For any set of symbols, denotes the set of all words of symbols over , including
the empty string . The set denotes the free monoid generated by under the operation of
concatenation with the empty string serving as identity.
Let R be a nite set. We de ne the binary relation )R
as follows, where u; v 2 : u)R
v
if there exist x; y 2 and (l;m) 2 R with u = xly and v = xmy. The structure
;)R
is a
reduction system of words and the relation )R
is the reduction relation. Let
;)R
be a reduction
system of words. The relation )R
is Noetherian if there is no in nite sequence w0;w1; ::: 2 such
that for all i 0;wi)R
wi+1.
In this paper, we study properties of reduction systems of words and give conditions under which
a reduction system of word is Noetherian.