Polyadic Algebras Research Articles

Finitary Polyadic Algebras from Cylindric Algebras

It is known that every α-dimensional quasi polyadic equality algebra (QPEAα) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties \((CA^{+}_\alpha, \alpha \geq 4)\). The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that from every algebra in \(CA^{+}_\alpha\) a β-dimensional algebra can be obtained in QPEAβ where \(\beta < \alpha (\beta \geq 4)\) , moreover the algebra obtained is representable in a sense.

Conservative extension of polyadic MV-algebras to polyadic pavelka algebras

In this paper we prove polyadic counterparts of the Hajek, Paris and Shepherdson's conservative extension theorems of Łukasiewicz predicate logic to rational Pavelka predicate logic. We also discuss the algebraic correspondents of the provability and truth degree for polyadic MV-algebras and prove a representation theorem similar to the one for polyadic Pavelka algebras.

2214 N-Sの式誘導の際の4階等方性テンソルのポリアディクスによる記述について(G05-3 解析モデルと解析手法,G05 流体工学)

This paper describes the isotropic tensor of rank 4 in three-dimensional space using polyadic algebra. The isotropic tensor of rank 4 appears in the relation between strain and stress for the well-known Navier-Stokes equation. When the deformation of viscous fluid is assumed to be slight and isotropic, the isotropic tensor of rank 4, that is Cijkl, is reconfirmed explicitely to be described as C_<ijkl>=Aδ_<ij>δ_<kl>+Bδ_<ik>δ_<jl>+Cδ_<il>δ_<jk>, where δ denotes the Kronecker delta. A, B and C denote the constants due to fiuid properties. In this case the polyadics is newly found to be suitable for the description of the rotation of the orthogonal curvilinear coordinates. The polyadics may be useful to describe the fluid properties further for microhydrodynamics and non-Newtonian fluid flow.

Algebraic Logic, Where Does it Stand Today?

AbstractThis is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel's incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic to omitting types of first order logic is given. Proofs are included when they serve to illustrate certain concepts. Several open problems are posed. We have tried as much as possible to avoid exploring territory already explored in the survey articles of Monk [93] and Németi [97] in the subject.

Omitting types for algebraizable extensions of first order logic

We prove an Omitting Types Theorem for certain algebraizable extensions of first order logic without equality studied in [SAI 00] and [SAY 04]. This is done by proving a representation theorem preserving given countable sets of infinite meets for certain reducts of ω- dimensional polyadic algebras, the so-called G polyadic algebras (Theorem 5). Here G is a special subsemigroup of (ω, ω o) that specifies the signature of the algebras in question. We state and prove an independence result connecting our representation theorem to Martin's axiom (Theorem 6). Also we show that the countable atomic G polyadic algebras are completely representable (Corollary 16) contrasting results on cylindric algebras. Several related results are surveyed.

Categorical abstract algebraic logic categorical algebraization of first-order logic without terms

An algebraization of multi-signature first-order logic without terms is presented. Rather than following the traditional method of choosing a type of algebras and constructing an appropriate variety, as is done in the case of cylindric and polyadic algebras, a new categorical algebraization method is used: The substitutions of formulas of one signature for relation symbols in another are treated in the object language. This enables the automatic generation via an adjunction of an algebraic theory. The algebras of this theory are then used to algebraize first-order logic.

On amalgamation of reducts of polyadic algebras

Following research initiated by Tarski, Craig and Nemeti, and further pursued by Sain and others, we show that for certain subsets G of ωw, G polyadic algebras have the strong amalgamation property. G polyadic algebras are obtained by restricting the (similarity type and) axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. Using algebraic logic, we infer that some theorems of Beth, Craig and Robinson hold for certain proper extensions of first order logic (without equality).

The class of 2-dimensional neat reducts is not elementary

SC, CA, QA and QEA stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras and Halmos' quasi- polyadic algebras with equality, respectively. Generalizing a result of Andr eka and N emeti on cylindric algebras, we show that for K2fSC;QA;CA;QEAg and any > 2 the class of 2-dimensional neat reducts of -dimensional algebras in K is not closed under forming elementary subalgebras, hence is not elementary. Whether this result extends to higher dimensions is open.

Algebraic Logic for Rational Pavelka Predicate Calculus

In this paper we define the polyadic Pavelka algebras as algebraic structures for Rational Pavelka predicate calculus (RPL∀). We prove two representation theorems which are the algebraic counterpart of the completness theorem for RPL∀.

On neat reducts of algebras of logic

SC α, CA α, QA α and QEA α stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension α, respectively. Generalizing a result of Nemeti on cylindric algebras, we show that for K ∈ {SC, CA, QA, QEA} and ordinals α < β, the class Nr α K β of α-dimensional neat reducts of β-dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if α > 1. From this it easily follows that for 1 < α < β, the operation of forming α-neat reducts of algebras in K β does not commute with forming subalgebras, a notion to be made precise. We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety. We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not. Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.

Polyadic Methods in Elastodynamics

Abstract–To see the similarities and differences with eletromagnetics, basic concepts and equations of elastodynamics are formulated in coordinate-free form applying concepts from Polyadic Algebra. Planewave propagation is studied for time-harmonic equations in isotropic and simple uniaxially anisotropic media and the Green dyadic is derived for the isotropic medium. As an extension, the Green dyadic for the anisotropic elastic medium is derived in perturbational approximation. In the Appendix A, some basic properties for tetradics, useful for practical analysis both in elastodynamics as well as in electromagnetics, are derived.

