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Operations on Submodules

Operations on Submodules

Subsections

Membership and Equality

The following operations are only available for submodules of R^((n)), Hom_R(M, N) and R[G], where R is a Euclidean Domain. If the modules involved are R[G]-modules, the operators refer to the underlying R-module.

u in M : ModTupRngElt, ModTupRng -> BoolElt
u in M : ModMatElt, ModMat -> BoolElt
True if the element u lies in the R-module M, where u and M belong to the same R-module.
u notin M : ModTupRngElt, ModTupRng -> BoolElt
u notin M : ModMatElt, ModMat -> BoolElt
True if the element u does not lie in the R-module M, where u and M belong to the same R-module.
N subset M : ModTupRng, ModTupRng -> BoolElt
N subset M : ModMat, ModMat -> BoolElt
True if the R-module N is contained in the R-module M, where M and N belong to a common R-module.
N notsubset M : ModTupRng, ModTupRng -> BoolElt
N notsubset M : ModMat, ModMat -> BoolElt
True if the R-module N is not contained in the R-module M, where M and N belong to a common R-module.
N eq M : ModTupRng, ModTupRng -> BoolElt
M eq N : ModMat, ModMat -> BoolElt
True if the R-modules N and M are equal, where N and M belong to a common R-module.
N ne M : ModTupRng, ModTupRng -> BoolElt
M ne N : ModMat, ModMat -> BoolElt
True if the R-modules N and M are not equal, where N and M belong to a common R-module.

Operations on Submodules

The following operations are only available for submodules of R^((n)), Hom_R(M, N) and R[G], where R is a Euclidean Domain. If the modules involved are R[G]-modules, the operators refer to the underlying R-module.

M + N : ModTupRng, ModTupRng -> ModTupRng
M + N : ModMat, ModMat -> ModMat
Sum of the submodules M and N, where M and N belong to a a common R-module.
M meet N : ModTupRng, ModTupRng -> ModTupRng
M meet N : ModMat, ModMat -> ModMat
Intersection of the submodules M and N, where M and N belong to a common R-module.
Fix(M): Mod -> Mod
Given an R-module M, construct the largest submodule of M on which R acts trivially, i.e. the fixed-point space of M.
Include(W, v) : ModTupRng, ModTupRngElt -> ModTupRng, BoolElt
(Function.) Given a vector v from a tuple module V and a submodule W of V, return the space obtained by including v in W, together with a boolean stating whether including v actually increased the space (i.e. v was not in W).
Include(W, S) : ModTupRng, ModTupRngElt -> ModTupRng, BoolElt
Include(W, S) : ModTupRng, [ ModTupRngElt ] -> ModTupRng, BoolElt
Include(W, S) : ModTupRng, @ ModTupRngElt @ -> ModTupRng, BoolElt
(Function.) Given a set or sequence S of vectors from a tuple module V and a submodule W of V, return the space obtained by including all vectors of S in W, together with a boolean stating whether including the vectors actually increased the space (i.e. any element of S was not in W).
Include(~W, v) : ModTupRng, ModTupRngElt ->
(Procedure.) Given a vector v from a tuple module V and a submodule W of V, destructively insert v in W.
Include(~W, S) : ModTupRng, ModTupRngElt ->
Include(~W, S) : ModTupRng, [ ModTupRngElt ] ->
Include(~W, S) : ModTupRng, @ ModTupRngElt @ ->
(Procedure.) Given a set or sequence S of vectors from a tuple module V and a submodule W of V, destructively insert all the elements of S in W.
Include(~W, v, ~f) : ModTupRng, ModTupRngElt ->
(Procedure.) Given a vector v from a tuple module V and a submodule W of V, destructively insert v in W. Set f to whether W increases.
Include(~W, S, ~f) : ModTupRng, ModTupRngElt ->
Include(~W, S, ~f) : ModTupRng, [ ModTupRngElt ] ->
Include(~W, S, ~f) : ModTupRng, @ ModTupRngElt @ ->
(Procedure.) Given a set or sequence S of vectors from a tuple module V and a submodule W of V, destructively insert all the elements of S in W. Set f to whether W increases.
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