By Leeb B.

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**Extra resources for 3-manifolds with(out) metrics of nonpositive curvature**

**Example text**

10 Theorem It is possible to enlarge κ to an inﬁnite signature σ so that Gp is graph-tame with respect to this signature [6]. The added implicit operations are those of the form ϕω−1 (vi ) where ϕ ∈ End Ωn S is deﬁned by the n-tuple (w1 , . . , wn ) and the vi and wi are κ-terms such that Gp |= vi = wi and the determinant of the matrix (|wi |xj )i,j is invertible in Z/pZ. Here for a κ-term w and a letter x, |w|x is the integer obtained by viewing w as a group word and counting the signed number of occurrences of x in w.

2) is saturated by the congruence ∼N . Hence a language L ⊆ A+ is piecewise testable if and only if it is saturated by some congruence ∼N . 1). 1 Theorem (Simon [80]) A language L ⊆ A+ over a ﬁnite alphabet is piecewise testable if and only if it is J-recognizable. In terms of Eilenberg’s correspondence, Simon’s Theorem says that the variety of languages associated with J is the variety of piecewise testable languages. 2) suﬃce to separate points (cf. 8). In other words, implicit operations over J can be distinguished by looking at their subwords.

We are thus led to an alternative construction of ΩA V. 10 Theorem For a ﬁnite set A, the completion of the semigroup ΩA V with respect to the natural metric is a proﬁnite semigroup isomorphic to ΩA V. 9, ΩA V is a metric space under the natural metric, and its restriction to the dense subspace ΩA V is the natural metric of ΩA V. 4], it follows that the metric space (ΩA V, d) is the completion of (ΩA V, d). Proﬁnite semigroups and applications 17 It remains to show that the multiplication of the completion as deﬁned above coincides with ✷ the multiplication of ΩA V which follows from the continuity of multiplication in ΩA V.

### 3-manifolds with(out) metrics of nonpositive curvature by Leeb B.

by Donald

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