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Semigroup cohomologies: a survey. (Russian. English summary) Zbl 1014.20039
The paper contains a survey of results on the theory of semigroup cohomologies and their applications to homological algebra and semigroup theory. Preferences are given to methods that allow to raise knowledge on semigroup cohomologies from certain subsemigroups of a given semigroup. At the end of the survey some open problems are presented, in particular, the following ones: (1) Is it true that the cohomological dimension of a subsemigroup \(S\) of a finitely generated free semigroup is at most the maximum of the ranks of all free semigroups that contain \(S\)? (2) The same question for subsemigroups of a free semigroup. (3) Similarly with group theory is it possible to define homogeneous cohomologies for semigroups? What are their properties? Does there exist a nice interpretation for them in small dimensions? (4) What are the properties for 0-cohomologies of finite nilpotent semigroups?

20M50 Connections of semigroups with homological algebra and category theory