On finite complete rewriting systems and large subsemigroups

Abstract

Let S be a semigroup and T be a subsemigroup of finite index in S (that is, the set S \ T is finite). The subsemigroup T is also called a large subsemigroup of S. It is well known that if T has a finite complete rewriting system, then so does S. In this paper, we will prove the converse, that is, if S has a finite complete rewriting system, then so does T . Our proof is purely combinatorial and also constructive. Introduction.Let S be a semigroup and T be a subsemigroup of finite index in S (that is, the set S \ T is finite). Then T is called a large subsemigroup of S, and S is called a small extension of T .In [4], Ruškuc asked if S is a small extension of T , whether S has a finite complete rewriting system if and only if T has a finite complete rewriting system (see [4, Problem 11.1 (iii)] and [6, Remark 4.2]). This problem was partially solved by Wang in [5, Theorem 1], who proved that if T has a finite complete rewriting system, then so does S. However it is still not known whether T has a finite complete rewriting system or not, when S has a finite complete rewriting system. In this paper we shall prove that this is true, i.e., we shall prove the following: Theorem 1.1. Suppose S is a small extension of T . If S has a finite complete rewriting system, then so does T . By Theorem 1.1 and the result of Wang [5, Theorem 1], we have completely answered the problem posed by Ruškuc (see [4, Problem 11.1 (iii)]).Corollary 1.2. Suppose S is a small extension of T . Then S has a finite complete rewriting system if and only if T has a finite complete rewriting system.Let A be a non-empty set. This set A is called the alphabet and the elements of A are called letters. We shall denote the free semigroup and free monoid on A by A + and A * , respectively.The elements of A + and A * are called words. Note that A * = A + ∪ {ǫ}, where ǫ is the empty 2010 Mathematics Subject Classification. Primary 20M05, 20M35, 68Q42.

On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids

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Finite Complete Rewriting Systems for the Monoids M, ρ, and M/ρ

“…String rewriting systems can be regarded as the basis in the development of theoretical computer science and have been widely studied by researchers recently (Squier et al, 1994;Wang, 1998, Wong et al, 2010Gray, Malheiro, 2011). Max Dehn in (Dehn, 1911) introduced the word problem, whose solvability is one of the fundamental questions in combinatorial semigroup theory, for finitely presented groups.…”

Ideals and Finiteness Conditions for Subsemigroups

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