Atlas: A representation theorem for $d$-rings by Karim Boulabiar

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Sixth Annual Conference in Ordered Algebraic Structures
March 5-8, 2003
Department of Mathematics, Vanderbilt University
Nashville, TN, USA

Organizers
Jorge Martinez, University of Florida and Constantine Tsinakis, Vanderbilt University

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A representation theorem for d-rings
by
Karim Boulabiar
University of Carthage (Tunisia)

Let C( X) be the f-ring of all real-valued continuous functions on a locally compact Hausdorff topological space X and let * be an associative multiplication in C( X) . It is shown that C( X) is a d-ring with respect to * if and only if there exist functions h, k from X into itself, which are continuous on the cozero-set of e * e, such that

( f * g) ( x) = ( e * e) (x) ( f o h) ( x) ( g o k)( x)

for all f, g in C( X) and x in X (where e(x) = 1 for all x in X). As a generalization, the following result is obtained. Let R be an archimedean f-ring with identity element e and let * be another associative multiplication in R. Then R is a d-ring with respect to * if and only if there exist l-ring homomorphisms j and \psi from R into the maximal ring of quotients Q( R) of R such that

f * g=( e * e) j( f) \psi(bg)

for all f, g in R.

Date received: December 3, 2002