‣ IsAntiassociative( M ) | ( property ) |
Returns: true or false
identifies whether magma M is antiassociative.
gap> IsAntiassociative(OneSmallGroup(16)); false gap> IsAntiassociative(OneSmallAntimagma(2)); true gap> IsAntiassociative(OneSmallAntimagma(3)); true
‣ IsLeftCyclic( M ) | ( property ) |
Returns: true or false
if magma is left cyclic m.
‣ IsRightCyclic( M ) | ( property ) |
Returns: true or false
if magma is left cyclic m.
‣ IsLeftCancellative( M ) | ( property ) |
Returns: true or false
if magma is left cancellative m.
gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> Display( MultiplicationTable(M) ); [ [ 2, 1 ], [ 2, 1 ] ] gap> IsRightCancellative(M); false gap> IsLeftCancellative(M); true gap> List(AllSmallAntimagmas(2), M -> IsLeftCancellative(M)); [ true, false ]
‣ IsRightCancellative( M ) | ( property ) |
Returns: true or false
if magma is right cancellative m.
gap> List(AllSmallAntimagmas(2), M -> IsRightCancellative(M)); [ false, true ]
‣ IsCancellative( M ) | ( property ) |
Returns: true or false
if magma is cancellative m.
gap> List(AllSmallAntimagmas(2), M -> IsCancellative(M)); [ false, false ]
‣ IsLeftFPFInducted( M ) | ( property ) |
Returns: true or false
is a left-hand sided fixed-point free inducted m.
gap> Display( MultiplicationTable( SmallAntimagma(2, 2) ) ); [ [ 2, 2 ], [ 1, 1 ] ] gap> IsLeftFPFInducted( SmallAntimagma(2, 2) ); true
‣ IsRightFPFInducted( M ) | ( property ) |
Returns: true or false
is a right-hand sided fixed-point free inducted m.
gap> Display( MultiplicationTable( SmallAntimagma(2, 1) ) ); [ [ 2, 1 ], [ 2, 1 ] ] gap> IsRightFPFInducted( SmallAntimagma(2, 1) ); true
‣ IsLeftDerangementInducted( M ) | ( property ) |
Returns: true or false
is a left-hand sided derangment inducted m.
gap> M := SmallAntimagma(2, 2); <magma with 2 generators> gap> IsLeftFPFInducted(M); true gap> IsRightFPFInducted(M); false gap> IsRightDerangementInducted(M); false
‣ IsRightDerangementInducted( M ) | ( property ) |
Returns: true or false
is a right-hand sided derangment inducted m.
gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> IsLeftFPFInducted(M); false gap> IsRightFPFInducted(M); true gap> IsRightDerangementInducted(M); true
‣ IsLeftAlternative( M ) | ( property ) |
Returns: true or false
is a left-alternatve magma M.
‣ IsRightAlternative( M ) | ( property ) |
Returns: true or false
is a right-alternatve magma M.
‣ AssociativityIndex( M ) | ( attribute ) |
identifies associativity index of M.
gap> OneSmallAntimagma(2); <magma with 2 generators> gap> AssociativityIndex(OneSmallAntimagma(2)); 0 gap> OneSmallGroup(4); <pc group of size 4 with 2 generators> gap> AssociativityIndex(OneSmallGroup(4)); 64 gap> AssociativityIndex(OneSmallGroup(4)) = 4 ^ 3; true
‣ DiagonalOfMultiplicationTable( M ) | ( attribute ) |
computes diaognal of multiplication table of M.
gap> List(AllSmallAntimagmas(3), M -> DiagonalOfMultiplicationTable((M))); [ [ 2, 1, 1 ], [ 2, 1, 1 ], [ 2, 3, 2 ], [ 2, 1, 1 ], [ 2, 1, 1 ], [ 2, 1, 2 ], [ 2, 3, 2 ], [ 2, 1, 2 ], [ 2, 3, 1 ], [ 2, 3, 1 ] ]
‣ CommutativityIndex( M ) | ( attribute ) |
identifies commutativity index of M.
‣ AnticommutativityIndex( M ) | ( attribute ) |
calculates anticommutativity index of M.
‣ SquaresIndex( M ) | ( attribute ) |
computes squares index of M so the order of \(\left\{ m^2 | m \in M \right\}\).
gap> List(AllSmallAntimagmas(2), M -> List(M, m -> m * m) ); [ [ m2, m1 ], [ m2, m1 ] ] gap> List(AllSmallAntimagmas(2), M -> SquaresIndex(M )); [ 2, 2 ] gap> List(AllSmallAntimagmas(3), M -> SquaresIndex(M )); [ 2, 2, 2, 2, 2, 2, 2, 2, 3, 3 ]
‣ IdSmallAntimagma( M ) | ( attribute ) |
identifies class of antiassociative magma M.
gap> IsAntiassociative(OneSmallGroup(16)); false gap> IsAntiassociative(OneSmallAntimagma(2)); true gap> IsAntiassociative(OneSmallAntimagma(3)); true
‣ LeftOrder( [m] ) | ( attribute ) |
returns a left order of element m.
