A note on equational theories

2000 ◽  
Vol 65 (4) ◽  
pp. 1705-1712 ◽  
Author(s):  
Markus Junker
Keyword(s):  
Equational Theory ◽  
Order Theory ◽  
Point Of View ◽  
Unpublished Manuscript ◽  
Positive Information ◽  
Closed Sets ◽  
First Order Theory ◽  
Equational Theories ◽  
Definable Sets ◽  
Image Position

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example of a stable non-equational theory, the first and only one so far, is described in the unfortunately unpublished manuscript [HS].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].

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).

More on definable sets of p-adic numbers

1988 ◽  
Vol 53 (3) ◽  
pp. 912-920 ◽  
Author(s):  
Philip Scowcroft
Keyword(s):  
Cross Section ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Model Completeness ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Definable Sets ◽  
The Cross ◽  
Primitive Recursive

To eliminate quantifiers in the first-order theory of the p-adic field Qp, Ax and Kochen use a language containing a symbol for a cross-section map n → pn from the value group Z into Qp [1, pp. 48–49]. The primitive-recursive quantifier eliminations given by Cohen [2] and Weispfenning [10] also apply to a language mentioning the cross-section, but none of these authors seems entirely happy with his results. As Cohen says, “all the operations… introduced for our simple functions seem natural, with the possible exception of the map n → pn” [2, p. 146]. So all three authors show that various consequences of quantifier elimination—completeness, decidability, model-completeness—also hold for a theory of Qp not employing the cross-section [1, p. 453; 2, p. 146; 10, §4]. Macintyre directs a more specific complaint against the cross-section [5, p. 605]. Elementary formulae which use it can define infinite discrete subsets of Qp; yet infinite discrete subsets of R are not definable in the language of ordered fields, and so certain analogies between Qp and R suggested by previous model-theoretic work seem to break down.To avoid this problem, Macintyre gives up the cross-section and eliminates quantifiers in a theory of Qp written just in the usual language of fields supplemented by a predicate V for Qp's valuation ring and by predicates Pn for the sets of nth powers in Qp (for all n ≥ 2).

On models of the elementary theory of (Z, +, 1)

1990 ◽  
Vol 55 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Mark Nadel ◽  
Jonathan Stavi
Keyword(s):  
Stability Theory ◽  
Elementary Theory ◽  
Additive Group ◽  
Order Theory ◽  
Point Of View ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory ◽  
Distinguished Element ◽  
Unbounded Subset

Let T1 be the complete first-order theory of the additive group of the integers with 1 as distinguished element (in symbols, T1 = Th(Z, +, 1)). In this paper we prove that all models of T1 are ℵ0-homogeneous (§2), classify them (and lists of elements in them) up to isomorphism or L∞κ-equivalence (§§3 and 4) and show that they may be as complex as arbitrary sets of real numbers from the point of view of admissible set theory (§5). The results of §§2 and 5 together show that while the Scott heights of all models of T1 are ≤ ω (by ℵ0-homogeneity) their HYP-heights form an unbounded subset of the cardinal .In addition to providing this unusual example of the relation between Scott heights and HYP-heights, the theory T1 has served (using the homogeneity results of §2) as an example for certain combinations of properties that people had looked for in stability theory (see end of §4). In §6 it is shown that not all models of T = Th(Z, +) are ℵ0-homogeneous, so that the availability of the constant for 1 is essential for the result of §2.The two main results of this paper (2.2 and essentially Theorem 5.3) were obtained in the summer of 1979. Later we learnt from Victor Harnik and Julia Knight that T1 is of some interest for stability theory, and were encouraged to write up our proofs.During 1982/3 we improved the proofs and added some results.

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.

HODL(ℝ) is a Core Model Below Θ

1995 ◽  
Vol 1 (1) ◽  
pp. 75-84 ◽  
Author(s):  
John R. Steel
Keyword(s):  
Set Theory ◽  
Transitive Closure ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Inner Model ◽  
Definable Sets ◽  
The Axiom Of Choice ◽  
The Universe ◽  
Do So

In this paper we shall answer some questions in the set theory of L(ℝ), the universe of all sets constructible from the reals. In order to do so, we shall assume ADL(ℝ), the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L(ℝ) are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L(ℝ), and for reasons we cannot discuss here, ZFC + ADL(ℝ) yields the most interesting “completion” of the ZFC-theory of L(ℝ).ADL(ℝ) implies that L(ℝ) satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L(ℝ). Nevertheless, there is a natural inner model of L(ℝ), namely HODL(ℝ), which satisfies ZFC. (HOD is the class of all hereditarily ordinal definable sets, that is, the class of all sets x such that every member of the transitive closure of x is definable over the universe from ordinal parameters (i.e., “OD”). The superscript “L(ℝ)” indicates, here and below, that the notion in question is to be interpreted in L(R).) HODL(ℝ) is reasonably close to the full L(ℝ), in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL(ℝ): what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's. (See [4, p. 223]; also [12, p. 573] for variants involving finer notions of definability.)

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.

LOGICAL ASPECTS OF CAYLEY-GRAPHS: THE MONOID CASE

2006 ◽  
Vol 16 (02) ◽  
pp. 307-340 ◽  
Author(s):  
DIETRICH KUSKE ◽  
MARKUS LOHREY
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Order Theory ◽  
Second Order ◽  
Point Of View ◽  
Direct Products ◽  
Graph Products ◽  
Arbitrary Graph ◽  
First Order ◽  
First Order Theory

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

Sign in / Sign up

Export Citation Format

Share Document