Almost finite expansions of arbitrary semigroups.

*(English)*Zbl 0546.20055The 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.

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. |

##### Keywords:

almost finite semigroups; expansions; J-classes; almost finite expansions; free expansion; machine expansion
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\textit{J.-C. Birget} and \textit{J. Rhodes}, J. Pure Appl. Algebra 32, 239--287 (1984; Zbl 0546.20055)

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##### References:

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