Let be the variety of all commutative semigroups satisfying . Let be the subvariety generated by . If is a proper subvariety of , then there is a law that holds in that does not hold throughout . Using the identities of , we may reduce any such law to one of the form where , , and for all . Here a power of the form , with exponent , should be interpreted as the identity element of , which is .
Since does not hold in , there must be some index where the variables in these words have different exponents, say . Substitute the identity element for all variables except the jth, and substitute for . You obtain from that . But the possible powers of are all distinct: . This makes it impossible to have , , and .
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