  
  [1X1 [33X[0;0YLocal nearrings[133X[101X
  
  [33X[0;0YA  set  [23XR[123X with two binary operations [23X+[123X and [23X\cdot[123X is called a [13X(left) nearring[113X
  if the following statements hold:[133X
  
  [31X1[131X   [33X[0;6Y[23X(R,+)=R^+[123X is a (not necessarily abelian) group with neutral element [23X0[123X;[133X
  
  [31X2[131X   [33X[0;6Y[23X(R,\cdot)[123X is a semigroup;[133X
  
  [31X3[131X   [33X[0;6Y[23Xx(y+z)=xy+xz[123X for all [23Xx[123X, [23Xy[123X, [23Xz\in R[123X.[133X
  
  [33X[0;0YIf [23XR[123X is a nearring, then the group [23XR^+[123X is called the [13Xadditive group[113X of [23XR[123X. If
  in addition [23X0\cdot x=0[123X, then the nearring [23XR[123X is called [13Xzero-symmetric[113X, and if
  the semigroup [23X(R,\cdot)[123X is a monoid, i.e. it has an identity element [23Xi[123X, then
  [23XR[123X  is  a  [13Xnearring  with identity[113X [23Xi[123X. In the latter case the group [23XR^*[123X of all
  invertible  elements  of  the  monoid [23X(R,\cdot)[123X is called the [13Xmultiplicative
  group[113X of [23XR[123X.[133X
  
  [33X[0;0YThe concepts of a subnearring and a nearring homomorphism are defined by the
  same  way as for rings. In particular, if [23X\lambda[123X is a nearring homomorphism
  of  [23X(R,+,  \cdot)[123X,  then  its  kernel  [23XKer \lambda[123X is a subnearring of [23X(R,+,
  \cdot)[123X whose additive subgroup is normal in [23XR^+[123X.[133X
  
  [33X[0;0YA  subnearring  [23XI[123X  of  [23X(R,+,  \cdot)[123X  is an ideal of [23X(R,+, \cdot)[123X if [23XI = Ker
  \lambda[123X for some [23X\lambda[123X.[133X
  
  [33X[0;0YIt  can  simply  be  verified  that  [23XI[123X  is  an ideal of [23XR[123X if and only if its
  additive  group [23XI^+[123X is a normal subgroup of [23XR^+[123X and for any elements [23Xr[123X, [23Xs\in
  R[123X  and  [23Xa\in  I[123X  the  inclusions  [23Xra\in  I[123X and [23X(r + a)s − rs\in I[123X hold. Main
  results  accumulated for local nearrings can be found in the surveys [Sys08]
  and [RR25].[133X
  
  [33X[0;0YA  nearring  [23XR[123X with identity is said to be [13Xlocal[113X if the set [23XL=R\setminus R^*[123X
  of all non-invertible elements of [23XR[123X is a subgroup of [23XR^+[123X.[133X
  
  [33X[0;0YIt  is  clear  that if [23XL[123X is an ideal of [23XR[123X, then the factor nearring [23XR/L[123X is a
  [13Xnearfield[113X.  For  example,  every  local  ring  [23XR[123X  is  a zero-symmetric local
  nearring  whose  subgroup  [23XL[123X  coincides  with  the  Jacobson  radical  of [23XR[123X.
  Reference: [Max68].[133X
  
  
  [1X1.1 [33X[0;0YThe local nearrings library[133X[101X
  
  [1X1.1-1 AdditiveGroupsOfLibraryOfLNRsOfOrder[101X
  
  [33X[1;0Y[29X[2XAdditiveGroupsOfLibraryOfLNRsOfOrder[102X( [3Xn[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YThe  argument  is  [23Xn[123X.  The  output  is  a  list  of additive groups of local
  nearrings in the library of this package of order [23Xn[123X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList(AdditiveGroupsOfLibraryOfLNRsOfOrder(81),IdGroup);[127X[104X
    [4X[28X[ [ 81, 1 ], [ 81, 2 ], [ 81, 3 ], [ 81, 5 ], [ 81, 6 ], [ 81, 11 ], [128X[104X
    [4X[28X  [ 81, 12 ], [ 81, 13 ], [ 81, 15 ] ][128X[104X
  [4X[32X[104X
  
