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Normal Forms for Free Aperiodic Semigroups
 Int. J. Algebra Comput
, 1999
"... The implicit operation # is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates. Using # there is a welldefined algebra which is known as the free aperiodic semigroup. In this article we show that for each n, the ..."
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The implicit operation # is the unary operation which sends each element of a finite semigroup to the unique idempotent contained in the subsemigroup it generates. Using # there is a welldefined algebra which is known as the free aperiodic semigroup. In this article we show that for each n, the n generated free aperiodic semigroup is defined by a finite list of pseudoidentities and has a decidable word problem. In the language of implicit operations, this shows that the pseudovariety of finite aperiodic semigroups is #recursive. This completes a crucial step towards showing that the KrohnRhodes complexity of every finite semigroup is decidable.
Maximal Groups in Free Burnside Semigroups
, 1998
"... . We prove that any maximal group in the free Burnside semigroup defined by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a gr ..."
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. We prove that any maximal group in the free Burnside semigroup defined by the equation x n = x n+m for any n 1 and any m 1 is a free Burnside group satisfying x m = 1. We show that such group is free over a well described set of generators whose cardinality is the cyclomatic number of a graph associated to the J class containing the group. For n = 2 and for every m 2 we present examples with 2m \Gamma 1 generators. Hence, in these cases, we have infinite maximal groups for large enough m. This allows us to prove important properties of Burnside semigroups for the case n = 2, which was almost completely unknown until now. Surprisingly, the case n = 2 presents simultaneously the complexities of the cases n = 1 and n 3: the maximal groups are cyclic of order m for n 3 but they can have more generators and be infinite for n 2; there are exactly 2 jAj J classes and they are easily characterized for n = 1 but there are infinitely many J classes and they are difficult to c...
Free Burnside Semigroups
, 1999
"... This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years. ..."
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This paper surveys the area of Free Burnside Semigroups. The theory of these semigroups, as is the case for groups, is far from being completely known. For semigroups, the most impressive results were obtained in the last 10 years.