In this paper we define the hyper operations ⊗, ∨ and ∧ on a hyper MV -algebra and we obtain some related results. After that by considering the notions ofhyper MV -ideals and weak hyper MV -ideals, we prove some theorems. Then we determine relationships between hyper MV -ideals in a hyper MV -algebra and hyper K -ideals in a hyper K -algebra . Finally we give a characterization of hyper MV -algebras (...) of order 3 or 4 based on the hyper MV -ideals. (shrink)

Various representation results have been established for logics of belief revision, in terms of remainder sets, epistemic entrenchment, systems of spheres and so on. In this paper I present another representation for logics of belief revision, as an algebra of theories. I show that an algebra of theories, enriched with a set of rejection operations, provides a suitable algebraic framework to characterize the theory change operations of systems of belief revision. The theory change operations arise as power operations of the (...) conjunction and disjunction connectives of the underlying logic. (shrink)

It is shown that a Clifford algebra formalism provides a unifying description of spin-0, -1/2, and-1 fields. Since the operators and operands are both expressed in terms of the same Clifford algebra, the formalism obtains some results which are considerably different from those of the standard formalisms for these fields. In particular, the conservation laws are obtained uniquely and unambiguously from the equations of motion in this formalism and do not suffer from the ambiguities and inconsistencies of the standard methods.

We study the structure of algebraic -closed subsets of an algebraic lattice L, where is some Browerian binary relation on L, in the special case when the lattice of such subsets is an atomistic lattice. This gives an approach to investigate the atomistic lattices of congruence-Noetherian quasivarieties.

We study the structure of algebraic τ-closed subsets of an algebraic lattice L, where τ is some Browerian binary relation on L, in the special case when the lattice of such subsets is an atomistic lattice. This gives an approach to investigate the atomistic lattices of congruence-Noetherian quasivarieties.

Application of recently developed non-Archimedean algebra to a flat and finite universe of total mass M 0 and radius R 0 is described. In this universe, mass m of a body and distance R between two points are bounded from above, i.e., 0≤m≤M 0, 0≤R≤R 0. The universe is characterized by an event horizon at R 0 (there is nothing beyond it, not even space). The radial distance metric is compressed toward horizon, which is shown to cause the phenomenon of (...) red shift. The corresponding modified Minkowski's metric and Lorentz transforms are obtained. Applications to Newtonian gravity shows a weakening at large scales (R→R 0) and a regular behavior as R→0. (shrink)

The notion of a Sasaki projectionon an orthomodular lattice is generalized to a mapping Φ: E × E → E, where E is an effect algebra. If E is lattice ordered and Φ is symmetric, then E is called a Φ-symmetric effect algebra.This paper launches a study of such effect algebras. In particular, it is shown that every interval effect algebra with a lattice-ordered ambient group is Φ-symmetric, and its group is the one constructed by Ravindran in his proof that (...) every effect algebra that has the Riesz decomposition property is an interval algebra. It is shown that the doubling construction introduced in the paper is connected to the conditional event algebrasof Goodman, Nguyen, and Walker. (shrink)

Elementary particles are considered as local oscillators under the influence of zeropoint fields. Such oscillatory behavior of the particles leads to the deviations in their path of motion. The oscillations of the particle in general may be considered as complex rotations in complex vector space. The local particle harmonic oscillator is analyzed in the complex vector formalism considering the algebra of complex vectors. The particle spin is viewed as zeropoint angular momentum represented by a bivector. It has been shown that (...) the particle spin plays an important role in the kinematical intrinsic or local motion of the particle. From the complex vector formalism of harmonic oscillator, for the first time, a relation between mass $m$ and bivector spin $S$ has been derived in the form $\varvec{\sigma }_3 mc^2{\mathcal {J}}_{\pm } =\lambda \Omega _{\mathbf{s}} \cdot \mathrm{{S}} {\mathcal {J}}_{\pm }$ . Where, $\Omega _{s}$ is the angular velocity bivector of complex rotations, $c$ is the velocity of light. The unit vector $\varvec{\sigma }_3$ acts as an operator on the idempotents ${\mathcal {J}}_{+}$ and ${\mathcal {J}}_{-}$ to give the eigen values $\lambda =\pm 1.$ The constant $\lambda $ represents two fold nature of the equation corresponding to particle and antiparticle states. Further the above relation shows that the mass of the particle may be interpreted as a local spatial complex rotation in the rest frame. This gives an insight into the nature of fundamental particles. When a particle is observed from an arbitrary frame of reference, it has been shown that the spatial complex rotation dictates the relativistic particle motion. The mathematical structure of complex vectors in space and spacetime is developed. (shrink)

