# Imbedding of semi-groups

*into groups*

A monomorphism of a semi-group into a group. A semi-group is imbeddable in a group if is isomorphic to a sub-semi-group of . Necessary and sufficient conditions for imbeddability of a semi-group into a group were found by A.I. Mal'tsev [1] (see also [3]). These conditions form an infinite system of conditional identities (or quasi-identities, cf. Quasi-identity), in particular, the following ones:

(cancellation laws);

where are elements of the semi-group. The class of semi-groups imbeddable in groups cannot be characterized by a finite system of conditional identities [2]. A number of sufficient conditions for imbeddability of a semi-group into a group are known. The most important ones are the following. If is a semi-group with cancellation and if for any elements of there exist elements such that (Ore's condition), then is imbeddable in a group. If is a semi-group with cancellation in which it always follows from the equality that either or for some element , then is imbeddable in a group [4]. Sufficient conditions for imbeddability, formulated in the language of graph theory (cf., for example, [5]), are known.

#### References

[1] | A.I. Mal'tsev, "On inclusion of associative systems in a group" Mat. Sb. , 6 (48) : 2 (1939) pp. 331–336 (In Russian) (German abstract) |

[2] | A.I. Mal'tsev, "On inclusion of associative systems in a group II" Mat. Sb. , 8 (50) : 2 (1940) pp. 251–264 (In Russian) |

[3] | P.M. Cohn, "Universal algebra" , Reidel (1981) |

[4] | R. Doss, "Sur l'immersion d'une semi-groupe dans une groupe" Bull. Sci. Math. (2) , 72 (1948) pp. 139–150 |

[5] | S.I. [S.I. Adyan] Adjan, "Defining relations and algorithmic problems for groups and semigroups" Proc. Steklov Inst. Math. , 85 (1967) Trudy Mat. Inst. Steklov. , 85 (1966) |

#### Comments

Instead of imbedding the word embedding is also used.

#### References

[a1] | A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. (1961–1967) |

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Imbedding of semi-groups.

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