unary predicate
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2019 ◽  
Vol 84 (3) ◽  
pp. 1136-1167 ◽  
Author(s):  
PIETRO GALLIANI

AbstractIn Team Semantics, a dependency notion is strongly first order if every sentence of the logic obtained by adding the corresponding atoms to First-Order Logic is equivalent to some first-order sentence. In this work it is shown that all nontrivial dependency atoms that are strongly first order, downwards closed, and relativizable (in the sense that the relativizations of the corresponding atoms with respect to some unary predicate are expressible in terms of them) are definable in terms of constancy atoms.Additionally, it is shown that any strongly first-order dependency is safe for any family of downwards closed dependencies, in the sense that every sentence of the logic obtained by adding to First-Order Logic both the strongly first-order dependency and the downwards closed dependencies is equivalent to some sentence of the logic obtained by adding only the downwards closed dependencies.


2019 ◽  
Vol 84 (1) ◽  
pp. 343-358 ◽  
Author(s):  
FEDOR PAKHOMOV ◽  
ALBERT VISSER

AbstractIn this paper, we study finitely axiomatizable conservative extensions of a theory U in the case where U is recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.Consider a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extension α of U in the expanded language that is conservative over U, there is a conservative extension β of U in the expanded language, such that $\alpha \vdash \beta$ and $\beta \not \vdash \alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U instead of conservative extensions. Moreover, the result is preserved when we replace $\dashv$ as ordering on the finitely axiomatized extensions in the expanded language by a relevant kind of interpretability, to wit interpretability that identically translates the symbols of the U-language.We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.


2017 ◽  
Vol 19 (6) ◽  
pp. 61-73 ◽  
Author(s):  
N.L. Polyakov ◽  
M.V. Shamolin

The activity presents an efficient description of symmetric closed classes of discrete functions preserving every unary predicate.


2013 ◽  
Vol 78 (1) ◽  
pp. 185-194 ◽  
Author(s):  
Michael C. Laskowski

AbstractWe introduce the notions of a mutually algebraic structures and theories and prove many equivalents. A theory T is mutually algebraic if and only if it is weakly minimal and trivial if and only if no model M of T has an expansion (M, A) by a unary predicate with the finite cover property. We show that every structure has a maximal mutually algebraic reduct. and give a strong structure theorem for the class of elementary extensions of a fixed mutually algebraic structure.


2010 ◽  
Vol 75 (3) ◽  
pp. 785-801
Author(s):  
Luis Miguel ◽  
Villegas Silva

AbstractLet ℒ be a first-order language of cardinality κ++ with a distinguished unary predicate symbol U. In this paper we prove, working on L, the two cardinal transfer theorem (κ+,κ) ⇒ (κ++, κ+) for this language. This problem was posed by Chang and Keisler more than twenty years ago.


2007 ◽  
Vol 72 (1) ◽  
pp. 123-137 ◽  
Author(s):  
Pavel Hrubeš

AbstractWe give four examples of theories in which Kreisel's Conjecture is false: (1) the theory PA(-) obtained by adding a function symbol minus, ‘—’, to the language of PA, and the axiom ∀x∀y∀z (x − y = z) ≡ (x = y + z ∨ (x < y ∧ z = 0)); (2) the theory L of integers; (3) the theory PA(q) obtained by adding a function symbol q (of arity ≥ 1) to PA, assuming nothing about q; (4) the theory PA(N) containing a unary predicate N(x) meaning ‘x is a natural number’. In Section 6 we suggest a counterexample to the so called Sharpened Kreisel's Conjecture.


2005 ◽  
Vol 70 (1) ◽  
pp. 29-60 ◽  
Author(s):  
Marcus Tressl

§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pL ⋃ pR = M and pL < pR, i.e., a < b for all a ∈ pL, b ∈ pR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L(D)-structure (M, pL) which is model complete.The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {a ∈ M ∣ a > Z} and Z− for the cut q with qL = {a ∈ M ∣ a < Z}.


2002 ◽  
Vol 12 (2) ◽  
pp. 203-235 ◽  
Author(s):  
B. COURCELLE ◽  
J. A. MAKOWSKY

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.


2001 ◽  
Vol 66 (3) ◽  
pp. 1127-1140 ◽  
Author(s):  
Enrique Casanovas ◽  
Martin Ziegler

Let M be an L-structure and A be an infinite subset of M. Two structures can be defined from A:• The induced structure on A has a name Rφ for every ∅-definable relation φ(M) ∩ An on A. Its language isA with its Lind-structure will be denoted by Aind.• The pair (M, A) is an L(P)-structure, where P is a unary predicate for A and L(P) = L ∪{P}.We call A small if there is a pair (N, B) elementarily equivalent to (M, A) and such that for every finite subset b of N every L–type over Bb is realized in N.A formula φ(x, y) has the finite cover property (f.c.p) in M if for all natural numbers k there is a set of φ–formulaswhich is k–consistent but not consistent in M. M has the f.c.p if some formula has the f.c.p in M. It is well known that unstable structures have the f.c.p. (see [6].) We will prove the following two theorems.Theorem A. Let A be a small subset of M. If M does not have the finite cover property then, for every λ ≥ ∣L∣, if both M andAindare λ–stable then (M, A) is λ–stable.Corollary 1.1 (Poizat [5]). If M does not have the finite cover property and N ≺ M is a small elementary substructure, then (M, N) is stable.Corollary 1.2 (Zilber [7]). If U is the group of wots of unity in the field ℂ of complex numbers the pair (ℂ, U) isω–stable.Proof. (See [4].) As a strongly minimal set ℂ is ω–stable and does not have the f.c.p. By the subspace theorem of Schmidt [3] every algebraic set intersects U in a finite union of translates of subgroups definable in the group structure of U alone. Whence Uind is nothing more than a (divisible) abelian group, which is ω–stable.


2001 ◽  
Vol 66 (3) ◽  
pp. 1321-1341 ◽  
Author(s):  
P. V. Andreev ◽  
E. I. Gordon

AbstractWe present an axiomatic framework for nonstandard analysis—the Nonstandard Class Theory (NCT) which extends von Neumann–Gödel–Bernays Set Theory (NBG) by adding a unary predicate symbol St to the language of NBG (St(X) means that the class X is standard) and axioms—related to it—analogs of Nelson's idealization, standardization and transfer principles. Those principles are formulated as axioms, rather than axiom schemes, so that NCT is finitely axiomatizable. NCT can be considered as a theory of definable classes of Bounded Set Theory by V. Kanovei and M. Reeken. In many aspects NCT resembles the Alternative Set Theory by P. Vopenka. For example there exist semisets (proper subclasses of sets) in NCT and it can be proved that a set has a standard finite cardinality iff it does not contain any proper subsemiset. Semisets can be considered as external classes in NCT. Thus the saturation principle can be formalized in NCT.


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