FOM: More on algebras

[Insall]

In my 1989 dissertation at the University of Houston, I devised the
following notions, which I had not previously seen in print, and which I
have not since seen elsewhere in print:

1.  An algebra A is locally embeddable in an algebra B if and only if each
of the finitely generated subalgebras of A is embeddable in B.

2.  Algebras A and B are locally isomorphic if and only if each is locally
embeddable in the other.

3.  Let A and B be algebras, with nonstandard enlargements *A and *B.  Then
A is internally embeddable in *B if A has an internal extension S in *A for
which there is an internal embedding of S into *B.

The following Theorems were proved:

Theorem 5 from page 24:  An algebra A is locally embeddable in an algebra B
if and only if A is internally embeddable in *B.

Theorem 6 from page 25:  Algebras A and B are locally isomorphic if and only
if there are hyperfinitely generated isomorphic subalgebras S of *A and T of
*B such that A is internally embeddable in S and B is internally embeddable
in T.

Theorem 7 from page 26:  If A and B are locally isomorphic, then *A and *B
are locally isomorphic.


(Note:  These are stated for algebras with only finitely many finitary
operations.  It is not difficult to verify that appropriate modifications
can be made to obtain similar results for algebras with infinitely many
operations, whether or not the list of those operations is considered to be
a well-ordered list.  In my dissertation is a short discussion of the
general problems that must be addressed to make such changes.)


Matt Insall
insall at umr.edu
(573)341-4901
montez at rollanet.org
(573)467-0948

Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla Missouri - USA

Where I ocassionally see a rainbow, or a bluebird, and the smiles on
children's faces are on the backdrop of the rolling Ozark foothills.