We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

We show that the equational theory of representable lower semilattice-ordered residuated semigroups is finitely based. We survey related results.

This paper provides characterisations of the equational theory of the PER model of a typed lambda calculus with inductive types. The characterisation may be cast as a full abstraction result; in other words, we show that the equations between terms valid in this model coincides with a certain syntactically defined equivalence relation. Along the way we give other characterisations of this equivalence; from below, from above, and from a domain model, a version of the Kreisel-Lacombe-Shoenfield theorem allows us (...) to transfer the result from the domain model to the PER model. (shrink)

For $\Bbb {F}$ the field of real or complex numbers, let $CG(\Bbb {F})$ be the continuous geometry constructed by von Neumann as a limit of finite dimensional projective geometries over $\Bbb {F}$ . Our purpose here is to show the equational theory of $CG(\Bbb {F})$ is decidable.

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ n × n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ n × n ) is shown to be undecidable.

It is well known that the model categories of universal Horn theories are locally presentable, hence essentially algebraic . In the special case of quasivarieties a direct translation of the implicational syntax into the essentially equational one is known . Here we present a similar translation for the general case, showing at the same time that many relationally presented Horn classes are in fact quasivarieties.

A first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields and of the theory of separably closed fields of arbitrary imperfection degree.

The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts (...) of \ ’s, as well. We prove that the equational theory of \ is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

We consider equational theories for functions defined via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S 1 2 . In particular this solves an open problem formulated by TAKEUTI (c.f. [5, p.5 problem 9.]).

Several attempts have been done to distinguish “positive” information in an arbitrary first order theory, i.e., to find a well behaved class of closed sets among the definable sets. In many cases, a definable set is said to be closed if its conjugates are sufficiently distinct from each other. Each such definition yields a class of theories, namely those where all definable sets are constructible, i.e., boolean combinations of closed sets. Here are some examples, ordered by strength:Weak normality describes (...) a rather small class of theories which are well understood by now (see, for example, [P]). On the other hand, normalization is so weak that all theories, in a suitable context, are normalizable (see [HH]). Thus equational theories form an interesting intermediate class of theories. Little work has been done so far. The original work of Srour [S1, S2, S3] adopts a point of view that is closer to universal algebra than to stability theory. The fundamental definitions and model theoretic properties can be found in [PS], though some easy observations are missing there. Hrushovski's example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS]. In fact, it is an expansion of the free pseudospace constructed independently by Baudisch and Pillay in [BP] as an example of a strictly 2-ample theory. Strong equationality, defined in [Hr], is also investigated in [HS]. (shrink)

Among others we will prove that the equational theory of $\omega$ dimensional representable polyadic equality algebras is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable. We will also show that the complexity of the equational theory of RPEA$_\omega$ is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko (...) Sain and Viktor Gyuris [12], the following methodological conclusions will be drawn: The negative properties of polyadic algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". (shrink)

Among others we will see that the equational theory of ω dimensional representable polyadic equality algebras. We will also see that the complexity of the equational theory of RPEAω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko Sain and Viktor Gyuris [10], the following methodological conclusions will be drawn: the negative properties of polyadic algebras can be removed by switching from what we call (...) the `polyadic algebraic paradigm' to the `cylindric algebraic paradigm'. (shrink)

We show the undecidability of the equational theories of some classes of BAOs with a non-associative, residuated binary extra-Boolean operator. These results solve problems in Jipsen [9], Pratt [21] and Roorda [22], [23]. This paper complements Andréka-Kurucz-Németi-Sain-Simon [3] where the emphasis is on BAOs with an associative binary operator.

Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA ω 's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable, as well). We will also show that the complexity of the equational theory of RPEA ω is also (...) extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildiko Sain and Viktor Gyuris [12], the following methodological conclusions will be drawn: The negative properties of polyadic (equality) algebras can be removed by switching from what we call the "polyadic algebraic paradigm" to the "cylindric algebraic paradigm". (shrink)

The class $$\mathsf{TPA}$$ TPA of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $$\mathsf{TPA}$$ TPA nor the first order theory of $$\mathsf{TPA}$$ TPA are decidable. Moreover, we show that the set of all equations valid in $$\mathsf{TPA}$$ TPA is exactly on the $$\Pi ^1_1$$ Π 1 1 level. We consider (...) the class $$\mathsf{TPA}^-$$ TPA - of the relation algebra reducts of $$\mathsf{TPA}$$ TPA ’s, as well. We prove that the equational theory of $$\mathsf{TPA}^-$$ TPA - is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and (...) Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic. (shrink)