  [1X1.1-2 LibraryOfLNRsOnGroup[101X
  
  [33X[1;0Y[29X[2XLibraryOfLNRsOnGroup[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YThe argument is a group [23XG[123X. The output is a list of catalogue entries for the
  local  nearrings  in  the  library  of  this package whose additive group is
  isomorphic to [23XG[123X.[133X
  
  [33X[0;0YThe local nearrings are sorted by their multiplicative groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(81,2);[127X[104X
    [4X[28X<pc group of size 81 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XLibraryOfLNRsOnGroup(G);[127X[104X
    [4X[28X[ "AllLocalNearRings(81,2,54,3)", "AllLocalNearRings(81,2,54,6)", [128X[104X
    [4X[28X  "AllLocalNearRings(81,2,54,9)", "AllLocalNearRings(81,2,54,10)", [128X[104X
    [4X[28X  "AllLocalNearRings(81,2,54,11)", "AllLocalNearRings(81,2,54,15)", [128X[104X
    [4X[28X  "AllLocalNearRings(81,2,72,14)", "AllLocalNearRings(81,2,72,19)", [128X[104X
    [4X[28X  "AllLocalNearRings(81,2,72,24)", "AllLocalNearRings(81,2,72,26)" ][128X[104X
  [4X[32X[104X
  
  [1X1.1-3 LocalNearRing[101X
  
  [33X[1;0Y[29X[2XLocalNearRing[102X( [3Xk[103X, [3Xl[103X, [3Xm[103X, [3Xn[103X, [3Xw[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya nearring[133X
  
  [33X[0;0YThe  arguments are [23Xk[123X, [23Xl[123X, [23Xm[123X, [23Xn[123X, [23Xw[123X. The output is the [23Xw[123X-th local nearring from
  the library of this package whose additive group has [10XIdGroup[110X value [10X[k,l][110X and
  whose  multiplicative  group  has  [10XIdGroup[110X value [10X[m,n][110X. No validation of the
  arguments is performed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=LocalNearRing(81,12,54,8,3);[127X[104X
    [4X[28XExplicitMultiplicationNearRing ( <pc group of size 81 with [128X[104X
    [4X[28X4 generators> , multiplication )[128X[104X
  [4X[32X[104X
  
  [1X1.1-4 AllLocalNearRings[101X
  
  [33X[1;0Y[29X[2XAllLocalNearRings[102X( [3Xk[103X, [3Xl[103X, [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya list[133X
  
  [33X[0;0YThe  arguments are [23Xk[123X, [23Xl[123X, [23Xm[123X, [23Xn[123X. The output is the list of all local nearrings
  from  the  library  of  this  package whose additive group has [10XIdGroup[110X value
  [10X[k,l][110X  and whose multiplicative group has [10XIdGroup[110X value [10X[m,n][110X. No validation
  of the arguments is performed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=AllLocalNearRings(81,12,54,8);;[127X[104X
    [4X[25Xgap>[125X [27XSize(L);[127X[104X
    [4X[28X30[128X[104X
  [4X[32X[104X
  
  [1X1.1-5 NumberLocalNearRings[101X
  
  [33X[1;0Y[29X[2XNumberLocalNearRings[102X( [3Xk[103X, [3Xl[103X, [3Xm[103X, [3Xn[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10Ya number[133X
  
  [33X[0;0YThe arguments are [23Xk[123X, [23Xl[123X, [23Xm[123X, [23Xn[123X. The output is the number of local nearrings in
  the library of this package whose additive group has [10XIdGroup[110X value [10X[k,l][110X and
  whose  multiplicative  group  has  [10XIdGroup[110X value [10X[m,n][110X. No validation of the
  arguments is performed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNumberLocalNearRings(81,15,54,8);[127X[104X
    [4X[28X10[128X[104X
  [4X[32X[104X
  
  [1X1.1-6 IsAdditiveGroupOfLibraryOfLNRs[101X
  
  [33X[1;0Y[29X[2XIsAdditiveGroupOfLibraryOfLNRs[102X( [3XG[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Ya boolean[133X
  
  [33X[0;0YThe argument is a group [23XG[123X. The output is [10Xtrue[110X if the library of this package
  contains a local nearring whose additive group is isomorphic to [23XG[123X, and [10Xfalse[110X
  otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(25,2);[127X[104X
    [4X[28X<pc group of size 25 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsAdditiveGroupOfLibraryOfLNRs(G);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsAdditiveGroupOfLibraryOfLNRs(SmallGroup(81,14));[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