The purpose of this paper is to problematize the place of religious authority in politics and education. Specifically, this essay highlights the role of religious authority in establishing a moral order that values compliance and conformity at the expense of liberty and critique. In doing so, the essay reflects on Israeli politics and Islamic political thought. Pondering on both, the essay explains how the authority conferred through the use of religious language creates a condition of hyper-solidarity. Under conditions of (...) hyper-solidarity, religious authority becomes pervasive, permeating, and more entrenched and difficult to challenge. It becomes increasingly difficult to be critical, because to criticize is to forgo that hyper-solidarity, and to assume a position of exteriority. Therefore, obedience becomes a choice and a position taken in favor of stability, consistency, and continuity of the group. (shrink)

In this paper a uniform methodology to perform natural\ndeduction over the family of linear, relevance and intuitionistic\nlogics is proposed. The methodology follows the Labelled\nDeductive Systems (LDS) discipline, where the deductive process\nmanipulates {\em declarative units} -- formulas {\em labelled}\naccording to a {\em labelling algebra}. In the system described\nhere, labels are either ground terms or variables of a given {\em\nlabelling language} and inference rules manipulate formulas and\nlabels simultaneously, generating (whenever necessary)\nconstraints on the labels used in the rules. A set of natural\ndeduction style (...) inference rules is given, and the notion of a\n{\em derivation} is defined which associates a labelled natural\ndeduction style ``structural derivation'' with a set of generated\nconstraints. Algorithmic procedures, based on a technique called\n{\em resource abduction\/}, are defined to solve the constraints\ngenerated within a structural derivation, and their termination\nconditions discussed. A natural deduction derivation is then\ndefined to be {\em correct} with respect to a given substructural\nlogic, if, under the condition that the algorithmic procedures\nterminate, the associated set of constraints is satisfied with\nrespect to the underlying labelling algebra. Finally, soundness\nand completeness of the natural deduction system are proved with\nrespect to the LKE tableaux system \cite{daga:LAD}.\footnote{Full\nversion of a paper presented at the {\em 3rd Workshop on Logic,\nLanguage, Information and Computation\/} ({\em WoLLIC'96\/}), May\n8--10, Salvador (Bahia), Brazil, organised by Univ.\ Federal de\nPernambuco (UFPE) and Univ.\ Federal da Bahia (UFBA), and\nsponsored by IGPL, FoLLI, and ASL.}. (shrink)

Prominent experts present papers which discuss problems regarding this subject. Coverage includes case studies in the translation of logics for second-order and propositional dynamic logic; many-sorted algebras and equational logic; logical foundations of artificial intelligence along with a variety of methods that exist to encode information; program verification techniques such as Floyd-Hoare, intermittent assertion and temporal logic of programs.

Quantum mechanics is troubled by the problem of nonlocality inherent in the theory. In a series of papers we explore the possibility of an algebraic formulation of quantum mechanics based on local observables which would incorporate nonlocality when small distances are involved but would be separable at large distances. This paper reviews some of the basic ideas and theories developed recently. These include a unified localization scheme, the introduction of local comoving evolution, local comoving observables, and related conservation laws. Technical (...) considerations and mathematical jargon are kept to a minimum to avoid obscuring physical reasoning. (shrink)

This article introduces the notion of duplex elements of the finite rings and corresponding neutrosophic rings. The authors establish duplex ring Dup(R) and neutrosophic duplex ring Dup(R)I)) by way of various illustrations. The tables of different duplicities are constructed to reveal the comparison between rings Dup(Zn), Dup(Dup(Zn)) and Dup(Dup(Dup(Zn ))) for the cyclic ring Zn . The proposed duplicity structures have several algebraic systems with dissimilar consequences. Author’s characterize finite rings with R + R is different from the duplex ring (...) Dup(R). However, this characterization supports that R + R = Dup(R) for some well known rings, namely zero rings and finite fields. (shrink)

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics.

