Suppose a semigroup (possibly infinite) presentation is given with generating set $S$ and relations $R$. I need to prove using Bergman's diamond lemma that the semigroup is non-zero i.e, I have to give normal forms of elements of the semigroup. Suppose I could guess the set of irreducible elements and I have also an order on the set of generators $S$. How do I prove that using the diamond lemma that this set is actually a set of reduced words ? How do I find all the ambiguities ?
2026-03-27 21:34:40.1774647280
semigroup presentation and Diamond lemma
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In general, the word problem for a semigroup $T$ defined by a presentation $\langle\: S\:|\:R\:\rangle$ can be undecidable. This means that, in some cases, if you are given two words over the generators $S$, there is no algorithm to decide whether or not these words represent the same element of $T$. Hence, in such cases, there is no way to find normal forms for the elements of $T$ (if there was, then this would be an algorithm for deciding if two words were equal, just calculate the normal forms and check if they are equal).
If you have a specific presentation for which you want to find the normal forms, then it might be possible to answer your question, but without the details of the presentation, it is not.