The near-ring library files can be used to systematically search for near-rings with (or without) certain properties.
For instance, the function LibraryNearring can be integrated
into a loop or occur as a part of the Filtered or the
First function to get all numbers of classes (or just the first class)
of near-rings on a specified group which have the specified properties.
In what follows, we shall give a few examples how this can be accomplished:
To get the number of the first class of near-rings on the group C_6 which have an identity, one could use a command like:
gap> First( [1..60], i -> > Identity( LibraryNearring( "C6", i ) ) <> [ ] ); 28
If we try the same with S_3, we will get an error message, indicating that there is no near-ring with identity on S_3:
gap> First( [1..39], i -> > Identity( LibraryNearring( "S3", i ) ) <> [ ] ); Error, at least one element of <list> must fulfill <func> in First( [ 1 .. 39 ], function ( i ) ... end ) called from main loop brk> gap>
To get all seven classes of near-rings with identity on the dihedral group D_8 of order 8, something like the following will work fine:
gap> l := Filtered( [1..1447], i -> > Identity( LibraryNearring( "D8", i ) ) <> [ ] ); [ 842, 844, 848, 849, 1094, 1096, 1097 ] gap> time; 122490
Note that a search like this may take a few minutes.
Here is another example that provides the class numbers of the four boolean near-rings on D_8:
gap> l := Filtered( [1..1447], i -> > IsBooleanNearring( LibraryNearring( "D8", i ) ) ); [ 1314, 1380, 1446, 1447 ]
The search for class numbers of near-rings can also be accomplished in a loop like the following:
gap> l:=[ ];; gap> for i in [1..1447] do > n := LibraryNearring( "D8", i ); > if IsDgNearring( n ) and > not IsDistributiveNearring( n ) then > Add( l, i ); > fi; > od; gap> time; 261580 gap> l; [ 765, 1094, 1098, 1306 ]
This provides a list of those class numbers of near-rings on D_8 which are distributively generated but not distributive.
Two or more conditions for library near-rings can also be combined with
or:
gap> l := [ ];;
gap> for i in [1..1447] do
> n := LibraryNearring( "D8", i );
> if Size( ZeroSymmetricElements( n ) ) < 8 or
> Identity( n ) <> [ ] or
> IsIntegralNearring( n ) then
> Add( l, i );
> fi;
> od;
gap> time;
124480
gap> l;
[ 842, 844, 848, 849, 1094, 1096, 1097, 1314, 1315, 1316, 1317, 1318,
1319, 1320, 1321, 1322, 1323, 1324, 1325, 1326, 1327, 1328, 1329,
1330, 1331, 1332, 1333, 1334, 1335, 1336, 1337, 1338, 1339, 1340,
1341, 1342, 1343, 1344, 1345, 1346, 1347, 1348, 1349, 1350, 1351,
1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1361, 1362,
1363, 1364, 1365, 1366, 1367, 1368, 1369, 1370, 1371, 1372, 1373,
1374, 1375, 1376, 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384,
1385, 1386, 1387, 1388, 1389, 1390, 1391, 1392, 1393, 1394, 1395,
1396, 1397, 1398, 1399, 1400, 1401, 1402, 1403, 1404, 1405, 1406,
1407, 1408, 1409, 1410, 1411, 1412, 1413, 1414, 1415, 1416, 1417,
1418, 1419, 1420, 1421, 1422, 1423, 1424, 1425, 1426, 1427, 1428,
1429, 1430, 1431, 1432, 1433, 1434, 1435, 1436, 1437, 1438, 1439,
1440, 1441, 1442, 1443, 1444, 1445, 1446, 1447 ]
gap> Length( l );
141
This provides a list of all 141 class numbers of near-rings on D_8 which are non-zerosymmetric or have an identity or are integral. (cf. Pil83, pp. 416ff).