We further develop a recent new proof (by Greenberger, Horne, and Zeilinger—GHZ) that local deterministic hidden-variable theories are inconsistent with certain strict correlations predicted by quantum mechanics. First, we generalize GHZ's proof so that it applies to factorable stochastic theories, theories in which apparatus hidden variables are causally relevant to measurement results, and theories in which the hidden variables evolve indeterministically prior to the particle-apparatus interactions. Then we adopt a more general measure-theoretic approach which requires that GHZ's argument be modified (...) in order to produce a valid proof. Finally, we motivate our more general proof's assumptions in a somewhat different way from previous authors in order to strengthen the implications of our proof as much as possible. After developing GHZ's proof along these lines, we then consider the analogue, for our proof, of Bohr's reply to the EPR argument, and conclude (pace GHZ) that in at least one respect (viz. that of most concern to Bohr) the proof is no more powerful than Bell's. Nevertheless, we point out some new advantages of our proof over Bell's, and over other algebraic proofs of nonlocality. And we conclude by giving a modified version of our proof that, like Bell's, does not rely on experimentally unrealizable strict correlations, but still leads to a testable “quasi-algebraic” locality inequality.“... to admit things not visible to the gross creatures that we are is, in my opinion, to show a decent humility, and not just a lamentable addiction to metaphysics.”J. S. Bell. (shrink)

The following logics are the most noteworthy from the perspective of the calculus of combinators: the Hilbert’s positive implicational logic , the Church’s weak theory of implication , the BCK-logic, and the BCI-logic. Their significance is due to a certain correspondence between combinators and implicational formulas . The first three logics mentioned have been immensely investigated but it was not so in case of the remaining one. The BCI-logics was mentioned by A. N. Prior in the second edition of his (...) Formal Logic of 1962 where it was credited to C. A. Meredith and dated in 1956 . A. (shrink)

In this paper we describe the well-founded initial segment of the free Heyting algebra ������α on finitely many, α, generators. We give a complete classification of initial sublattices of ������₂ isomorphic to ������₁ (called 'low ladders'), and prove that for 2 < α < ω, the height of the well-founded initial segment of ������α.

Algebraic Art explores the invention of a peculiarly Victorian account of the nature and value of aesthetic form, and it traces that account to a surprising source: mathematics. Drawing on literature, art, and photography, it explores how the Victorian mathematical conception of form still resonates today.

Left distributive algebras arise in the study of classical structures such as groups, knots, and braids, as well as more exotic objects like large cardinals. A long-standing open question is whether the set of left divisors of every term in the free left distributive algebra on any number of generators is well-ordered. A conjecture of J. Moody describes a halting condition for descending sequences of left divisors in the free left distributive algebra on an arbitrary number of generators. In this (...) paper we present progress toward an affirmative answer to the open question mentioned above, namely we prove that the many generator form of Moody’s conjecture holds if each member of the free left distributive algebra has a “division form” representation in an expanded algebra. (shrink)

It is shown that all the countable superatomic boolean algebras of finite rank have the small index property.

Split preorders are preordering relations on a domain whose composition is defined in a particular way by splitting the domain into two disjoint subsets. These relations and the associated composition arise in categorial proof theory in connection with coherence theorems. Here split preorders are represented isomorphically in the category whose arrows are binary relations and whose composition is defined in the usual way. This representation is related to a classical result of representation theory due to Richard Brauer.

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

In the late 1960s and early 1970s, Dana Scott introduced a kind of generalization (or perhaps simplification would be a better description) of the notion of inference, familiar from Gentzen, in which one may consider multiple conclusions rather than single formulas. Scott used this idea to good effect in a number of projects including the axiomatization of many-valued logics (of various kinds) and a reconsideration of the motivation of C.I. Lewis. Since he left the subject it has been vigorously prosecuted (...) by a number of authors under the heading of abstract entailment relations where it has found an important role in both algebra and theoretical computer science. In this essay we go back to the beginnings, as presented by Scott, in order to make some comments about Scott’s cut rule, and show how much of Scott’s main result may be applied to the case of single-conclusion logic. (shrink)

A new characterization of all the MV-algebras embedded in a CL-algebra has been presented. A new sequent calculus for Lukasiewicz ℵ0-valued logic is introduced. Some links between this calculus and the sequent calculus for multiplicative additive linear logic are established. It has been shown that Lukasiewicz ℵ0-valued logic can be embedded in a suitable extension of MALL.

With attractive and scantily clad female characters, Zeus as a philandering womanizer, the First Born as a hyper‐masculine war monger, and Hera as a jealous wife blaming other women for her husband's infidelities, Wonder Woman (the New 52 series) confirms some age old stereotypes about men and women. But, Wonder Woman (the New 52) also challenges some traditional gender stereotypes. The end of the twentieth and beginning of the twenty‐first century have witnessed great strides in gender equality for, among (...) others, heterosexual women and members of the LGBTQ community. In one of the most important early books in twentieth‐century feminist philosophy, The Second Sex, philosopher and novelist Simone de Beauvoir explores women's inequality to men. The chapter considers the transformation of Hera throughout Wonder Woman (the New 52) as an existential movement from immanence to transcendence. (shrink)

Building on a base of simple economic theory and elementary linear algebra and calculus, this broad treatment of static and dynamic optimization methods discusses the importance of shadow prices and functions defined by solutions of optimization problems.

