Innnite Convergent String-rewriting Systems and Cross-sections for Monoids

A nitely presented monoid has a decidable word problem if and only if it can be presented by some left-recursive convergent string-rewriting system if and only if it has a recursive cross-section. However, context-free cross-sections or even regular cross-sections do not suuce. This is shown by presenting examples of nitely presented monoids with decidable word problems that do not admit regular cross-sections, and that, hence, cannot be presented by left-regular convergent string-rewriting systems. Also examples of nitely presented monoids with decidable word problems are presented that do not even admit context-free cross-sections. In the concluding section we shortly address the notion of left-regular convergent string-rewriting systems that are tractable.