Let (X, *) be a semigroup in which, for every a, b in X, a ≠ b implies a * b ≠ b * a. Which equalities hold? A. a * a = a for every a. B. a * b * a = a for all a, b. C. a * b * c = a * c for all a, b, c.
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Correct answer: D. A, B and C
Explanation
x = a * a commutes with a, so a * a = a (A). (aba)·a = a·(aba), so aba = a (B). Using A and B, abc and ac commute with each other and so are equal (C). All three hold (a rectangular band).
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