How Effects Efficiency on Word Problem for Monoids?

Abstract

In this paper, we partially find an answer to the question \lq\lq \textit{Is there a relationship between the algebraic properties efficiency (or inefficiency) and solvability of the word problem?}". In fact, by considering the semi-direct product on special monoids, we show that efficiency and inefficiency are not completely independent properties to prove the solvability of the word problem over monoids.