A mathematically valuable but technical contribution to the algebraic study of the semantics of formalized languages.--L. K. B.

We will prove in Zermelo-Fraenkel set theory without axiom of choice that the transitive hull R* of a relation R is not much “bigger” than R itself. As a measure for the size of a relation we introduce the notion of κ+-narrowness using surjective Hartogs numbers rather than the usul injective Hartogs values. The main theorem of this paper states that the transitive hull of a κ+-narrow relation is κ+-narrow. As an immediate corollary we obtain that, for every infinite cardinal (...) κ, the class HCκ of all κ-hereditary sets is a set with von Neumann rank ϱ(HCκ) ≤ κ+. Moreover, ϱ(HCκ) = κ+ if and only if κ is singular, otherwise ϱ(HCκ) = κ. The statements of the corollary are well known in the presence of the axiom of choice (AC). To prove them without AC - as carried through here - is, however, much harder. A special case of the corollary (κ = ω1, i.e., the class HCω1 of all hereditarily countable sets) has been treated independently by T. JECH. (shrink)

We will prove in Zermelo-Fraenkel set theory without axiom of choice that the transitive hull R* of a relation R is not much “bigger” than R itself. As a measure for the size of a relation we introduce the notion of κ+-narrowness using surjective Hartogs numbers rather than the usul injective Hartogs values. The main theorem of this paper states that the transitive hull of a κ+-narrow relation is κ+-narrow. As an immediate corollary we obtain that, for every infinite cardinal (...) κ, the class HCκ of all κ-hereditary sets is a set with von Neumann rank ϱ ≤ κ+. Moreover, ϱ = κ+ if and only if κ is singular, otherwise ϱ = κ. The statements of the corollary are well known in the presence of the axiom of choice . To prove them without AC - as carried through here - is, however, much harder. A special case of the corollary has been treated independently by T. JECH. (shrink)

Simone Weil is widely recognized today as one of the profound religious thinkers of the twentieth century. Yet while her interpretation of natural science is critical to Weil's overall understanding of religious faith, her writings on science have received little attention compared with her more overtly theological writings. The present essay, which builds on Vance Morgan's Weaving the World: Simone Weil on Science, Necessity, and Love (2005), critically examines Weil's interpretation of the history of science. Weil believed that mathematical science, (...) for the ancient Pythagoreans a mystical expression of the love of God, had in the modern period degenerated into a kind of reification of method that confuses the means of representing nature with nature itself. Beginning with classical (Newtonian) science's representation of nature as a machine, and even more so with the subsequent assimilation of symbolic algebra as the principal language of mathematical physics, modern science according to Weil trades genuine insight into the order of the world for symbolic manipulation yielding mere predictive success and technological domination of nature. I show that Weil's expressed desire to revive a Pythagorean scientific approach, inspired by the "mysterious complicity" in nature between brute necessity and love, must be recast in view of the intrinsically symbolic character of modern mathematical science. I argue further that a genuinely mystical attitude toward nature is nascent within symbolic mathematical science itself. (shrink)

It is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel-Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a `generic' extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we (...) also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure. (shrink)

In this paper we consider the we known method by E. Post of solving the problem of construction of recursively enumerable sets that have a degree intermediate between the degrees of recursive and complete sets with respect to a given reducibility. Post considered reducibilities ≤m, ≤btt, ≤tt and ≤T and solved the problem for al of them except ≤T. Here we extend Post's original method of construction of incomplete sets onto two wide classes of sub-Turing reducibilities what were studying in (...) [1, 2]. (shrink)

This is a translation of Jacob Klein's study "Die Griechische Logistik und die Entstehung der Algebra" which appeared in 1934-1936. His principal thesis is that the Renaissance mathematicians of the sixteenth century did not simply continue the work of the Greek and Arab mathematicians but in the process of developing ancient mathematics introduced a radically new conception of number which has since guided modern mathematical thought. The central figure in this revolution is Vieta. Klein traces the influence of Vieta's ideas (...) upon Stevin, Descartes, Wallis, and other figures of the scientific revolution, after discussing the conception of number and arithmetic in Plato, Aristotle, and other Greek sources. Persons reading this book with a primary interest in the philosophical ideas involved will be frustrated by the mass of historical detail which often obscures rather than illuminates the philosophical issues. But the book deserves its reputation as an important historical study.--R. H. K. (shrink)