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The Burnside problem for groups asks whether the variety $x^n=1$ is locally finite. By work of Adian and Novikov they are not locally finite for $n$ odd and large enough (I think at least 667) and in the even case results are by Ivanov and Lysenok. For n=2,3,4,6 local finiteness is known. For n=5 it is unknown. Mark Sapir classified locally finite semigroup varieties modulo the group case.

Varieties generated by a finite algebra are locally finite by a result of Birkhoff.

Added. By Zelmanov's solution to the restricted Burnside problem a variety of groups is locally finite iff it is generated by a set of finite groups with uniformly bounded exponent. The analogue is false for semigroups.

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The Burnside problem for groups asks whether the variety $x^n=1$ is locally finite. By work of Adian and Novikov they are not locally finite for $n$ odd and large enough (I think at least 667) and in the even case results are by Ivanov and Lysenok. For n=2,3,4,6 local finiteness is known. For n=5 it is unknown. Mark Sapir classified locally finite semigroup varieties modulo the group case.

Varieties generated by a finite algebra are locally finite by a result of Birkhoff.