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--- status: moved December 2020
--- includes ++, directSum, isDirectSum, components, indices, IndexComponents
undocumented {
(symbol ++,RingElement,ZZ),
(symbol ++,ZZ,RingElement),
(symbol ++,Matrix,ZZ),
(symbol ++,ZZ,Matrix)
}
document {
Key => {
isDirectSum,
(isDirectSum, Module),
},
Headline => "whether something is a direct sum",
"Works for modules, chain complexes, etc.
The components of the sum can be recovered with ", TO "components", ".",
EXAMPLE lines ///
isDirectSum ZZ^6
F = ZZ^2 ++ ZZ^3
isDirectSum F
components F
///
}
document {
Key => {
(symbol ++,Module,Module),
},
Headline => "direct sum of modules",
TT "M++N", " -- computes the direct sum of two modules (or coherent sheaves).",
EXAMPLE {
"R = ZZ/101[a..c];",
"image vars R ++ kernel vars R",
},
PARA {
"The projection and inclusion maps for direct sums can be obtained with the following methods."
},
UL {
TO (symbol ^,Module,Array),
TO (symbol _,Module,Array)
},
PARA{
"The components can be recovered later with ", TO "components", " or with ", TO "formation", ".",
},
SeeAlso => directSum}
document {
Key => {
(symbol ++,Matrix,Matrix),
(symbol ++,RingElement,Matrix),
(symbol ++,Matrix,RingElement),
(symbol ++,RingElement,RingElement)
},
Headline => "direct sum of maps",
TT "f++g", " -- computes the direct sum of two maps between modules.",
PARA{},
"If an argument is a ring element or integer, it is promoted
to a one by one matrix.",
EXAMPLE {
"R = ZZ/101[a..c];",
"vars R ++ transpose vars R",
"oo^[1]",
"a++b++c",
},
"Selecting rows or columns of blocks:",
UL {
TO (symbol ^,Matrix,Array),
TO (symbol _,Matrix,Array)
},
SeeAlso => {directSum, (symbol |, Matrix, Matrix), (symbol ||, Matrix, Matrix)}}
document {
Key => {
directSum,
(directSum, Module),
(directSum, Matrix),
(directSum, List),
(directSum, Option),
(directSum, Sequence),
(symbol++, Option, Option)
},
Headline => "direct sum of modules or maps",
TT "directSum(M,N,...)", " -- forms the direct sum of matrices or modules.",
PARA{
"The components can be recovered later with ", TO "components", " or with ", TO "formation", ".",
},
PARA{},
"Projection and inclusion maps for direct sums:",
UL {
TO (symbol ^,Module,Array),
TO (symbol _,Module,Array),
TO (symbol ^,Matrix,List),
TO (symbol _,Matrix,List)
},
PARA{},
"It sometimes happens that the user has indices for the components of
a direct sum preferable to the usual consecutive small integers. In
this case the preferred indices can be specified with code
like ", TT "directSum(a=>M,b=>N,...)", ", as in the following example.",
EXAMPLE {
///F = directSum(a=>ZZ^1, b=>ZZ^2, c=>ZZ^3)///,
///F_[b]///,
///F^[c]///,
},
"Similar syntax works with ", TO "++", ".",
EXAMPLE {
///F = (a => ZZ^1) ++ (b => ZZ^2)///,
///F_[b]///,
},
SeeAlso => {"++", "indices", formation},
Subnodes => {
TO isDirectSum,
TO components,
}}
document {
Key => {
components,
(components, Module),
(components, Matrix),
},
Headline => "list the components of a direct sum",
TT "components x", " -- produces a list of the components of an element of a
free module.",
BR{},
TT "components M", " -- the list of components for a module ", TT "M", " which was
formed as a direct sum, or ", TT "{M}", " if ", TT "M", " was not formed as a direct sum.
Works also for matrices, chain complexes, etc.",
SeeAlso => {"vector", "directSum", "++"},
Subnodes => { TO indexComponents },
}
-- TODO:
-- document {
-- Key => (indices,HashTable),
-- Headline => "preferred indices of a direct sum",
-- Usage => "",
-- Inputs => {
-- },
-- Outputs => {
-- },
-- Consequences => {
-- },
-- "description",
-- EXAMPLE {
-- },
-- Caveat => {},
-- SeeAlso => {"directSum", "components", "indexComponents"}
-- }
document {
Key => indexComponents,
Headline => "specify keys for components of a direct sum",
TT "indexComponents", " -- a symbol used as a key in a direct sum
under which to store a hash table in which to register preferred keys used
to index the components of the direct sum.",
PARA{},
SeeAlso => {"directSum", "components", "indices"}}
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