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Journal of the London Mathematical Society Advance Access originally published online on March 4, 2009
Journal of the London Mathematical Society 2009 79(2):465-477; doi:10.1112/jlms/jdn083
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© 2009 London Mathematical Society

Rational codes and free profinite monoids

Jorge Almeida

Departamento de Matemática Pura
Faculdade de Ciências
Universidade do Porto
Rua do Campo Alegre 687
4169-007 Porto
Portugal
jalmeida@fc.up.pt

Benjamin Steinberg

School of Mathematics and Statistics
Carleton University
1125 Colonel By Drive
Ottawa, ON
Canada
K1S 5B6

It is well known that clopen subgroups of finitely generated free profinite groups are again finitely generated free profinite groups. Clopen submonoids of free profinite monoids need not be finitely generated nor free. Margolis, Sapir and Weil proved that the closed submonoid generated by a finite code (which is, in fact, clopen) is a free profinite monoid generated by that code. In this note we show that a clopen submonoid is free profinite if and only if it is the closure of a rational free submonoid. In this case its unique closed basis is clopen, and is, in fact, the closure of the corresponding rational code. More generally, our results apply to free pro-Formula monoids for H an extension-closed pseudovariety of groups.

2000 Mathematics Subject Classification 20M05 (primary), 20M07, 20M35, 68Q45, 94A45 (secondary).

The first author was partially supported by the Centro de Matemática da Universidade do Porto, financed by FCT through the programmes POCTI and POSI, with Portuguese and European Community structural funds, and by the FCT project PTDC/MAT/65481/2006. The second author gratefully acknowledges the support of an NSERC discovery grant.

Received May 24, 2008; revised October 21, 2008; published online March 4, 2009.


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