A characterization of companionable, universal theories

1978 ◽  
Vol 43 (3) ◽  
pp. 402-429 ◽  
Author(s):  
William H. Wheeler
Keyword(s):  
Amalgamation Property ◽  
Finiteness Condition ◽  
Model Companion ◽  
Coherence Property ◽  
Finite Presentation ◽  
First Order Theory ◽  
Closed Fields ◽  
Inductive Theory ◽  
Model Completion ◽  
Finitely Presented

A first-order theory is companionable if it is mutually model-consistent with a model-complete theory. The latter theory is then called a model-companion for the former theory. For example, the theory of formally real fields is a companionable theory; its model-companion is the theory of real closed fields. If a companionable, inductive theory has the amalgamation property, then its model-companion is actually a model-completion. For example, the theory of fields is a companionable, inductive theory with the amalgamation property; its model-completion is the theory of algebraically closed fields.The goal of this paper is the characterization, by “algebraic” or “structural” properties, of the companionable, universal theories which satisfy a certain finiteness condition. A theory is companionable precisely when the theory consisting of its universal consequences is companionable. Both theories have the same model-companion if either has one. Accordingly, nought is lost by the restriction to universal theories. The finiteness condition, finite presentation decompositions, is an analogue for an arbitrary theory of the decomposition of a radical ideal in a Noetherian, commutative ring into a finite intersection of prime ideals for the theory of integral domains. The companionable theories with finite presentation decompositions are characterized by two properties: a coherence property for finitely generated submodels of finitely presented models and a homomorphism lifting property for homomorphisms from submodels of finitely presented models.

The model-companion of a class of structures

1972 ◽  
Vol 37 (3) ◽  
pp. 546-556 ◽  
Author(s):  
G. L. Cherlin
Keyword(s):  
Order Theory ◽  
Model Companion ◽  
Elementary Substructure ◽  
Closed Model ◽  
First Order ◽  
First Order Theory ◽  
Logical Equivalence ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Model Completion

If Σ is the class of all fields and Σ* is the class of all algebraically closed fields, then it is well known that Σ* is characterized by the following properties:(i) Σ* is the class of models of some first order theory K*.(ii) If m1m2 are in Σ* and m1 ⊆ m2 then m1 ≺ m2 (m1 is an elementary substructure of m2, i.e. any first order sentence true in m1 is true in m2).(iii) If m1 is in Σ then there is a structure m2 in Σ* such that m1 ⊆ m2.If Σ is some other class of models of a first order theory K and a subclass Σ* of Σ exists satisfying (i)–(iii) then Σ* is uniquely determined and K* (which is unique up to logical equivalence) is called the model-companion of K. This notion is a generalization of the fundamental notion of model-completion introduced and extensively studied by A. Robinson [6], When the model-companion exists it provides the basis for a satisfactory treatment of the notion of an algebraically closed model of K.Recently A. Robinson has developed a more general formulation of the notion of “algebraically closed” structures in Σ, which is applicable to any inductive elementary class Σ of structures (by elementary we always mean ECΔ). Condition (i) must be weakened to(i′) Σ* is closed under elementary substructure (i.e. if m1 is in Σ* and m2 ≺ m1 then m2 is in Σ*).

A generalized model companion for a theory of partially ordered fields

1979 ◽  
Vol 44 (4) ◽  
pp. 643-652
Author(s):  
Werner Stegbauer
Keyword(s):  
General Model ◽  
Order Theory ◽  
Model Companion ◽  
Generalized Model ◽  
First Order ◽  
First Order Theory ◽  
Ordered Fields ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Model Completion

The notion of a model companion for a first-order theory T was introduced and discussed in [1] and [2] as a generalization of the concept of a model completion of a theory. Both concepts reflect, on a general model theoretic level, properties of the theory of algebraically closed fields. The literature provides many examples of first-order theories with and without model companions—see [3] for a survey of these results. In this paper, we give a further generalization of the notion of a model companion.Roughly speaking, we allow instead of embeddings more general classes of maps (e.g. homomorphisms) and we allow any set of formulas which is preserved by these maps instead of existential formulas. This plan is worked out in detail in [5], where we discuss also several examples. One of these examples is given in this paper.In order to clarify the model theoretic background, we now introduce the relevant concepts and theorems from [5].