On the equational theory of representable polyadic equality algebras

AbstractAmong others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEAω'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 RPEAω is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildikó 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”.

An autobiography of polyadic algebras

An autobiography of polyadic algebras

Polyadic Methods in Elastodynamics - Abstract *

To see the similarities and differences with eletromagnetics, basic concepts and equations of elastodynamics are formulated in coordinate-free form applying concepts from Polyadic Algebra. Plane-wave propagation is studied for time-harmonic equations in isotropic and simple uniaxially anisotropic media and the Green dyadic is derived for the isotropic medium. As an extension, the Green dyadic for the anisotropic elastic medium is derived in perturbational approximation. In the Appendix A, some basic properties for tetradics, useful for practical analysis both in elastodynamics as well as in electromagnetics, are derived.

Decidability of cylindric set algebras of dimension two and first-order logic with two variables

AbstractThe aim of this paper is to give a new proof for the decidability and finite model property of first-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse2). The new proof also shows the known results that the universal theory of Pse2 is decidable and that every finite Pse2 can be represented on a finite base. Since the class Cs2 of cylindric set algebras of dimension 2 forms a reduct of Pse2, these results extend to Cs2 as well.

On the equational theory of representable polyadic equality algebras (extended abstract)

Among others we will see that the equational theory of ! dimensional representable polyadic equality algebras (RP EA!’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 nite schema-axiomatizable (and hence the equational theory of this class is nite schemaaxiomatizable, as well). We will also see that the complexity of the equational theory of RP EA! is also extremely high in the recursion theoretic sense. Finally, comparing the present negative results with the positive results of Ildik o Sain and Viktor Gyuris [10], 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". 1

Lambda Abstraction Algebras: Coordinatizing Models of Lambda Calculus

Lambda abstraction algebras are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order logic; they are intended as an alternative to combinatory algebras in this regard. Like combinatory algebras they can be defined by true identities and thus form a variety in the sense of universal algebra. One feature of lambda abstraction algebras that sets them apart from combinatory algebras is the way variables in the lambda calculus are abtracted; this provides each lambda abstraction algebra with an implicit coordinate system. Another peculiar feature is the algebraic reformulation of (β)-conversion as the definition of abstract substitution. Functional lambda abstraction algebras arise as the “coordinatizations” of environment models or lambda models, the natural combinatory models of the lambda calculus. As in the case of cylindric and polyadic algebras, questions of the functional representation of various subclasses of lambda abstraction algebras are an important part of the theory. The main result of the paper is a stronger version of the functional representation theorem for locally finite lambda abstraction algebras, the algebraic analogue of the completeness theorem of lambda calculus. This result is used to study the connection between the combinatory models of the lambda calculus and lambda abstraction algebras. Two significant results of this kind are the existence of a strong categorical equivalence between lambda algebras and locally finite lambda abstraction algebras, and between lambda models and rich, locally finite lambda abstraction algebras.

Relation algebras from cylindric and polyadic algebras

Relation algebras from cylindric and polyadic algebras

Lambda Abstraction Algebras: Coordinatizing Models of Lambda Calculus

Lambda abstraction algebras are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order logic; they are intended as an alternat...

Perfect extensions and derived algebras

Jónsson and Tarski [1951] introduced the notion of a Boolean algebra with (additive) operators (for short, a Bo). They showed that every Bo can be extended to a complete and atomic Bo satisfying certain additional conditions, and that any two complete, atomic extensions of satisfying these conditions are isomorphic over . Henkin [1970] extended these results to Boolean algebras with generalized (i.e., weakly additive) operators. The particular complete, atomic extension of studied by Jónsson and Tarski is called the perfect extension of , and is denoted by +. It is very useful in algebraic investigations of classes of algebras that are associated with logics.Interesting examples of Bos abound in algebraic logic, and include relation algebras, cylindric algebras, and polyadic and quasi-polyadic algebras (with or without equality). Moreover, there are several important constructions that, when applied to certain Bos, lead to other, derived Bos. Obvious examples include the formation of subalgebras, homomorphic images, relativizations, and direct products. Other examples include the Boolean algebra of ideal elements of a Bo, the neat β;-reduct of an α-dimensional cylindric algebra (β; &lt; α), and the relation algebraic reduct of a cylindric algebra (of dimension at least 3). It is natural to ask about the relationship between the perfect extension of a Bo and the perfect extension of one of its derived algebras ′: Is the perfect extension of the derived algebra just the derived algebra of the perfect extension? In symbols, is (′)+ = (+)′? For example, is the perfect extension of a subalgebra, homomorphic image, relativization, or direct product, just the corresponding subalgebra, homomorphic image, relativization, or direct product of the perfect extension (up to isomorphisms)? Is the perfect extension of the Boolean algebra of ideal elements, or the neat reduct of a cylindric algebra, or the relation algebraic reduct of a cylindric algebra just the Boolean algebra of ideal elements, or the neat β;-reduct, or the relation algebraic reduct, of the perfect extension? We shall prove a general result in this direction; namely, if the derived algebra is constructed as the range of a relatively multiplicative operator, then the answer to our question is “yes”. We shall also give examples to show that in “infinitary” constructions, our question can have a spectacularly negative answer.