A set \(R\) with two binary operations \(+\) and \(\cdot\) is called a *(left) nearring* if the following statements hold:

\((R,+)=R^+\) is a (not necessarily abelian) group with neutral element \(0\);

\((R,\cdot)\) is a semigroup;

\(x(y+z)=xy+xz\) for all \(x\), \(y\), \(z\in R\).

If \(R\) is a nearring, then the group \(R^+\) is called the *additive group* of \(R\). If in addition \(0\cdot x=0\), then the nearring \(R\) is called *zero-symmetric*, and if the semigroup \((R,\cdot)\) is a monoid, i.e. it has an identity element \(i\), then \(R\) is a *nearring with identity* \(i\). In the latter case the group \(R^*\) of all invertible elements of the monoid \((R,\cdot)\) is called the *multiplicative group* of \(R\).

A nearring \(R\) with identity is said to be *local* if the set \(L=R\setminus R^*\) of all non-invertible elements of \(R\) is a subgroup of \(R^+\).

It is clear that if \(L\) is an ideal of \(R\), then the factor nearring \(R/L\) is a *nearfield*. For example, every local ring \(R\) is a zero-symmetric local nearring whose subgroup \(L\) coincides with the Jacobson radical of \(R\).

`‣ TheAdditiveGroupsOfLibraryOfLNRsOfOrder` ( n ) | ( function ) |

Returns: a list

The argument is \(n\). The output a list of `IdGroup`

of the additive groups of local nearrings from `Library`

of order \(n\).

gap> List(TheAdditiveGroupsOfLibraryOfLNRsOfOrder(81),IdGroup); [ [ 81, 1 ], [ 81, 2 ], [ 81, 3 ], [ 81, 5 ], [ 81, 6 ], [ 81, 11 ], [ 81, 12 ], [ 81, 13 ], [ 81, 15 ] ]

`‣ TheLibraryOfLNRsOnGroup` ( G ) | ( function ) |

Returns: a list

The argument is a group \(G\). The output a list of the catalogues of local nearrings from `Library`

on \(G\).

The local nearrings are sorted by their multiplicative groups.

gap> G:=SmallGroup(81,2); <pc group of size 81 with 4 generators> gap> TheLibraryOfLNRsOnGroup(G); [ "AllLocalNearRings(81,2,54,3)", "AllLocalNearRings(81,2,54,6)", "AllLocalNearRings(81,2,54,9)", "AllLocalNearRings(81,2,54,10)", "AllLocalNearRings(81,2,54,11)", "AllLocalNearRings(81,2,54,15)", "AllLocalNearRings(81,2,72,14)", "AllLocalNearRings(81,2,72,19)", "AllLocalNearRings(81,2,72,24)", "AllLocalNearRings(81,2,72,26)" ]

`‣ LocalNearRing` ( k, l, m, n, w ) | ( operation ) |

Returns: a nearring

The arguments are \(k\), \(l\), \(m\), \(n\), \(w\). The output is local nearring from `Library`

without check. The arguments \(k\), \(l\), \(m\), \(n\) are from IdGroup of the additive group and the multiplicative group, respectively, \(w\) is the position in the list.

gap> L:=LocalNearRing(81,12,54,8,3); ExplicitMultiplicationNearRing ( <pc group of size 81 with 4 generators> , multiplication )

`‣ AllLocalNearRings` ( k, l, m, n ) | ( operation ) |

Returns: a list

The arguments are \(k\), \(l\), \(m\), \(n\). The output are all local nearrings from `Library`

without check. The arguments \(k\), \(l\), \(m\), \(n\) are as above.

gap> L:=AllLocalNearRings(81,12,54,8);; gap> Size(L); 30

`‣ IsAdditiveGroupOfLibraryOfLNRs` ( G ) | ( function ) |

Returns: a boolean

The argument is a group \(G\). The output is `true`

if in `Library`

there exists a local nearring whose additive group is isomorphic to \(G\) otherwise the output is `false`

.

gap> G:=SmallGroup(25,2); <pc group of size 25 with 2 generators> gap> IsAdditiveGroupOfLibraryOfLNRs(G); true gap> IsAdditiveGroupOfLibraryOfLNRs(SmallGroup(81,14)); false

`‣ InfoLocalNearRing` ( G ) | ( function ) |

Returns: information

The argument is a group \(G\). The output some information about local nearrings from `Library`

on \(G\).

gap> InfoLocalNearRing(SmallGroup(361,2)); The local nearrings are sorted by their multiplicative groups. [ "AllLocalNearRings(361,2,342,1) (2)", "AllLocalNearRings(361,2,342,2) (2)", \ "AllLocalNearRings(361,2,342,4) (1)", "AllLocalNearRings(361,2,342,6) (1)", "AllLocalNearRings(361,2,342,7) (7)",\ "AllLocalNearRings(361,2,342,8) (6)", "AllLocalNearRings(361,2,360,4) (1)", "AllLocalNearRings(361,2,360,15) (1)"\ ]

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