#! @Arguments M
#! @Description
#! identifies whether magma <A>M</A> is antiassociative.
#!
#! @BeginExampleSession
#! gap> IsAntiassociative(OneSmallGroup(16));
#! false
#! gap> IsAntiassociative(OneSmallAntimagma(2));
#! true
#! gap> IsAntiassociative(OneSmallAntimagma(3));
#! true
#! @EndExampleSession
#!
DeclareProperty("IsAntiassociative", IsMagma);
#! @Arguments M
#! @Description
#! identifies associativity index of <A>M</A>.
#!
#! @BeginExampleSession
#! 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
#! @EndExampleSession
#!
DeclareAttribute("AssociativityIndex", IsMagma);
#! @Arguments M
#! @Description
#! computes diaognal of multiplication table of <A>M</A>.
#!
#! @BeginExampleSession
#! 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 ]
#! ]
#! @EndExampleSession
#!
DeclareAttribute("DiagonalOfMultiplicationTable", IsMagma);
#! @Arguments M
#! @Description
#! identifies commutativity index of <A>M</A>.
#!
#! @BeginExampleSession
#!
#! @EndExampleSession
#!
DeclareAttribute("CommutativityIndex", IsMagma);
#! @Arguments M
#! @Description
#! calculates anticommutativity index of <A>M</A>.
#!
#! @BeginExampleSession
#! @EndExampleSession
#!
DeclareAttribute("AnticommutativityIndex", IsMagma);
#! @Arguments M
#! @Description
#! computes squares index of <A>M</A> so the order of $\left\{ m^2 | m \in M \right\}$.
#!
#! @BeginExampleSession
#! 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 ]
#! @EndExampleSession
#!
DeclareAttribute("SquaresIndex", IsMagma);
#! @Arguments M
#! @Description
#! builds a collection of non-isomorphic submagmas of <A>M</A>.
#!
#! @BeginExampleSession
#! 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> ] ]
#! @EndExampleSession
#!
DeclareGlobalFunction("AllSubmagmas");
#! @Arguments M
#! @Description
#! identifies class of antiassociative magma <A>M</A>.
#!
#! @BeginExampleSession
#! gap> IsAntiassociative(OneSmallGroup(16));
#! false
#! gap> IsAntiassociative(OneSmallAntimagma(2));
#! true
#! gap> IsAntiassociative(OneSmallAntimagma(3));
#! true
#! @EndExampleSession
#!
DeclareAttribute("IdSmallAntimagma", IsMagma);
#! @Arguments M
#! @Description
#! computes isomorphism invariants of <A>M</A>.
DeclareGlobalFunction("MagmaIsomorphismInvariantsMatch");
#! @Arguments M, N
#! @Description
#! computes an isomoprhism between magmas <A>M</A>, <A>N</A>.
#!
#! @BeginExampleSession
#! gap> M := SmallAntimagma(2, 1);
#! <magma with 2 generators>
#! gap> N := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]);
#! <magma with 2 generators>
#! gap> MagmaIsomorphism(M, N);
#! <general mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
#! @EndExampleSession
#!
DeclareOperation("MagmaIsomorphism", [IsMagma, IsMagma]);
#! @Arguments M, N
#! @Description
#! creates an antiisomoprhism between magmas <A>M</A>, <A>N</A>.
#!
#! @BeginExampleSession
#! 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 ]) >
#! @EndExampleSession
#!
DeclareOperation("MagmaAntiisomorphism", [IsMagma, IsMagma]);
#! @Arguments M, N
#! @Description
#! identifies whether magmas <A>M</A>, <A>N</A> are isomorphic.
#!
#! @BeginExampleSession
#! 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
#! @EndExampleSession
#!
DeclareGlobalFunction("IsMagmaIsomorphic");
#! @Arguments [M, N]
#! @Description
#! identifies whether magmas <A>M</A>, <A>N</A> are antiisomorphic.
#!
#! @BeginExampleSession
#! 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
#! @EndExampleSession
#!
DeclareGlobalFunction("IsMagmaAntiisomorphic");
#! @Arguments [M]
#! @Description
#! generates transposed magma <A>M</A>.
#!
#! @BeginExampleSession
#! 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 ] ]
#! @EndExampleSession
#!
DeclareGlobalFunction("TransposedMagma");
#! @Arguments [m, k]
#! @Description
#! returns a left $k$-power of element <A>m</A>.
DeclareGlobalFunction("LeftPower");
#! @Arguments [m, k]
#! @Description
#! returns a right $k$-power of element <A>m</A>.
DeclareGlobalFunction("RightPower");
#! @Arguments [m]
#! @Description
#! returns a left order of element <A>m</A>.
DeclareAttribute("LeftOrder", IsExtLElement);
#! @Arguments [m]
#! @Description
#! returns a right order of element <A>m</A>.
