Talk:Nowhere commutative semigroup
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Band not semigroup
[edit]"A semigroup S is nowhere commutative if and only if any two elements of S are inverses of each other.[1]"
Perhaps this should say "band" not semigroup. The citation of C&P p26 is incorrect. Exercise 1.8.1 on that page says "A rectangular band is nowhere commutative".
On p33 Excercise 1.9.3 says "A semigroup S has the property that any two elements are inverses of each other if and only if S is nowhere commutative". If that exercise can be proved the citation should be changed from p26 to Exercise 1.9.3 on p33.
I have just edited the section on Rectangular Bands in page on Band (algebra) to include the proof (from Howie 1976 proposition 3.2 p96). But note that one direction requires idempotence (the equivalence from D to A in prop 3.2 depends on idempotence in D). p97 remarks "Exercise 9 below gives an example of a semigroup (containing six elements) in which abc = ac for every a, b and c but which is not a band.
Exercise 9 is at p126. I do not know where to find any corresponding material in Howie 1995.
Hopefully somebody else could fix my clumsiness and should consider whether to move stuff from that section on Rectangular Bands to Nowhere commutative semigroups (or bands) or merge the two or move stuff from both to a page on Rectangular Bands. — Preceding unsigned comment added by ArthurD8 (talk • contribs) 18:11, 25 November 2021 (UTC)