Nd-hypersubstitutions of many-sorted algebras

We presuppose the terminology of [1], and we give a negative answer to the following problem ([1], p. 19): Does every essentially undecidable axiomatizable theory have an essentially undecidable finitely axiomatizable subtheory?We use the following theorem of Kleene ([2], p. 311). There exist two recursively enumerable sets α and β such that (1) α and β are disjoint (2) there is no recursive set η for which α ⊂ η, β ⊂ η′. By the definition of recursive enumerability, there are recursive predicates Φ and Ψ for whichWe now specify a theory T which will afford a counter-example to the given problem of Tarski. The only non-logical constants of T are two binary predicates P and Q, one unary operation symbol S, and one individual constant 0. As in ([1], p. 52) we defineThe only non-logical axioms of T are the formulae P(Δm, Δn) for all pairs of integers m, n satisfying Δ(m, n); the formulae Q(Δm, Δn) for all pairs of integers m, n satisfying Ψ(m, n); and the formulaT is consistent, since it has a model. It remains to show that (1) every consistent extension of T is undecidable (2) if T1 is a finitely axiomatizable subtheory of T, there exists a consistent and decidable extension of T1 which has the same constants as T1.

Download Full-text

Decidable properties of finite sets of equations in trivial languages

Journal of Symbolic Logic ◽

10.2307/2274282 ◽

1984 ◽

Vol 49 (4) ◽

pp. 1333-1338

Author(s):

Cornelia Kalfa

Keyword(s):

Equational Theory ◽

Order Theory ◽

Operation Symbol ◽

First Order ◽

First Order Theory ◽

Order Language ◽

Finite Set ◽

Joint Embedding ◽

Algebraic Language ◽

Joint Embedding Property

In [4] I proved that in any nontrivial algebraic language there are no algorithms which enable us to decide whether a given finite set of equations Σ has each of the following properties except P2 (for which the problem is open):P0(Σ) = the equational theory of Σ is equationally complete.P1(Σ) = the first-order theory of Σ is complete.P2(Σ) = the first-order theory of Σ is model-complete.P3(Σ) = the first-order theory of the infinite models of Σ is complete.P4(Σ) = the first-order theory of the infinite models of Σ is model-complete.P5(Σ) = Σ has the joint embedding property.In this paper I prove that, in any finite trivial algebraic language, such algorithms exist for all the above Pi's. I make use of Ehrenfeucht's result [2]: The first-order theory generated by the logical axioms of any trivial algebraic language is decidable. The results proved here are part of my Ph.D. thesis [3]. I thank Wilfrid Hodges, who supervised it.Throughout the paper is a finite trivial algebraic language, i.e. a first-order language with equality, with one operation symbol f of rank 1 and at most finitely many constant symbols.

Download Full-text

Green’s Relations on Regular Elements of Semigroup of Relational Hypersubstitutions for Algebraic Systems of Type ((m), (n))

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.53.2022.3436 ◽

2021 ◽

Vol 53 ◽

Author(s):

Sorasak Leeratanavalee ◽

Jukkrit Daengsaen

Keyword(s):

Green’S Relations ◽

Operation Symbol ◽

Algebraic Systems ◽

Regular Elements ◽

Relational Term ◽

Green's Relations ◽

Algebraic Properties

Any relational hypersubstitution for algebraic systems of type (τ,τ′) = ((mi)i∈I,(nj)j∈J) is a mapping which maps any mi-ary operation symbol to an mi-ary term and maps any nj - ary relational symbol to an nj-ary relational term preserving arities, where I,J are indexed sets. Some algebraic properties of the monoid of all relational hypersubstitutions for algebraic systems of a special type, especially the characterization of its order and the set of all regular elements, were first studied by Phusanga and Koppitz[13] in 2018. In this paper, we study the Green’srelationsontheregularpartofthismonoidofaparticulartype(τ,τ′) = ((m),(n)), where m, n ≥ 2.

Download Full-text

Green's Relations on the Monoid of Regular Hypersubstitutions

Algebra Colloquium ◽

10.1142/s1005386706000563 ◽

2006 ◽

Vol 13 (04) ◽

pp. 623-632 ◽

Cited By ~ 1

Author(s):

Th. Changphas ◽

S. L. Wismath

Keyword(s):

Galois Connection ◽

Green’S Relations ◽

Operation Symbol ◽

Green's Relations ◽

Fixed Type

The theory of hyperidentities and hypervarieties is based on the fact that the set Hyp (τ) of all hypersubstitutions of a fixed type τ forms a monoid, with a Galois connection between submonoids of this monoid and complete sublattices of the lattice of all varieties of type τ. For this reason, there is interest in studying the semigroup or monoid properties of Hyp (τ) and its submonoids. One approach is to study the five relations known as Green's relations definable on any semigroup. In this paper, we consider the type τ = (n) with one n-ary operation symbol for n≥ 1, and the submonoid Reg (n) of regular hypersubstitutions. We characterize Green's relations on every subsemigroup of Reg (n); then using this characterization we describe which subsemigroups of Reg (n) are 𝒢-subsemigroups of Reg (n) defined by Levi.

