Lattice of Submodules

Let \(M\) be an \(A\)-module. Magma can construct the lattice \(L\) of all submodules of \(M\) if this is not too large. Various properties of the lattice \(L\) may then be examined. The elements of \(L\) are called submodule-lattice elements and are numbered from 1 to \(n\) where \(n\) is the cardinality of \(L\). Once the lattice has been constructed, the result of various lattice operations, such as meet and intersection, are available without the need for any module-theoretic calculation. Certain information about \(M\) and its submodules may then be obtained by analyzing \(L\). Given an element of \(L\), one can easily create the submodule \(N\) of \(M\) corresponding to it and one can also create the element of \(L\) corresponding to any submodule of \(M\). The functions is this section apply only in the case in which \(A\) is an algebra over a finite field.

Creating Lattices

SubmoduleLattice(M): ModRng -> SubModLat, BoolElt

Limit           : RngIntElt                    Default: 0
CodimensionLimit: RngIntElt                    Default: 

Given a \(K[G]\)-module \(M\) defined over \(F_q\), construct the lattice \(L\) of submodules of \(M\). If the parameter Limit is set to \(n\), at most \(n\) submodules are calculated, and the second return value indicates whether the returned lattice \(L\) is the full lattice of submodules of \(M\).

SubmoduleLatticeAbort(M, n): ModRng, RngIntElt -> BoolElt, SubModLat: Given a \(K[G]\)-module \(M\) defined over \(F_q\) and a positive integer \(n\), construct the lattice \(L\) of submodules of \(M\), provided that the number of submodules does not exceed \(n\). In this case the value true and the lattice \(L\) are returned. If \(M\) has more than \(n\) submodules, the function aborts and returns the value false.

SetVerbose("SubmoduleLattice", i): MonStgElt, RngIntElt: Control verbose printing for the submodule lattice algorithm. The level \(i\) can be 2 for maximal printing or 1 for moderate printing. The algorithm works down a composition series of the module and a summary is printed for each level.

Submodules(M): ModRng -> [ModRng]

CodimensionLimit: RngIntElt                    Default: Dimension(M)

Given a \(K[G]\)-module \(M\) defined over \(F_q\), return a sequence containing all the submodules of \(M\) sorted by dimension.

Example: Creating a Submodule Lattice (ex-05bd83)

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We create the lattice of submodules for the \(A\)-module \({\bf F}_{3}[{\mathbb{Z}}_6]\) with level 1 verbose printing turned on.

> M := PermutationModule(CyclicGroup(6), GF(3));
> SetVerbose("SubmoduleLattice", 1);
> L := SubmoduleLattice(M);
Submodule Lattice; Dimension: 6, Composition length: 6
Starting level 4; Current number of modules: 2
Starting level 3; Current number of modules: 3
Starting level 2; Current number of modules: 6
Starting level 1; Current number of modules: 9
Starting level 0; Current number of modules: 12
Change basis time: 0.010
Jacobson radical time: 0.060
Complement time: 0.070
Total time: 0.250
> #L;
16

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Operations on Lattices

In the following, \(L\) is the lattice of submodules for a module \(M\).

# L: SubModLat -> RngIntElt: The cardinality of \(L\), i.e. the number of submodules of \(M\).

L ! i: SubModLat, RngIntElt -> SubModLatElt: Create the \(i\)-th element of the lattice \(L\). The number \(i\) is insignificant (i.e. the elements of \(L\) are not numbered in any special way), but this allows one to uniquely identify each element of the lattice \(L\).

L ! S: SubModLat, ModRng -> SubModLatElt: Create the element of the lattice \(L\) corresponding to the submodule \(S\) of \(M\).

Bottom(L): SubModLat -> SubModLatElt: Create the bottom of the lattice \(L\), i.e. the element of \(L\) corresponding to the zero-submodule of \(M\). If the lattice was created with a limit on the number of submodules and the lattice is partial, the bottom of the lattice may not be the zero submodule.

Random(L): SubModLat -> SubModLatElt: Create a random element of \(L\).

Top(L): SubModLat -> SubModLatElt: Create the top of the lattice \(L\), i.e. the element of \(L\) corresponding to \(M\).

Operations on Lattice Elements

In the following, \(L\) is the lattice of submodules for a module \(M\). Elements of \(L\) are identified with the integers \([1..\#L]\) but not in any particular order.

