# Properties preserved under algebraic constructions

@article{Lyndon1959PropertiesPU, title={Properties preserved under algebraic constructions}, author={Roger Lyndon}, journal={Bulletin of the American Mathematical Society}, year={1959}, volume={65}, pages={287-299} }

The study of properties of algebras belongs equally to Algebra and to Metamathematics. But here we consider only questions that concern all abstract algebras, and we attempt to discuss them without invoking any unfamiliar metamathematical machinery. In consequence, we can not do justice to the many interesting matters that concern important special classes of algebraic systems, nor to the broad metamathematical context in which the problems mentioned here belong. Our topic was approached in…

## 15 Citations

Symposium on Metamathematics of Algebra and Geometry: Metamathematics and Algebra: An Example

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- 1963

The semantical concepts, such as satisfaction, truth, and model, form the subject matter of a field known as the theory of models. I am going to discuss today several related recent developments in…

Some Applications of Infinitely Long Formulas

- MathematicsJ. Symb. Log.
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The notion of a finite approximation to an infinitely long formula, in a language L with infinitely long expressions of the type considered by Henkin in [2], and a number of theorems about ordinary finitely long sentences are introduced.

Bases for First-Order Theories and Subtheories

- PhilosophyJ. Symb. Log.
- 1960

What degree of economy, or at least regularity, can be achieved for all recursively axiomatizable first-order theories is investigated.

Rings of integer-valued continuous functions

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- 1961

During the past twenty years extensive work has been done on the ring C(X). The pioneer papers in the subject are [8] for compact X and [3] for arbitrary X. A significant part of this work has…

Free Vector Lattices

- MathematicsCanadian Journal of Mathematics
- 1968

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal…

Defining subdirect product closed classes in infinitary logic

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- 2013

We note that a class of models is subdirect product closed if and only if it is definable by a class of L∞∞-sentences of a special form.

Model-theoretic and algorithmic questions in group theory

- Mathematics
- 1985

This paper provides a survey of results of an algorithmic nature. Model-theoretic methods and results in group theory are discussed.

## References

SHOWING 1-10 OF 36 REFERENCES

Some interconnections between modern algebra and mathematical logic

- Philosophy
- 1953

Mathematical logic originated when mathematical methods were brought to bear on traditional questions of logic, especially the problem of what constitutes a valid proof. For the most part work in…

Two Concepts From the Theory of Models

- MathematicsJ. Symb. Log.
- 1956

It is pointed out that two concepts which have been introduced separately into the theory of models are extensionally equivalent and can be expressed conveniently in the terminology of Tarski's theory of arithmetical classes.

On Sentences Which are True of Direct Unions of Algebras

- MathematicsJ. Symb. Log.
- 1951

A wide class of invariant sentences is determined which are true of a direct union provided they aretrue of some factor algebra and it will be shown that these criteria are the only ones of their kind.

Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory and Proof Theory

- MathematicsJ. Symb. Log.
- 1957

The Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms, showing that the expressive power of each first-order system is rounded out, or the system is functionally complete.

Existential Horn sentences

- Mathematics
- 1959

A. Horn [2] has shown that every sentence of a certain class, which we define precisely below, is true in a direct product of algebras whenever it is true in each component algebra. McKinsey [4] had…

On unions of chains of models

- Mathematics
- 1959

(*) IIxi, • • • , x„Syi, • • • , ym v where v is quantifier-free, if and only if K is closed under unions of chains of models.1 We have adopted most of the notational features of Tarski [7], in which…

Horn Sentences in Identity Theory

- MathematicsJ. Symb. Log.
- 1959

It is shown that, if consideration is restricted to identity theory, that is, a first-order predicate calculus with equality but no other relation symbols or operation symbols, Horn's condition is necessary and sufficient and every Horn sentence is preserved under the direct product of one or more systems.

Note on a Problem of L. Henkin

- Mathematics, PhilosophyJ. Symb. Log.
- 1956

Henkin has shown that, in order that a statement X be absolutely persistent, it is necessary and sufficient that there exist a statement Y ϵ E such that is provable.

Theory of Models with Generalized Atomic Formulas

- MathematicsJ. Symb. Log.
- 1960

The following theorem is proved that if K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

On the Structure of Abstract Algebras

- Mathematics
- 1935

The following paper is a study of abstract algebras qua abstract algebras. As no vocabulary suitable for this purpose is current, I have been forced to use a number of new terms, and extend the…