Download Full-text

Undecidable properties of finite sets of equations

Journal of Symbolic Logic ◽

10.1017/s0022481200051161 ◽

1976 ◽

Vol 41 (3) ◽

pp. 589-604 ◽

Cited By ~ 8

Author(s):

George F. McNulty

Keyword(s):

Decision Problems ◽

Equational Logic ◽

Relation Algebras ◽

Finite Sets ◽

Operation Symbol ◽

First Order ◽

Finite Models ◽

Order Language

Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical.Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive.Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

Download Full-text

WHEN IS A CATEGORY OF MANY-SORTED PARTIAL ALGEBRAS CARTESIAN-CLOSED?

International Journal of Foundations of Computer Science ◽

10.1142/s0129054195000056 ◽

1995 ◽

Vol 06 (01) ◽

pp. 51-66 ◽

Cited By ~ 5

Author(s):

M. MONSERRAT ◽

F. ROSSELLÓ ◽

J. TORRENS

Keyword(s):

Operation Symbol ◽

Functor Category ◽

Natural Categories ◽

First Case ◽

Cartesian Closedness ◽

Partial Algebras ◽

The One ◽

Cartesian Closed

In this paper we study the cartesian closedness of the five most natural categories with objects all partial many-sorted algebras of a given signature. In particular, we prove that, from these categories, only the usual one and the one having as morphisms the closed homomorphisms can be cartesian closed. In the first case, it is cartesian closed exactly when the signature contains no operation symbol, in which case such a category is a slice category of sets. In the second case, it is cartesian closed if and only if all operations are unary. In this case, we identify it as a functor category and we show some relevant constructions in it, such as its subobjects classifier or the exponentials.

Download Full-text

Decision problems concerning properties of finite sets of equations

Journal of Symbolic Logic ◽

10.2307/2273945 ◽

1986 ◽

Vol 51 (1) ◽

pp. 79-87 ◽

Cited By ~ 2

Author(s):

Cornelia Kalfa

Keyword(s):

Order Theory ◽

List Type ◽

Finite Sets ◽

Operation Symbol ◽

First Order ◽

First Order Theory ◽

Rank One ◽

Joint Embedding ◽

Algebraic Language ◽

Joint Embedding Property

In this paper a general method of proving the undecidability of a property P, for finite sets Σ of equations of a countable algebraic language, is presented. The method is subsequently applied to establish the undecidability of the following properties, in almost all nontrivial such languages:1. The first-order theory generated by the infinite models of Σ is complete.2. The first-order theory generated by the infinite models of Σ is model-complete.3. Σ has the joint-embedding property.4. The first-order theory generated by the models of Σ with more than one element has the joint-embedding property.5. The first-order theory generated by the infinite models of Σ has the joint-embedding property.A countable algebraic language ℒ is a first-order language with equality, with countably many nonlogical symbols but without relation symbols, ℒ is trivial if it has at most one operation symbol, and this is of rank one. Otherwise, ℒ is nontrivial. An ℒ-equation is a sentence of the form , where φ and ψ are ℒ-terms. The set of ℒ-equations is denoted by Eqℒ. A set of sentences is said to have the joint-embedding property if any two models of it are embeddable in a third model of it.If P is a property of sets of ℒ-equations, the decision problem of P for finite sets of ℒ-equations is the problem of the existence or not of an algorithm for deciding whether, given a finite Σ ⊂ Eqℒ, Σ has P or not.

Download Full-text

Konsepsi siswa kelas tiga sekolah dasar tentang bilangan bulat

PYTHAGORAS Jurnal Pendidikan Matematika ◽

10.21831/pg.v15i2.37645 ◽

2021 ◽

Vol 15 (2) ◽

pp. 201-215

Author(s):

Syafdi Maizora ◽

Rizky Rosjanuardi

Keyword(s):

