Applying rewriting methods to special monoids

Introduction A special monoid is a monoid presented by generators and defining relations of the form w = e, where w is a non-empty word on generators and e is the empty word. Groups are special monoids. But there exist special monoids that are not groups. Special monoids have been extensively studied by Adjanfl] and Makanin[7] (see also [2]). The present paper is a sequel to [11]. In [11], we showed that the conjugacy problem for a special monoid is reducible to the conjugacy problem for its group of units. Since the conjugacy problem is decidable for one-relator groups with torsion, we obtain that the conjugacy problem is decidable for one-relator special monoids with torsion. In this paper, we develop rewriting methods for approaching other standard problems of 'combinatorial semigroup theory'. The key point of rewriting methods is to consider defining relations of a monoid as a set of ordered rules rather than a set of equations and to find well behaved defining relations from the given ones. We apply the methods to obtain a structure theorem for finitely presented (f.p.) special monoids. In [10], Squier showed that the submonoid of right (left) units of a f.p. special Church-Rosser monoid is a free product of its group of units and a free monoid with finite rank. Here we relax the condition of being Church-Rosser. We also obtain simple proofs of two theorems about the word problem and the right (left) divisibility problem for f.p. special monoids, which are contained implicitly in [7]. The first one says that the decidability of the word problem for a f.p. special monoid coincides with that of its groups of units. Another says that right (left) divisibility problem is reducible to the word problem for f.p. special monoids. This paper is divided into six parts. In Section 1, we introduce some basic notions which are used in this paper. After proving some basic results in Section 2, we give a presentation theorem about the groups of units of f.p. special monoids (Theorem 3-7). It says that, from a finite presentation of a special monoid M, we can obtain a finite presentation of the group G(M) of units of M that has the same number of defining relations as that oiM. This result is used in the following sections. In Section 4, we exhibit a structure result of f.p. special monoids (Theorem 4-4 and Theorem 4-5). …