# Structure Operations

## Related Structures

The main structure related to an algebra of symmetric functions is its coefficient ring. An algebra of symmetric functions belongs to the Magma category `AlgSym`.

### `BaseRing(L): AlgSym -> Rng`

### `CoefficientRing(L): AlgSym -> Rng`

Return the coefficient ring of an algebra of symmetric functions $L$.

### `Category(L): AlgSym -> Cat`

### `Parent(L): AlgSym -> Pow`

### `PrimeRing(L): AlgSym -> Rng`

## Ring Predicates and Booleans

The usual ring functions returning boolean values are available on algebras of symmetric functions.

### `IsCommutative(L): AlgSym -> BoolElt`

### `IsUnitary(L): AlgSym -> BoolElt`

### `IsFinite(L): AlgSym -> BoolElt`

### `IsOrdered(L): AlgSym -> BoolElt`

### `IsField(L): AlgSym -> BoolElt`

### `IsEuclideanDomain(L): AlgSym -> BoolElt`

### `IsPID(L): AlgSym -> BoolElt`

### `IsUFD(L): AlgSym -> BoolElt`

### `IsDivisionRing(L): AlgSym -> BoolElt`

### `IsEuclideanRing(L): AlgSym -> BoolElt`

### `IsDomain(L): AlgSym -> BoolElt`

### `IsPrincipalIdealRing(L): AlgSym -> BoolElt`

### `L eq M: AlgSym, AlgSym -> BoolElt`

### `L ne M: AlgSym, AlgSym -> BoolElt`

Return `true` if $L$ and $M$ are equal (respectively, not equal). Magma considers two algebras to be equal if they are over the same ring.

## Predicates on Basis Types

### `HasSchurBasis(A): AlgSym -> BoolElt`

Returns `true` if $A$ is an algebra with a Schur basis.

### `HasHomogeneousBasis(A): AlgSym -> BoolElt`

### `HasElementaryBasis(A): AlgSym -> BoolElt`

### `HasPowerSumBasis(A): AlgSym -> BoolElt`

### `HasMonomialBasis(A): AlgSym -> BoolElt`

Returns `true` if $A$ is an algebra with a homogeneous, elementary, power sum or monomial basis respectively.