Elementary School ◽

Eighth Graders ◽

Number Line ◽

Formal Learning ◽

Case Study Approach ◽

Operation Symbol ◽

Study Approach ◽

The Subject ◽

The City

Artikel ini menggambarkan konsepsi salah seorang siswa kelas 3 Sekolah Dasar di Kota Bengkulu tentang bilangan bulat di luar pembelajaran formal. Siswa ini mengalami banyak intervensi tanpa skenario dalam pembelajarannya, di antaranya dari keluarga (kakak kelas 8 yang memiliki prestasi baik dalam matematika, kedua orang tua pengajar matematika) dan pelatihan sempoa. Konsepsi yang digali adalah arti bilangan negatif, bilangan bulat, serta operasi penjumlahan dan pengurangan pada bilangan bulat. Jenis penelitian ini adalah penelitian kualitatif dengan pendekatan studi kasus. Subjek diberikan beberapa pertanyaan seputar konsepsi bilangan bulat. Hasil penelitian menunjukkan bahwa subjek memiliki konsepsi sebagai berikut: 1) menggunakan istilah “kurang”, “utang” atau “posisi di bawah permukaan” untuk memaknai bilangan bulat negatif, 2) bilangan bulat negatif diartikan sebagai invers penjumlahan bilangan asli, 3) ada perbedaan antara simbol negatif dengan simbol operasi pengurangan, 4) bilangan bulat negatif berada di sebelah kiri bilangan 0 pada garis bilangan, 5) bilangan bulat negatif terkecil berada di sebelah kiri bilangan bulat negatif lainnya, seperti bilangan-bilangan pada penggaris, dan mampu menggunakan dinding sebagai pengganti garis bilangan, 6) menggunakan istilah “maju” atau “mundur” untuk mengoperasikan penjumlahan bilangan bulat, 7) menggunakan kata “jarak”, “lompatan di atas garis bilangan”, dan “lompatan di bawah garis bilangan” untuk mengoperasikan pengurangan bilangan bulat. Conceptions of third-grader elementary school about integersAbstractThis article described the conception of a third-grader elementary school in the City of Bengkulu about integers outside formal learning. This student experienced many interventions without scenarios in their learning, including their families (a brother in eighth-graders who had good mathematics achievements, parents were mathematics education lecturer) and an abacus training. The explored conceptions were the meaning of negative numbers, integers, and addition and subtraction operations on integers. This research was qualitative research with a case study approach. The subject was asked several questions regarding the conception of integers. The results of this research indicated that the subject had the following conception: 1) using the term “less”, “debt”, or “position under the surface” to interpret negative integers; 2) interpreting negative integers as the inverse of the addition of natural numbers; 3) differentiating the negative symbol and the subtraction operation symbol, 4) locating negative integers to the left of “0” on a number line; 5) locating smaller negative integers to the left of other negative integers like numbers on a ruler and having an ability to use a wall as a substitute of a number line; 6) using terms “forward” or “backward” to operate integer additions; and 7) using the term “distance”, “jumps over the number line”, and “jumps under the number line” to operate integer subtractions.

Download Full-text

Green's Relations on HypG(2)

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/v10309-012-0016-5 ◽

2012 ◽

Vol 20 (1) ◽

pp. 249-264

Author(s):

Wattapong Puninagool ◽

Sorasak Leeratanavalee

Keyword(s):

Binary Operation ◽

Green’S Relations ◽

Operation Symbol ◽

Green's Relations ◽

Natural Way

AbstractA generalized hypersubstitution of type τ = (2) is a mapping which maps the binary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ = (2), and any two such extensions can be composed in a natural way. Thus, the set HypG(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. Green's relations on the monoid of all hypersubstitutions of type τ = (2) were studied by K. Denecke and Sh.L. Wismath. In this paper we describe the classes of generalized hypersubstitutions of type τ = (2) under Green's relations.

Download Full-text

Some undecidability results in strong algebraic languages

Journal of Symbolic Logic ◽

10.2307/2274149 ◽

1984 ◽

Vol 49 (3) ◽

pp. 951-954

Author(s):

Cornelia Kalfa

Keyword(s):

Decision Problem ◽

Decision Problems ◽

Turing Machines ◽

Constant Symbol ◽

Finite Sets ◽

Operation Symbol ◽

Halting Problem ◽

First Order ◽

Order Language ◽

Image Position

The recursively unsolvable halting problem for Turing machines is reduced to the problem of the existence or not of an algorithm for deciding whether a field is finite. The latter problem is further reduced to the decision problem of each of propertiesfor recursive sets Σ of equations of strong algebraic languages with infinitely many operation symbols.Decision problems concerning properties of sets of equations were first raised by Tarski [9] and subsequently examined by Perkins [6], McKenzie [4], McNulty [5] and Pigozzi [7]. Perkins is the only one who studied recursive sets; the others investigated finite sets. Since the undecidability of properties Pi for recursive sets of equations does not imply any answer to the corresponding decision problems for finite sets, the latter problems remain open.The work presented here is part of my Ph.D. thesis [2]. I thank Wilfrid Hodges, who supervised it.An algebraic language is a first-order language with equality but without relation symbols. It is here denoted by , where Qi is an operation symbol and cj, is a constant symbol.

Download Full-text