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# Finiteness Properties of Matrix Representations

John R. Stallings
Annals of Mathematics
Second Series, Vol. 124, No. 2 (Sep., 1986), pp. 337-346
DOI: 10.2307/1971282
Stable URL: http://www.jstor.org/stable/1971282
Page Count: 10
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## 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.

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