Talk:Completely regular semigroup


 * 1) Possible misrepresentation of the relation between completely regular semigroups and inverse semigroups

In the article you find:


 * "The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse semigroups being another such subclass."

This sounds as if the class of completely regular semigroups is more or less unrelated to the class of inverse semigroups, except that both are subclasses of the class of regular semigroups. However looking in the text referenced in the "Examples" section, I conclude that the class of completely regular semigroups is a subclass of the class of inverse semigroups:

If I don't misunderstand, the text says that inverse semigroups are characterized by the demand that every element has an inverse, while completely regular semigroups also fulfil that condition, but in addition require that an element commutes with its inverse.

Here's a quote of the relevant text from that link:


 * "The completely regular semigroups form a variety $$\underline{CR}$$ determined by the associative law for the multiplication together with the identities $$xx^{-1} x=x$$, $$(x^{-1})^{-1} =x$$ and $$x^{-1} x=xx^{-1}$$. The variety of inverse semigroups which also satisfies the identities $$xx^{-1} x=x$$ and $$(x^{-1})^{-1} =x$$ has been the focus of much attention in the past four decades."

If my interpretation is correct, I think the article text as currently written, while not actually wrong, is severely misleading and should therefore be changed. --132.199.99.50 (talk) 09:49, 9 January 2018 (UTC)