‣ RightOrder( [m] ) | ( attribute ) |
returns a right order of element m.
‣ LeftOrdersOfElements( [m] ) | ( attribute ) |
returns a left order of element m.
‣ RightOrdersOfElements( [m] ) | ( attribute ) |
returns a left order of element m.
‣ AllSubmagmas( M ) | ( function ) |
builds a collection of non-isomorphic submagmas of M.
gap> AllSmallAntimagmas(2); [ <magma with 2 generators>, <magma with 2 generators> ] gap> List(AllSmallAntimagmas(2), M -> AllSubmagmas(M)); [ [ <magma with 1 generator> ], [ <magma with 1 generator> ] ]
‣ MagmaIsomorphismInvariantsMatch( M ) | ( function ) |
computes isomorphism invariants of M.
‣ IsMagmaIsomorphic( M, N ) | ( function ) |
identifies whether magmas M, N are isomorphic.
gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> N := SmallAntimagma(2, 2); <magma with 2 generators> gap> T := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]); <magma with 2 generators> gap> IsMagmaIsomorphic(M, M); true gap> IsMagmaIsomorphic(M, T); true gap> IsMagmaIsomorphic(M, N); false
‣ IsMagmaAntiisomorphic( [M, N] ) | ( function ) |
identifies whether magmas M, N are antiisomorphic.
gap> N := SmallAntimagma(2, 1); <magma with 2 generators> gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> N := SmallAntimagma(2, 2); <magma with 2 generators> gap> IsMagmaAntiisomorphic(M, M); false gap> IsMagmaAntiisomorphic(M, N); true gap> IsMagmaAntiisomorphic(M, TransposedMagma(M)); true
‣ TransposedMagma( [M] ) | ( function ) |
generates transposed magma M.
gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> IsMagmaAntiisomorphic(M, TransposedMagma(M)); true gap> IsMagmaIsomorphic(M, TransposedMagma(TransposedMagma(M))); true gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> Display(MultiplicationTable(M)); [ [ 2, 1 ], [ 2, 1 ] ] gap> Display(MultiplicationTable(TransposedMagma(M))); [ [ 2, 2 ], [ 1, 1 ] ]
‣ LeftPower( [m, k] ) | ( function ) |
returns a left \(k\)-power of element m.
‣ RightPower( [m, k] ) | ( function ) |
returns a right \(k\)-power of element m.
‣ AllSmallAntimagmas( n ) | ( function ) |
returns all antiassociative magmas of specified size n (a number)
gap> AllSmallAntimagmas(2); [ <magma with 2 generators>, <magma with 2 generators> ] gap> AllSmallAntimagmas(3); [ <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators> ]
‣ NrSmallAntimagmas( n ) | ( function ) |
counts number of antiassociative magmas of specified size n (a number).
gap> NrSmallAntimagmas(2); 2 gap> NrSmallAntimagmas(3); 10 gap> NrSmallAntimagmas(4); 17780
‣ SmallAntimagma( n, i ) | ( function ) |
returns antiassociative magma of id [n, i].
gap> SmallAntimagma(2, 1); <magma with 2 generators> gap> SmallAntimagma(4, 5); <magma with 4 generators>
‣ OneSmallAntimagma( n ) | ( function ) |
returns a random antiassociative magma of size n.
gap> OneSmallAntimagma(2); <magma with 2 generators> gap> OneSmallAntimagma(3); <magma with 3 generators>
‣ ReallyAllSmallAntimagmas( n ) | ( function ) |
returns really-all antiassociative magmas, isomorphic, of specified size n (a number)
gap> ReallyAllSmallAntimagmas(2); [ <magma with 2 generators>, <magma with 2 generators> ]
‣ ReallyNrSmallAntimagmas( n ) | ( function ) |
counts number of antiassociative magmas of specified size n (a number)
gap> ReallyNrSmallAntimagmas(3); 52
‣ AntimagmaGeneratorPossibleDiagonals( n ) | ( function ) |
returns all possible diagonals of multiplication table for [n]-antimagma.
gap> AntimagmaGeneratorPossibleDiagonals(2); [ [ 2, 1 ] ] gap> AntimagmaGeneratorPossibleDiagonals(3); [ [ 2, 1, 1 ], [ 2, 1, 2 ], [ 2, 3, 1 ], [ 2, 3, 2 ], [ 3, 1, 1 ], [ 3, 1, 2 ], [ 3, 3, 1 ], [ 3, 3, 2 ] ]
‣ AntimagmaGeneratorFilterNonIsomorphicMagmas( Ms ) | ( function ) |
filters non-isomorphic magmas m.
‣ MagmaIsomorphism( M, N ) | ( operation ) |
computes an isomoprhism between magmas M, N.
gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> N := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]); <magma with 2 generators> gap> MagmaIsomorphism(M, N); <mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
‣ MagmaAntiisomorphism( M, N ) | ( operation ) |
creates an antiisomoprhism between magmas M, N.
gap> M := SmallAntimagma(2, 1); <magma with 2 generators> gap> N := SmallAntimagma(2, 2); <magma with 2 generators> gap> MagmaAntiisomorphism(M, N); <mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
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