Subject: Relating Rewriting Techniques on Monoids and Rings:
Congruences on Monoids and Ideals in Monoid Rings
ATTENTION JOUR ET HORAIRE SPECIAUX
Jour: 3/11/97 (Lundi 3 Novembre-16h)

           http://pauillac.inria.fr/bin/calendar/Seminaires


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                          I N R I A - Rocquencourt, 
                    Salle de confe'rence du batiment 11

                        Lundi 3 Novembre, 16h
                       ^^^^^^^^^^^^^^^^^^^^^^^

                         -------------------
                            Klaus Madlener
                         -------------------

          Fachbereich Informatik Universita"t Kaiserslautern

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         Relating Rewriting Techniques on Monoids and Rings:
          Congruences on Monoids and Ideals in Monoid Rings
     ============================================================

A first explicit connection between finitely presented commutative
monoids and ideals in polynomial rings was used 1958 by Emelichev
yielding a solution to the word problem in commutative monoids by
deciding the ideal membership problem.  The aim of this talk is to
show how congruences on monoids and groups can be characterized by
ideals in respective monoid and group rings.  These characterizations
allow to transfer well known results from the theory of string
rewriting systems for presenting monoids and groups to the algebraic setting
of subalgebras and ideals in monoid and group rings, respectively.
Moreover, natural one-sided congruences defined by subgroups of a
group are connected to one-sided ideals in the respective group ring and
hence the subgroup problem and the ideal membership problem are directly
related.  For several classes of finitely presented groups we show
explicitly how Gro"bner basis methods are related to existing solutions of
the subgroup problem that are based on rewriting methods.  For the
case of general monoids and submonoids weaker results are presented.
In fact it becomes clear that string rewriting methods for monoids and
groups can be lifted in a natural fashion to define reduction relations in
monoid and group rings.