Tameness of pseudovariety joins
نویسنده
چکیده
The concept of tameness of a pseudovariety was introduced by Almeida and Steinberg [1] as a tool for proving decidability of the membership problem for semidirect products of pseudovarieties. Recall that the join V ∨ W of two pseudovarieties V and W is the least pseudovariety containing both V and W. This talk is concerned with the problem of proving tameness of joins. This problem was considered in [3] for joins with the pseudovariety K of semigroups in which the idempotents are left zeros. In a joint work with Almeida and Zeitoun [2] the same problem was treated for joins involving the pseudovarieties J and R of all J -trivial and R-trivial semigroups respectively. In particular, it was shown that the pseudovariety V ∨ W is κ-tame when V is a subpseudovariety of J with decidable κ-word problem and W is κ-tame. Here, κ denotes the canonical signature, consisting of the semigroup multiplication and the (ω − 1)power. Moreover, if W is a κ-tame pseudovariety which satisfies the pseudoidentity x1 · · ·xryzt = x1 · · ·xryzt, then it was proved that R ∨ W is also κ-tame. Since tameness implies decidability, this proves for instance the decidability of R ∨ G, where G is the pseudovariety of groups. These results are discussed in this talk.
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