DeclareAttribute("RightOrder", IsExtRElement);
#! @Arguments [m]
#! @Description
#! returns a left order of element <A>m</A>.
DeclareAttribute("LeftOrdersOfElements", IsMagma);
#! @Arguments [m]
#! @Description
#! returns a left order of element <A>m</A>.
DeclareAttribute("RightOrdersOfElements", IsMagma);
#! @Arguments M
#! @Description
#! if magma is left cyclic <A>m</A>.
DeclareProperty("IsLeftCyclic", IsMagma);
#! @Arguments M
#! @Description
#! if magma is left cyclic <A>m</A>.
DeclareProperty("IsRightCyclic", IsMagma);
#! @Arguments M
#! @Description
#! if magma is left distributive <A>m</A>.
#!
#! @BeginExampleSession
#! gap> List(AllSmallAntimagmas(3), M -> IsLeftDistributive(M) );
#! [ true, false, false, false, false, false, false, false, false, true ]
#! @EndExampleSession
#!
DeclareProperty("IsLeftDistributive", IsMagma);
#! @Arguments M
#! @Description
#! if magma is right distributive <A>m</A>.
#!
#! @BeginExampleSession
#! gap> List(AllSmallAntimagmas(3), M -> IsRightDistributive(M) );
#! [ false, false, false, false, true, false, false, false, true, false ]
#! @EndExampleSession
#!
DeclareProperty("IsRightDistributive", IsMagma);
#! @Arguments M
#! @Description
#! if magma is left cancellative <A>m</A>.
#!
#! @BeginExampleSession
#! 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 ]
#! @EndExampleSession
#!
DeclareProperty("IsLeftCancellative", IsMagma);
#! @Arguments M
#! @Description
#! if magma is right cancellative <A>m</A>.
#!
#! @BeginExampleSession
#! gap> List(AllSmallAntimagmas(2), M -> IsRightCancellative(M));
#! [ false, true ]
#! @EndExampleSession
#!
DeclareProperty("IsRightCancellative", IsMagma);
#! @Arguments M
#! @Description
#! if magma is cancellative <A>m</A>.
#!
#! @BeginExampleSession
#! gap> List(AllSmallAntimagmas(2), M -> IsCancellative(M));
#! [ false, false ]
#! @EndExampleSession
#!
DeclareProperty("IsCancellative", IsMagma);
#! @Arguments M
#! @Description
#! is a left-hand sided fixed-point free inducted <A>m</A>.
#!
#! @BeginExampleSession
#! gap> Display( MultiplicationTable( SmallAntimagma(2, 2) ) );
#! [ [ 2, 2 ],
#! [ 1, 1 ] ]
#! gap> IsLeftFPFInducted( SmallAntimagma(2, 2) );
#! true
#! @EndExampleSession
#!
DeclareProperty("IsLeftFPFInducted", IsMagma);
#! @Arguments M
#! @Description
#! is a right-hand sided fixed-point free inducted <A>m</A>.
#!
#! @BeginExampleSession
#! gap> Display( MultiplicationTable( SmallAntimagma(2, 1) ) );
#! [ [ 2, 1 ],
#! [ 2, 1 ] ]
#! gap> IsRightFPFInducted( SmallAntimagma(2, 1) );
#! true
#! @EndExampleSession
#!
#!
DeclareProperty("IsRightFPFInducted", IsMagma);
#! @Arguments M
#! @Description
#! is a left-hand sided derangment inducted <A>m</A>.
#!
#! @BeginExampleSession
#! gap> M := SmallAntimagma(2, 2);
#! <magma with 2 generators>
#! gap> IsLeftFPFInducted(M);
#! true
#! gap> IsRightFPFInducted(M);
#! false
#! gap> IsRightDerangementInducted(M);
#! false
#! @EndExampleSession
#!
DeclareProperty("IsLeftDerangementInducted", IsMagma);
#! @Arguments M
#! @Description
#! is a right-hand sided derangment inducted <A>m</A>.
#!
#! @BeginExampleSession
#! gap> M := SmallAntimagma(2, 1);
#! <magma with 2 generators>
#! gap> IsLeftFPFInducted(M);
#! false
#! gap> IsRightFPFInducted(M);
#! true
#! gap> IsRightDerangementInducted(M);
#! true
#! @EndExampleSession
#!
#!
DeclareProperty("IsRightDerangementInducted", IsMagma);
#! @Arguments M
#! @Description
#! is a left-alternatve magma <A>M</A>.
#!
#! @BeginExampleSession
#! @EndExampleSession
#!
DeclareProperty("IsLeftAlternative", IsMagma);
#! @Arguments M
#! @Description
#! is a right-alternatve magma <A>M</A>.
#!
#! @BeginExampleSession
#! @EndExampleSession
#!
DeclareProperty("IsRightAlternative", IsMagma);