scholarly journals Model companions of theories with an automorphism

2000 ◽  
Vol 65 (3) ◽  
pp. 1215-1222 ◽  
Author(s):  
Hirotaka Kikyo
Keyword(s):  
Amalgamation Property ◽  
Complete Theory ◽  
Model Companion ◽  
Independence Property ◽  
Model Completion

AbstractFor a theory T in L, Tσ is the theory of the models of T with an automorphism σ. If T is an unstable model complete theory without the independence property, then Tσ has no model companion. If T is an unstable model complete theory and Tσ has the amalgamation property, then Tσ has no model companion. If T is model complete and has the fcp, then Tσ has no model completion.

The structure of algebraically and existentially closed Stone and double Stone algebras

1989 ◽  
Vol 54 (2) ◽  
pp. 363-375 ◽  
Author(s):  
David M. Clark
Keyword(s):  
Finite Algebra ◽  
Amalgamation Property ◽  
Explicit Construction ◽  
Structural Description ◽  
Model Companion ◽  
Existentially Closed ◽  
Boolean Spaces ◽  
Countably Infinite ◽  
Model Completion ◽  
Primitive Recursive

In this paper we study the varieties of Stone algebras (S, ∧, ∨, *, 0, 1) and double Stone algebras (D, ∧, ∨, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite ∀∃-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also ℵ0 categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra A, the universal Horn class ISP has a model companion. Weispfenning [24], [25] discovered that this model companion is always ℵ0 categorical and has a primitive recursive ∀∃-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of ISP is available. Burris and Werner [4] solve this problem in a special setting.

VERIFYING THAT A GROUP IS VIRTUALLY FREE

1991 ◽  
Vol 01 (03) ◽  
pp. 339-351
Author(s):  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Universal Group ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finitely Presented

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.

scholarly journals Coherency and Constructions for Monoids

2020 ◽  
Vol 71 (4) ◽  
pp. 1461-1488
Author(s):  
Yang Dandan ◽  
Victoria Gould ◽  
Miklós Hartmann ◽  
Nik Ruškuc ◽  
Rida-E Zenab
Keyword(s):  
Finiteness Condition ◽  
Direct Products ◽  
Finitely Generated ◽  
Rees Matrix Semigroups ◽  
The Relationship ◽  
Brandt Semigroups ◽  
Finitely Presented ◽  
Matrix Semigroups

Abstract A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

scholarly journals Some remarks on divisible polyhedral MV-algebras

2020 ◽  
Vol 70 (1) ◽  
pp. 51-60
Author(s):  
Serafina Lapenta
Keyword(s):  
Amalgamation Property ◽  
Finite Products ◽  
Finitely Presented ◽  
Mv Algebras

AbstractBuilding on similar notions for MV-algebras, polyhedral DMV-algebras are defined and investigated. For such algebras dualities with suitable categories of polyhedra are established, and the relation with finitely presented Riesz MV-algebras is investigated. Via hull-functors, finite products are interpreted in terms of hom-functors, and categories of polyhedral MV-algebras, polyhedral DMV-algebras and finitely presented Riesz MV-algebras are linked together. Moreover, the amalgamation property is proved for finitely presented DMV-algebras and Riesz MV-algebras, and for polyhedral DMV-algebras.

scholarly journals Groups of p-deficiency one

2013 ◽  
Vol 156 (1) ◽  
pp. 115-121
Author(s):  
ANITHA THILLAISUNDARAM
Keyword(s):  
Finite Index ◽  
Index Subgroup ◽  
P Deficiency ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finite Index Subgroup ◽  
Finitely Presented

AbstractIn a previous paper, Button and Thillaisundaram proved that all finitely presented groups of p-deficiency greater than one are p-large. Here we prove that groups with a finite presentation of p-deficiency one possess a finite index subgroup that surjects onto the integers. This implies that these groups do not have Kazhdan's property (T). Additionally, we show that the aforementioned result of Button and Thillaisundaram implies a result of Lackenby.

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.

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