ◽  
Category Structure ◽  
Arbitrary Type ◽  
Unique Identity ◽  
Equality Relation ◽  
Coherence Theorem ◽  
Initial Category

In type theory a proposition is represented by a type, the type of its proofs. As a consequence, the equality relation on a certain type is represented by a binary family of types. Equality on a type may be conventional or inductive. Conventional equality means that one particular equivalence relation is singled out as the equality, while inductive equality – which we also call identity – is inductively defined as the ‘smallest reflexive relation’. It is sometimes convenient to know that the type representing a proposition is collapsed, in the sense that all its inhabitants are identical. Although uniqueness of identity proofs for an arbitrary type is not derivable inside type theory, there is a large class of types for which it may be proved. Our main result is a proof that any type with decidable identity has unique identity proofs. This result is convenient for proving that the class of types with decidable identities is closed under indexed sum. Our proof of the main result is completely formalized within a kernel fragment of Martin-Löf's type theory and mechanized using ALF. Proofs of auxiliary lemmas are explained in terms of the category theoretical properties of identity. These suggest two coherence theorems as the result of rephrasing the main result in a context of conventional equality, where the inductive equality has been replaced by, in the former, an initial category structure and, in the latter, a smallest reflexive relation.

Equivalence of codes for countable sets of reals

2020 ◽  
pp. 1-11
Author(s):  
William Chan
Keyword(s):  
Equivalence Relation ◽  
Borel Set ◽  
Borel Sets ◽  
Generic Absoluteness ◽  
Equality Relation ◽  
Continuous Images

Abstract A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

scholarly journals Preface to the special issue: isomorphisms of types and invertibility of lambda terms

2008 ◽  
Vol 18 (4) ◽  
pp. 645-646
Author(s):  
RALPH MATTHES ◽  
SERGEI SOLOVIEV
Keyword(s):  
Left Inverse ◽  
Special Issue ◽  
Logical Equivalence ◽  
Natural Generalisation ◽  
Equality Relation ◽  
Computable Functions

Isomorphisms of types are computational witnesses of logical equivalence with additional properties. The types/formulas A and B are isomorphic if there are functions (in a certain formalism) f : A → B and g : B → A such that g ○ f and f ○ g are equal in a certain sense to the identity on A and B, respectively. Typical such formalisms are extensions of simply typed λ-calculus, with βη-convertibility as equality relation. Another view of a pair of functions f : A → B and g : B → A (besides establishing the logical equivalence of A and B) is that f is invertible with left-inverse g, and it is then natural to relax the above symmetric condition to just g ○ f being equal to the identity on A. In this situation, A is called a retract of B, which is thus a natural generalisation of the notion of an isomorphism, while both these notions are refinements of the concept of logical equivalence in operational terms, that is, in terms of computable functions.

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