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Almost finite expansions of arbitrary semigroups. (English) Zbl 0546.20055
The authors call general propositions global which are of the form: A semigroup S is a morphic image of a subsemigroup $$\bar S$$ of a semigroup X (i.e., S divides X and $$\bar S$$ is an expansion of S) and the oversemigroup X is built up from ”elementary” semigroups using ”elementary” operations.
Two expansions of S are considered. These constructions are formed so that some of the methods of finite semigroup theory are applicable. Firstly, a semigroup S is said to be almost finite or finite-J-above if for all s in S there exist only finitely many t in S with $$t\geq_ Js$$. In particular, every J-class of S is finite and each J-class has only finitely many J-classes above it. Two almost finite expansions are mentioned: The free expansion and the machine expansion. These are refined in Part II to expansions $$\bar S$$ which are almost finite and, moreover, close to the original S. In Part III some applications are mentioned.
Reviewer: T.J.Harju

##### MSC:
 20M10 General structure theory for semigroups 20M15 Mappings of semigroups 20M35 Semigroups in automata theory, linguistics, etc.
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