Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Finiteness Properties of Matrix Representations

John R. Stallings
Annals of Mathematics
Second Series, Vol. 124, No. 2 (Sep., 1986), pp. 337-346
Published by: Annals of Mathematics
DOI: 10.2307/1971282
Stable URL: http://www.jstor.org/stable/1971282
Page Count: 10
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Finiteness Properties of Matrix Representations
Preview not available

Abstract

Let \Phi be a collection of interger polynomials in the n2 variables $\lbrace x_{ij} \rbrace$. Then \Phi can be considered a collection of functions from Mn(A) to A, where A is a commutative ring, and Mn(A) is the set of n × n matrices over A. A set $X \subset M_n(A) is said to satisfy \Phi, when all these functions are 0 on X. The general result is that if S is any finitely generated monoid, $P \subset S an arbitrary subset, and if n is given, then there is a finite subset Q \subset P such that for every monoid homorphism: \alpha \colon S \rightarrow M_n(A), if \alpha (Q) satisfies \Phi, then \alpha(P) satisfies \Phi. This has several consequences. For instance, if S is a finitely generated submonoid of Mn(A), and α : S → S an endomorphism, then the set of elements left fixed by some (variable) power of \alpha is left fixed by some single power of \alpha. The Proof is an expansion of an idea due to V. S. Guba for proving Ehrebfeucht'sConjecture that the equalizer of maps of finitely generated free monoids is determined by a finite seet. It uses elementary constructions and the Hilbert Basis Theorem.

Page Thumbnails

  • Thumbnail: Page 
[337]
    [337]
  • Thumbnail: Page 
338
    338
  • Thumbnail: Page 
339
    339
  • Thumbnail: Page 
340
    340
  • Thumbnail: Page 
341
    341
  • Thumbnail: Page 
342
    342
  • Thumbnail: Page 
343
    343
  • Thumbnail: Page 
344
    344
  • Thumbnail: Page 
345
    345
  • Thumbnail: Page 
346
    346