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Principal quasi-ideals of cohomological dimension 1. (English) Zbl 1037.20060
The principal quasi-ideal generated by an element \(w\) of the semigroup \(S\) is the set \(\langle w\rangle_q=S^1w\cap wS^1\). The cohomological dimension of the semigroup \(S\) is the smallest integer \(n\) such that for any \(S\)-module \(A\) and \(k>n\) the \(k\)-th cohomology group \(H^k(S,A)\) is trivial. It is proved that the principal quasi-ideal \(\langle w\rangle_q\) of a free non-commutative semigroup has cohomological dimension 1 if and only if it is free as a semigroup.
MSC:
20M05 Free semigroups, generators and relations, word problems
20M50 Connections of semigroups with homological algebra and category theory
20M12 Ideal theory for semigroups
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