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A set Σ of pseudoidentities such that V = [[Σ]] is called a basis of pseudoidentities for V. The pseudovariety V will be called finitely based if it admits a finite basis of pseudoidentities. To give examples illustrating Reiterman’s Theorem, we now describe some important unary implicit operations on finite semigroups. There are several equivalent ways to describe them so we will choose one which is economical in the sense that it requires essentially no verification. +k )n becomes constant for n sufficiently large, namely n > max{|k|, |S|} suffices.

2 Theorem Let C be a recursively enumerable set of finite systems of equations with rational constraints and suppose V is a C-tame pseudovariety. Then it is decidable whether a given Σ ∈ C is V-inevitable. Proof Let V be C-tame with respect to an implicit signature σ. To prove the theorem it suffices to effectively enumerate those Σ ∈ C that are V-inevitable and those that are not. One can start by enumerating all systems Σ ∈ C. Since V is C-tame with respect to σ, if Σ is V-inevitable then there is a solution ψ : X → ΩA S for Σ in ΩA V that takes its values in ΩσA S.

1, it follows that there exists a factorization ψ(x) = uv ω w with u, v, w ∈ ΩA S. Since the constraint for x translates into a condition of the form ψ(x) belongs to a given clopen subset of ΩA S and ψ(x) is zero in ΩA N, we may replace u, v, w by words. This changes ψ(x) to a κ-term by maintaining a solution in ΩA N. See [7] for details. 4 Theorem (Almeida and Delgado [13]) The pseudovariety Ab of all finite Abelian groups is completely tame with respect to κ. 4 amounts to linear algebra over the profinite completion Z of the ring of integers, which we have already observed to be isomorphic with Ω1 G under multiplication and composition.

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