Functions for type MODZ

This type represents finitely generated Abelian groups or, equivalently, finitely generated Z-modules. They usually come up as results of integral homology computations, see for instance PCOMPLEX[homology]. To construct them directly, use the function modz.

&<>

&<>(M1::MODZ, M2::MODZ)::boolean
M1 &<> M2

Test whether M1 and M2 represent different Abelian groups.

&=

&=(M1::MODZ, M2::MODZ)::boolean
M1 &= M2

Test whether M1 and M2 represent the same Abelian group.

corank

corank(M::MODZ)::nonnegint

The corank of a Z-module is its minimal number of generators.

directsum

directsum(M1::MODZ, ...)::MODZ

The direct sum of the given modules.

Examples

	M1 := modz(1, [2]); M2 := modz([3]);

M1 := Z+Z[2]

M2 := Z[3]

	directsum(M1, M2);

Z+Z[6]

Without arguments, the function must be called in its long form:

	MODZ[directsum]();

0

modz

modz(r::nonnegint, t::list(posint))::MODZ

Creates an object of type MODZ representing a Z-module isomorphic to Z^r+Z /t[1]+...+Z /t[-1]. The torsion part is transformed into a chain of divisors, and 1's are omitted. If r is omitted, it is taken to be 0; if t is omitted, it is taken to be the empty list.

Examples

	modz(3, [2, 7, 8, 1]);

3*Z+Z[2]+Z[56]

Distinguish between an object of type MODZ and its visual representation:

	modz(); evalb(modz() = 0);

0

false

	M1 := modz(1, [2])+modz([2]); type(M1, MODZ);

M1 := Z+2*Z[2]

false

	M2 := directsum(modz(1, [2]), modz([2])); type(M2, MODZ);

M2 := Z+2*Z[2]

true

rank

rank(M::MODZ)::nonnegint

The rank of the (free) quotient of M by its torsion submodule.

torsion

torsion(M::MODZ)::list(posint)

The list of elementary divisors of the torsion submodule of M as a chain of divisors.