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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