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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