scholarly journals THE VARIETY OF COSET RELATION ALGEBRAS

2018 ◽  
Vol 83 (04) ◽  
pp. 1595-1609 ◽  
Author(s):  
STEVEN GIVANT ◽  
HAJNAL ANDRÉKA
Keyword(s):  
Large Class ◽  
Identity Element ◽  
Relation Algebra ◽  
Order Theory ◽  
Relation Algebras ◽  
First Order ◽  
First Order Theory ◽  
Atomic Pair ◽  
Finite Set ◽  
Image Position

AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Decidable properties of finite sets of equations in trivial languages

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.

A lattice of interpretability types of theories

1977 ◽  
Vol 42 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Order Theory ◽  
Partial Ordering ◽  
First Order ◽  
First Order Theory ◽  
Relative Consistency ◽  
Infinite Distributivity ◽  
Closed Fields ◽  
Image Position ◽  
Interpretability Type ◽  
Consistency Proofs

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:We do not know if the dual lawis true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 60-88 ◽  
Author(s):  
URI ANDREWS ◽  
STEFFEN LEMPP ◽  
JOSEPH S. MILLER ◽  
KENG MENG NG ◽  
LUCA SAN MAURO ◽  
...  
Keyword(s):  
Computable Function ◽  
Order Theory ◽  
Equivalence Relations ◽  
Index Set ◽  
First Order ◽  
First Order Theory ◽  
Bounded Poset ◽  
Image Position ◽  
Open Question ◽  
Computably Enumerable

Abstract We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

INTERPRETING ARITHMETIC IN THE FIRST-ORDER THEORY OF ADDITION AND COPRIMALITY OF POLYNOMIAL RINGS

2019 ◽  
Vol 84 (3) ◽  
pp. 1194-1214
Author(s):  
JAVIER UTRERAS
Keyword(s):  
Power Series ◽  
Entire Functions ◽  
Order Theory ◽  
Ring Structure ◽  
Polynomial Rings ◽  
Series Ring ◽  
First Order ◽  
First Order Theory ◽  
Power Series Rings ◽  
Image Position

AbstractWe study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, + , \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. This work enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself.

CONSTRUCTING MANY ATOMIC MODELS IN ℵ1

2016 ◽  
Vol 81 (3) ◽  
pp. 1142-1162 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
MICHAEL C. LASKOWSKI ◽  
SAHARON SHELAH
Keyword(s):  
Order Theory ◽  
Atomic Model ◽  
First Order ◽  
Atomic Models ◽  
First Order Theory ◽  
Image Position

AbstractWe introduce the notion of pseudoalgebraicity to study atomic models of first order theories (equivalently models of a complete sentence of ${L_{{\omega _1},\omega }}$). Theorem: Let T be any complete first-order theory in a countable language with an atomic model. If the pseudominimal types are not dense, then there are 2ℵ0 pairwise nonisomorphic atomic models of T, each of size ℵ1.

scholarly journals NIP FOR THE ASYMPTOTIC COUPLE OF THE FIELD OF LOGARITHMIC TRANSSERIES

2017 ◽  
Vol 82 (1) ◽  
pp. 35-61 ◽  
Author(s):  
ALLEN GEHRET
Keyword(s):  
Order Theory ◽  
Exchange Property ◽  
Independence Property ◽  
First Order ◽  
Valued Field ◽  
First Order Theory ◽  
Order Language ◽  
Complete Survey ◽  
Image Position

AbstractThe derivation on the differential-valued field Tlog of logarithmic transseries induces on its value group ${{\rm{\Gamma }}_{{\rm{log}}}}$ a certain map ψ. The structure ${\rm{\Gamma }} = \left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ is a divisible asymptotic couple. In [7] we began a study of the first-order theory of $\left( {{{\rm{\Gamma }}_{{\rm{log}}}},\psi } \right)$ where, among other things, we proved that the theory $T_{{\rm{log}}} = Th\left( {{\rm{\Gamma }}_{{\rm{log}}} ,\psi } \right)$ has a universal axiomatization, is model complete and admits elimination of quantifiers (QE) in a natural first-order language. In that paper we posed the question whether Tlog has NIP (i.e., the Non-Independence Property). In this paper, we answer that question in the affirmative: Tlog does have NIP. Our method of proof relies on a complete survey of the 1-types of Tlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions ${\rm{\Gamma }}\left\langle \alpha \right\rangle$ of ${\rm{\Gamma }}$. We also show that Tlog does not have the Steinitz exchange property and we weigh in on the relationship between models of Tlog and the so-called precontraction groups of [9].

More on an undecidability result of Bateman, Jockusch and Woods

1998 ◽  
Vol 63 (1) ◽  
pp. 50-50 ◽  
Author(s):  
M. Boffa
Keyword(s):  
Prime Numbers ◽  
Order Theory ◽  
Linear Case ◽  
Natural Numbers ◽  
Fixed Element ◽  
First Order ◽  
First Order Theory ◽  
Image Position ◽  
Dirichlet Theorem ◽  
Undecidability Result

Let P be the set of prime numbers. Theorem 1 of [1] shows that the linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,P〉 and therefore that the first-order theory of this structure is undecidable. Let m be any fixed natural number >2, let R be the set of natural numbers <m which are prime to m, and let r be any fixed element of R. The setis infinite (Dirichlet). Theorem 1 of [1] can be improved as follows:Proposition. The linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,Pm,r〉 and therefore that the first-order theory of this structure is undecidable.Proof. We follow [1] with the following new ingredients. Let k be the number of elements of R, i.e. k = ϕ(m) where ϕ is Euler's totient function. Since k is even, the polynomial g(n) = nk + n satisfies g(0) = g(−1) = 0, so (by Lemma 1 of [1]) it follows from the linear case of (H) that there are natural numbers al (l ϵ ω) such that al+g(0), al+g(1),…, al+g(l) are consecutive primes. Since R is finite, we may assume that all the al's have the same residue t in R, so that al+g(i) ≡ t+1+i (mod m) for i ϵ R. This implies that the function t+1+i (reduced mod m) gives a permutation of R, so we can find s ϵ R such that al+g(s) ≡ r (mod m). Consider the polynomial h(n) = g(s + mn) and let bl = as+ml. Then bl + h(0), bl + h(1),…, bl + h(l) are elements of Pm,r. They are not necessarily consecutive elements of Pm,r, but they are separated by a fixed number of elements of Pm,r. This implies that {h(n) ∣ n ϵ ω} is definable in 〈ω,+,Pm,r〉(by adapting the proof of Theorem 1 of [1]), and the result follows.

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