IntegerRing() ! e: RngInt, SubModLatElt -> RngIntElt: The integer corresponding to lattice element \(e\).

e + f: SubModLatElt, SubModLatElt -> SubModLatElt: The sum of lattice elements \(e\) and \(f\), i.e. the lattice element corresponding to the sum of the modules corresponding to \(e\) and \(f\).

e meet f: SubModLatElt, SubModLatElt -> SubModLatElt: The intersection of lattice elements \(e\) and \(f\).

e eq f: SubModLatElt, SubModLatElt -> SubModLatElt: Returns true if and only if lattice elements \(e\) and \(f\) are equal.

e subset f: SubModLatElt, SubModLatElt -> SubModLatElt: Returns true if and only if \(e\) is under \(f\) in the lattice \(L\), i.e. the submodule corresponding to \(e\) is a submodule of the submodule corresponding to \(f\).

MaximalSubmodules(e): SubModLatElt -> { SubModLatElt }: The maximal submodules of \(e\), returned as a set of lattice elements.

MinimalSupermodules(e): SubModLatElt -> { SubModLatElt }: The minimal supermodules of \(e\), returned as a set of lattice elements.

Module(e): SubModLatElt -> ModRng: The submodule of \(M\) corresponding to the element \(e\) of the lattice \(L\).

Properties of Lattice Elements

Dimension(e): SubModLatElt -> RngIntElt: The dimension of the submodule of \(M\) corresponding to \(e\).

JacobsonRadical(e): SubModLatElt -> SubModLatElt: The Jacobson radical of \(e\), i.e. the lattice element corresponding to the Jacobson radical of the submodule corresponding to \(e\).

Morphism(e): SubModLatElt -> ModMatRngElt: The morphism from the module corresponding to \(e\) to \(M\).

Example: Operations in a Submodule Lattice (ex-42abf7)

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We create the lattice of submodules for the \(A\)-module \({\bf F}_{3}[{\mathbb{Z}}_6]\).

> SetSeed(1);
> M := PermutationModule(CyclicGroup(6), GF(3));
> L := SubmoduleLattice(M);
> #L;
16
> T := Top(L);
> B := Bottom(L);
> T;
16
> B;
1
> // Check that element of L corresponding to M is T
> L ! M;
1
> (L ! M) eq T;
true
> // Check that module corresponding to B is zero-submodule of M
> Module(B);
GModule of dimension 0 with base ring GF(3)

We next find the minimal supermodules (immediate parents) of \(B\) in \(L\) and then determine the actual \(A\)-submodules to which they correspond.

> S := MinimalSupermodules(B);
> S;
{ 2, 3 }
> Module(L ! 2);
GModule of dimension 1 with base ring GF(3)
> Module(L ! 3);
GModule of dimension 1 with base ring GF(3)
> Dimension(L ! 2);
1
> Morphism(L ! 2);
[1 1 1 1 1 1]
> Morphism(L ! 3);
[1 2 1 2 1 2]
> // Set A to the sum of these elements
> A := L!2 + L!3;
> A;
5;
> // Note that A has dimension 2 and its morphism is the sum of the previous
> Dimension(A);
2
> Morphism(A);
[1 0 1 0 1 0]
[0 1 0 1 0 1]
> MaximalSubmodules(A);
{ 2, 3}
> S!2 subset A;
true

We now find the maximal submodules of \(L\), and examine one, \(S\), in detail.

> MaximalSubmodules(T);
{ 14, 15 }
> A := L ! 14;
> Dimension(A);
5
> Morphism(A);
[1 0 0 0 0 1]
[0 1 0 0 0 2]
[0 0 1 0 0 1]
[0 0 0 1 0 2]
[0 0 0 0 1 1]
> S := Module(A);
> S;
GModule S of dimension 5 with base ring GF(3)

Finally, we compute the Jacobson radical of \(S\) directly, and also obtain it from the lattice, checking that the two methods match.

> J := JacobsonRadical(S);
> J;
GModule J of dimension 3 with base ring GF(3)
> L ! J;
8
> JacobsonRadical(A);
8

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Lattice of Submodules#

Creating Lattices#

Operations on Lattices#

Operations on Lattice Elements#

Properties of Lattice Elements#

Lattice of Submodules

Creating Lattices

Operations on Lattices

Operations on Lattice Elements

Properties of Lattice Elements