# Nonstandard Methods and Finitness Conditions in Algebra

## Abstract

Let a be an algebra of a finite signature and S be a superstructure with the elements of a as urelements. The author gives a nonstandard characterization of properties of a involving finiteness conditions. For example, it is proved that a is locally finite iff it has a hyperfinite extension. He also proves that in certain (strong) varieties the sums of locally finite algebras are locally finite. Further, he introduces the notion of the extension monad of A, defined by Ab = T {B _ _A: B is an internal extension of A}. An alternate characterization of locally finite algebras and finitely generated algebras is given in terms of this notion; e.g., an algebra a is locally finite iff Ab = A. Homomorphisms and products of algebras are also studied in the context of extension monads; for example, it is proved that for any finite sequence of algebras A1,A2, • • • ,An, the equality (A1 × A2 ×• • •×An)b = Ab1 ×Ab2 ×• • •×Abn holds.

## Recommended Citation

M.
Insall,
"Nonstandard Methods and Finitness Conditions in Algebra," *Z. Math. Logik Grundlag. Math*, American Mathematical Society, Jan 1991.

## Department(s)

Mathematics and Statistics

## Keywords and Phrases

Algebra

## Document Type

Article - Journal

## Document Version

Citation

## File Type

text

## Language(s)

English

## Rights

© 1991 American Mathematical Society, All rights reserved.

## Publication Date

01 Jan 1991