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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