The following loop does the same for the near-rings on the quaternion group of order 8, Q_8:
gap> l := [ ];; gap> for i in [1..281] do > n := LibraryNearring( "Q8", i ); > if Size( ZeroSymmetricElements( n ) ) < 8 or > Identity( n ) <> [ ] or > IsIntegralNearring( n ) then > Add( l, i ); > fi; > od; gap> time; 17740 gap> l; [ 280, 281 ]
Once a list of class numbers has been computed (in this case here: l),
e.g. the function Library-Near-ring-Info can be used to print
some information about these near-rings:
gap> LibraryNearringInfo( "Q8", l ); ---------------------------------------------------------------------- >>> GROUP: Q8 elements: [ n0, n1, n2, n3, n4, n5, n6, n7 ] addition table:tt+
| n0 n1 n2 n3 n4 n5 n6 n7
------------------------------
n0 tt | n0 n1 n2 n3 n4 n5 n6 n7
n1 tt | n1 n2 n3 n0 n7 n4 n5 n6
n2 tt | n2 n3 n0 n1 n6 n7 n4 n5
n3 tt | n3 n0 n1 n2 n5 n6 n7 n4
n4 tt | n4 n5 n6 n7 n2 n3 n0 n1
n5 tt | n5 n6 n7 n4 n1 n2 n3 n0
n6 tt | n6 n7 n4 n5 n0 n1 n2 n3
n7 tt |n7 n4 n5 n6 n3 n0 n1 n2group endomorphisms: 1: [ n0, n0, n0, n0, n0, n0, n0, n0 ] 2: [ n0, n0, n0, n0, n2, n2, n2, n2 ] 3: [ n0, n1, n2, n3, n5, n6, n7, n4 ] 4: [ n0, n1, n2, n3, n6, n7, n4, n5 ] 5: [ n0, n1, n2, n3, n7, n4, n5, n6 ] 6: [ n0, n2, n0, n2, n0, n2, n0, n2 ] 7: [ n0, n2, n0, n2, n2, n0, n2, n0 ] 8: [ n0, n3, n2, n1, n4, n7, n6, n5 ] 9: [ n0, n3, n2, n1, n5, n4, n7, n6 ] 10: [ n0, n3, n2, n1, n6, n5, n4, n7 ] 11: [ n0, n3, n2, n1, n7, n6, n5, n4 ] 12: [ n0, n4, n2, n6, n1, n7, n3, n5 ] 13: [ n0, n4, n2, n6, n3, n5, n1, n7 ] 14: [ n0, n4, n2, n6, n5, n1, n7, n3 ] 15: [ n0, n4, n2, n6, n7, n3, n5, n1 ] 16: [ n0, n5, n2, n7, n1, n4, n3, n6 ] 17: [ n0, n5, n2, n7, n3, n6, n1, n4 ] 18: [ n0, n5, n2, n7, n4, n3, n6, n1 ] 19: [ n0, n5, n2, n7, n6, n1, n4, n3 ] 20: [ n0, n6, n2, n4, n1, n5, n3, n7 ] 21: [ n0, n6, n2, n4, n3, n7, n1, n5 ] 22: [ n0, n6, n2, n4, n5, n3, n7, n1 ] 23: [ n0, n6, n2, n4, n7, n1, n5, n3 ] 24: [ n0, n7, n2, n5, n1, n6, n3, n4 ] 25: [ n0, n7, n2, n5, n3, n4, n1, n6 ] 26: [ n0, n7, n2, n5, n4, n1, n6, n3 ] 27: [ n0, n7, n2, n5, n6, n3, n4, n1 ] 28: [ n0, n1, n2, n3, n4, n5, n6, n7 ]
NEARRINGS: ---------------------------------------------------------------------- 280) phi: [ 1, 28, 28, 28, 28, 28, 28, 28 ]; 28; -B----I--P-RW ---------------------------------------------------------------------- 281) phi: [ 28, 28, 28, 28, 28, 28, 28, 28 ]; 28; -B----I--P-RW ----------------------------------------------------------------------
GAP 3.4.4