# Inversion semi-group

*inverse semi-group*

A semi-group in which any element $a$ possesses a unique inverse element $a^{-1}$ (see Regular element). This property of a semi-group $S$ is equivalent to each of the following properties: $S$ is a regular semi-group and any two of its idempotents commute (thus the set of all idempotents of an inverse semi-group is a semi-lattice, see Idempotents, semi-group of); each left or right principal ideal of $S$ has a unique generating idempotent. Every group is an inverse semi-group; groups are the only inverse semi-groups with a unique idempotent. An important role in the study of inverse semi-groups is played by the following natural partial order relation ${\le}$ on an arbitrary inverse semi-group $S$: $a \le b$ if and only if $aa^{-1} = ab^{-1}$ ($a,b \in S$). On the semi-lattice of idempotents of an inverse semi-group this relation is the same as the natural partial order of this semi-lattice (see Idempotent). A semi-lattice of inverse semi-groups (see Band of semi-groups) is an inverse semi-group. The translation hull of an inverse semi-group (see Translations of semi-groups) is also an inverse semi-group [7]. Every congruence on an inverse semi-group is determined by the classes containing idempotents.

Let $J_X$ be the set of all one-to-one partial transformations of a set $X$ (including the "empty transformation" , taking the empty set to itself). Then $J_X$ is an inverse semi-group with respect to the operation of superposition, called the symmetric inverse semi-group on $X$. The Wagner–Preston theorem is of fundamental importance: Any inverse semi-group $S$ can be isomorphically imbedded in the symmetric inverse semi-group $J_S$.

The theory of inverse semi-groups is an important and deeply researched branch of the theory of semi-groups. Representations of inverse semi-groups by one-to-one partial transformations and matrices over a field have been studied (see [1]). Congruences on inverse semi-groups have been studied. Inverse semi-groups with finiteness conditions are being studied. Quite a number of important special types of inverse semi-groups have been singled out. The restrictions imposed on the majority of these bear the mark of simplicity in some sense (for example, bi-simplicity, see Simple semi-group), or relate to the semi-lattice of idempotents $E$, or are combinations of both types. The restrictions on $E$ may involve abstract properties of $E$ as a semi-lattice (for example, that $E$ be a certain type of chain) or certain relative properties of $E$ in the semi-group, in particular, the behaviour of $E$ with respect to certain congruences. There exists on any inverse semi-group $S$ a least congruence $\sigma$ with the property that $S/\sigma$ is a group (the least group congruence), namely $$ \sigma = \{ (a,b) : ae = be\ \text{for some}\ e \in E \} \ . $$

An inverse semi-group is called proper if $E$ constitutes a $\sigma$-class. There exists on any inverse semi-group $S$ a largest congruence $\mu$ separating idempotents, namely $$ \mu = \{ (a,b) : a^{-1}ea = b^{-1}eb\ \text{for any}\ e \in E \} $$ and $\mu$ is contained in the relation $\mathcal{H}$ (see Green equivalence relations); an inverse semi-group is called fundamental if $\mu$ is the same as the equality relation. Quite a number of structure theorems have been obtained for inverse semi-groups of the above-mentioned types, and in many instances the description of inverse semi-groups is effected "modulo groups" ; the groups emerge as blocks of various structures in which semi-lattices, group homomorphisms, etc. also participate. Of this type, for example, are the typical descriptions of Clifford inverse semi-groups (see Clifford semi-group) and the completely $O$-simple inverse semi-groups (see Brandt semi-group).

Inverse semi-groups can also be regarded as universal algebras with two operations: the binary operation of multiplication and the unary operation of taking the inverse element. A classification has been obtained of the monogenic (that is, generated by a single element) inverse semi-groups as universal algebras [6], [9]. With respect to the above operations the class of all inverse semi-groups is a variety; it can be defined, for example, by the following system of identities [8]: $$ x(yz) = (xy)z\,;\ \ (x^{-1})^{-1} = x\,;\ \ xx^{-1}x = x\,; $$ $$ (xy)^{-1} = y^{-1}x^{-1}\,;\ \ xx^{-1}yy^{-1} = yy^{-1}xx^{-1}\ . $$

#### References

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

[2] | E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) |

[3] | A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian) |

[4] | V.V. Vagner, "Generalized groups" Dokl. Akad. Nauk SSSR , 84 : 6 (1952) pp. 1119–1122 (In Russian) |

[5] | G.B. Preston, "Inverse semigroups" J. London Math. Soc. , 29 : 4 (1954) pp. 396–403 |

[6] | L.M. Gluskin, "Elementary generalized groups" Mat. Sb. , 41 : 1 (1957) pp. 23–36 (In Russian) |

[7] | I.S. Ponizovskii, "Remark on inverse semigroups" Uspekhi Mat. Nauk , 20 : 6 (1965) pp. 147–148 (In Russian) |

[8] | B.M. Shain, "On the theory of generalized groups and generalized heaps" , The theory of semigroups and its applications , 1 , Saratov (1965) pp. 286–324 (In Russian) |

[9] | T.I. Ershova, "Monogenic inverse semigroups" Mat. Zap. Ural'sk. Univ. , 8 : 1 (1971) pp. 30–33 (In Russian) |

[10] | W.D. Munn, "Some recent results on the structure of inverse semigroups" K.W. Folley (ed.) , Semigroups , Acad. Press (1969) pp. 107–123 |

[11] | L. O'Carroll, "Embedding theorems for proper inverse semigroups" J. of Algebra , 42 (1976) pp. 26–40 |

[12] | M. Petrich, "Inverse semigroups" , Wiley (1984) |

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Inversion semi-group.

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