Suppose that T is an equational theory of groups or of rings. If T is finitely axiomatizable, then there is a least number μ so that T can be axiomatized by μ equations. This μ can depend on the operation symbols that occur in T. In the 1960s, Tarski and Green completely determined the values of μ for arbitrary equational theories of groups and of rings. While Tarski and Green announced the results of their collaboration in 1970, (...) the only fuller publication of their work occurred as part of a seminar led by Tarski at Berkeley during the 1968–69 academic year. The present paper gives a full account of their findings and their proofs. (shrink)

The Church-Turing Thesis (CTT) is often paraphrased as ``every computable function is computable by means of a Turing machine.'' The author has constructed a family of equational theories that are not Turing-decidable, that is, given one of the theories, no Turing machine can recognize whether an arbitrary equation is in the theory or not. But the theory is called pseudorecursive because it has the additional property that when attention is limited to equations with a bounded number of (...) variables, one obtains, for each number of variables, a fragment of the theory that is indeed Turing-decidable. In a 1982 conversation, Alfred Tarski announced that he believed the theory to be decidable, despite this contradicting CTT. The article gives the background for this proclamation, considers alternate interpretations, and sets the stage for further research. (shrink)

This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy. The Riccati equation, a nonlinear equation that can be linearized, has the potential to link these two (...) worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in a consistent mathematical description of physical laws. (shrink)

This article considers claims that biology should seek general theories similar to those found in physics but argues for an alternative framework for biological theories as collections of prototypical interlevel models that can be extrapolated by analogy to different organisms. This position is exemplified in the development of the Hodgkin‐Huxley giant squid model for action potentials, which uses equations in specialized ways. This model is viewed as an “emergent unifier.” Such unifiers, which require various simplifications, involve the types of heuristics (...) discussed in Wimsatt’s writings on reduction, but with a twist. Here, the heuristics are used to generate emergent rather than reductive explanations. †To contact the author, please write to: Department of History and Philosophy of Science, University of Pittsburgh, 1017 Cathedral of Learning, Pittsburgh, PA 15260; e‐mail: [email protected]. (shrink)

Equational Boolean Relation Theory concerns the Boolean equations between sets and their forward images under multivariate functions. We study a particular instance of equational BRT involving two multivariate functions on the natural numbers and three infinite sets of natural numbers. We prove this instance from certain large cardinal axioms going far beyond the usual axioms of mathematics as formalized by ZFC. We show that this particular instance cannot be proved in ZFC, even with the addition of slightly (...) weaker large cardinal axioms, assuming the latter are consistent. (shrink)

We show that equational independence in the sense of Srour equals local non-forking. We then examine so-called almost equational theories where equational independence is a symmetric relation.

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to (...) corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations. (shrink)

Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness (...) of the first-order functional calculus and Completeness in propositional type theory. More precisely, from and we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \. From, we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system. (shrink)

Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness (...) of the first-order functional calculus and Completeness in propositional type theory. More precisely, from and we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \. From, we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system. (shrink)

Diagrams have played an important role throughout the entire history of differential equations. Geometrical intuition, visual thinking, experimentation on diagrams, conceptions of algorithms and instruments to construct these diagrams, heuristic proofs based on diagrams, have interacted with the development of analytical abstract theories. We aim to analyze these interactions during the two centuries the classical theory of differential equations was developed. They are intimately connected to the difficulties faced in defining what the solution of a differential equation is and (...) in describing the global behavior of such a solution. (shrink)

In this article some intriguing aspects of electromagnetic theory and its relation to mathematics and reality are discussed, in particular those related to the suppositions needed to obtain the wave equations from Maxwell equations and from there Helmholtz equation. The following questions are discussed. How is that equations obtained with so many irreal or fictitious assumptions may provide a description that is in a high degree verifiable? Must everything that is possible to deduce from a theoretical mathematical model occur (...) in the world? Does everything that takes place in the world have a mathematical description? (shrink)

This article is concerned with the role of fundamental principles in theoretical physics, especially quantum theory. The fundamental principles of relativity will be addressed as well, in view of their role in quantum electrodynamics and quantum field theory, specifically Dirac’s work, which, in particular Dirac’s derivation of his relativistic equation of the electron from the principles of relativity and quantum theory, is the main focus of this article. I shall also consider Heisenberg’s earlier work leading him to (...) the discovery of quantum mechanics, which inspired Dirac’s work. I argue that Heisenberg’s and Dirac’s work was guided by their adherence to and their confidence in the fundamental principles of quantum theory. The final section of the article discusses the recent work by D’Ariano and coworkers on the principles of quantum information theory, which extend quantum theory and its principles in a new direction. This extension enabled them to offer a new derivation of Dirac’s equations from these principles alone, without using the principles of relativity. (shrink)