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Basic Algebra Hierarchy


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{AbelianSemiGroup, SemiGroup, OrderedSet, Finite} -> SetCategory;
AbelianMonoid -> AbelianSemiGroup;
Rng -> {AbelianGroup, SemiGroup};
Monoid -> SemiGroup;
OrderedAbelianSemiGroup -> {OrderedSet, AbelianMonoid};
OrderedFinite -> {OrderedSet, Finite};
CancellationAbelianMonoid -> AbelianMonoid;
OrderedAbelianMonoid -> {AbelianMonoid,OrderedAbelianSemiGroup};
Group -> Monoid;
OrderedMonoid -> {Monoid, OrderedSet};
AbelianGroup -> CancellationAbelianMonoid;
OrderedCancellationAbelianMonoid -> {CancellationAbelianMonoid, OrderedAbelianMonoid};
LeftModule -> AbelianGroup;
RightModule -> AbelianGroup;
Ring -> {Rng, Monoid};
OrderedAbelianMonoidSup -> OrderedCancellationAbelianMonoid;
OrderedAbelianGroup -> {OrderedCancellationAbelianMonoid, AbelianGroup};
BiModule -> {LeftModule, RightModule};
PartialDifferentialRing -> Ring;
DifferentialRing -> Ring;
CommutativeRing -> {BiModule, Ring};
EntireRing -> {BiModule, Ring};
Module -> BiModule;
OrderedRing -> {OrderedAbelianGroup, Ring, OrderedMonoid};
LinearlyExplicitRingOver -> Ring;
CharacteristicZero -> Ring;
CharacteristicNonZero -> Ring;
DifferentialExtension -> {PartialDifferentialRing, DifferentialRing, CharacteristicZero, CharacteristicNonZero};
Algebra -> {CharacteristicNonZero, Module, Ring};
VectorSpace -> Module;
FullyLinearExplicitRingOver ->LinearlyExplicitRingOver;
DivisionRing -> {EntireRing, Algebra};
IntegralDomain -> {CommutativeRing, EntireRing, Algebra}
GcdDomain -> IntegralDomain;
UniqueFactorizationDomain -> GcdDomain;
PrincipalIdealDomain -> GcdDomain;
EuclideanDomain -> PrincipalIdealDomain;
PolynomialFactorizationExplicit -> UniqueFactorizationDomain;
Field -> {UniqueFactorizationDomain, DivisionRing, EuclideanDomain};
FieldOfPrimeCharacteristic -> CharacteristicNonZero;
FiniteFieldCategory -> {FieldOfPrimeCharacteristic, Finite};
}
 




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