gUdZddlmZmZmZmZmZddlmZGddZ Gdde Z Gdd e Z Gd d e Z d Z d S)a  Computations with homomorphisms of modules and rings. This module implements classes for representing homomorphisms of rings and their modules. Instead of instantiating the classes directly, you should use the function ``homomorphism(from, to, matrix)`` to create homomorphism objects. )Module FreeModuleQuotientModule SubModuleSubQuotientModule)CoercionFailedceZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z d Zd ZdZdZdZdZdZdZdZdZeZdZdZdZdZdZdZdZdZ dZ!dS) ModuleHomomorphisma" Abstract base class for module homomoprhisms. Do not instantiate. Instead, use the ``homomorphism`` function: >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [0, 1]]) Matrix([ [1, 0], : QQ[x]**2 -> QQ[x]**2 [0, 1]]) Attributes: - ring - the ring over which we are considering modules - domain - the domain module - codomain - the codomain module - _ker - cached kernel - _img - cached image Non-implemented methods: - _kernel - _image - _restrict_domain - _restrict_codomain - _quotient_domain - _quotient_codomain - _apply - _mul_scalar - _compose - _add c<t|tstd|zt|tstd|z|j|jkrt d|d|||_||_|j|_d|_d|_dS)NzSource must be a module, got %szTarget must be a module, got %sz0Source and codomain must be over same ring, got z != ) isinstancer TypeErrorring ValueErrordomaincodomain_ker_img)selfrrs Z/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/polys/agca/homomorphisms.py__init__zModuleHomomorphism.__init__8s&&)) H=FGG G(F++ J=HII I ;(- ' '*/5vvxxABB B   K   cP|j||_|jS)a Compute the kernel of ``self``. That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute `ker(\phi) = \{x \in M | \phi(x) = 0\}`. This is a submodule of `M`. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [x, 0]]).kernel() <[x, -1]> )r_kernelrs rkernelzModuleHomomorphism.kernelFs#$ 9  DIyrcP|j||_|jS)a Compute the image of ``self``. That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute `im(\phi) = \{\phi(x) | x \in M \}`. This is a submodule of `N`. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [x, 0]]).image() == F.submodule([1, 0]) True )r_imagers rimagezModuleHomomorphism.image\s#$ 9  DIyrct)zCompute the kernel of ``self``.NotImplementedErrorrs rrzModuleHomomorphism._kernelr!!rct)zCompute the image of ``self``.r rs rrzModuleHomomorphism._imagevr"rctz%Implementation of domain restriction.r rsms r_restrict_domainz#ModuleHomomorphism._restrict_domainzr"rctz'Implementation of codomain restriction.r r&s r_restrict_codomainz%ModuleHomomorphism._restrict_codomain~r"rctz"Implementation of domain quotient.r r&s r_quotient_domainz#ModuleHomomorphism._quotient_domainr"rct)$Implementation of codomain quotient.r r&s r_quotient_codomainz%ModuleHomomorphism._quotient_codomainr"rc|j|std|jd|||jkr|S||S)a? Return ``self``, with the domain restricted to ``sm``. Here ``sm`` has to be a submodule of ``self.domain``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.restrict_domain(F.submodule([1, 0])) Matrix([ [1, x], : <[1, 0]> -> QQ[x]**2 [0, 0]]) This is the same as just composing on the right with the submodule inclusion: >>> h * F.submodule([1, 0]).inclusion_hom() Matrix([ [1, x], : <[1, 0]> -> QQ[x]**2 [0, 0]]) zsm must be a submodule of , got )r is_submodulerr(r&s rrestrict_domainz"ModuleHomomorphism.restrict_domainsi@{''++ 2* $ RR122 2   K$$R(((rc||s'td|d|||jkr|S||S)a Return ``self``, with codomain restricted to to ``sm``. Here ``sm`` has to be a submodule of ``self.codomain`` containing the image. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.restrict_codomain(F.submodule([1, 0])) Matrix([ [1, x], : QQ[x]**2 -> <[1, 0]> [0, 0]]) z the image  must contain sm, got )r4rrrr+r&s rrestrict_codomainz$ModuleHomomorphism.restrict_codomainsr2tzz||,, 3* $ bb233 3   K&&r***rc||s'td|d||r|S||S)am Return ``self`` with domain replaced by ``domain/sm``. Here ``sm`` must be a submodule of ``self.kernel()``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.quotient_domain(F.submodule([-x, 1])) Matrix([ [1, x], : QQ[x]**2/<[-x, 1]> -> QQ[x]**2 [0, 0]]) zkernel r7)rr4ris_zeror.r&s rquotient_domainz"ModuleHomomorphism.quotient_domainsw0{{}}))"-- 2*"kkmmmmRR122 2 ::<< K$$R(((rc|j|std|jd||r|S||S)a: Return ``self`` with codomain replaced by ``codomain/sm``. Here ``sm`` must be a submodule of ``self.codomain``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.quotient_codomain(F.submodule([1, 1])) Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2/<[1, 1]> [0, 0]]) This is the same as composing with the quotient map on the left: >>> (F/[(1, 1)]).quotient_hom() * h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2/<[1, 1]> [0, 0]]) z#sm must be a submodule of codomain r3)rr4rr:r1r&s rquotient_codomainz$ModuleHomomorphism.quotient_codomainsk>}))"-- 4* $ rr344 4 ::<< K&&r***rct)zApply ``self`` to ``elem``.r relems r_applyzModuleHomomorphism._applyr"rc|j||j|SN)rconvertrArr?s r__call__zModuleHomomorphism.__call__s4}$$T[[1D1DT1J1J%K%KLLLrct)a  Compose ``self`` with ``oth``, that is, return the homomorphism obtained by first applying then ``self``, then ``oth``. (This method is private since in this syntax, it is non-obvious which homomorphism is executed first.) r roths r_composezModuleHomomorphism._composes "!rct)z8Scalar multiplication. ``c`` is guaranteed in self.ring.r rcs r _mul_scalarzModuleHomomorphism._mul_scalar'r"rct)zv Homomorphism addition. ``oth`` is guaranteed to be a homomorphism with same domain/codomain. r rGs r_addzModuleHomomorphism._add+s "!rcpt|tsdS|j|jko|j|jkS)zEHelper to check that oth is a homomorphism with same domain/codomain.F)r r rrrGs r _check_homzModuleHomomorphism._check_hom2s7#122 5zT[(JS\T]-JJrct|tr%|j|jkr||S ||j|S#t$r tcYSwxYwrC) r r rrrIrMrrDrNotImplementedrGs r__mul__zModuleHomomorphism.__mul__8s c- . . &4;#,3N3N<<%% % "##DI$5$5c$:$:;; ; " " "! ! ! ! "s,A))A=<A=c |d|j|z S#t$r tcYSwxYw)N)rMrrDrrSrGs r __truediv__zModuleHomomorphism.__truediv__CsW "##Adi&7&7&<&<$<== = " " "! ! ! ! "s/2AAcd||r||StSrC)rQrOrSrGs r__add__zModuleHomomorphism.__add__Is, ??3   "99S>> !rc||r@|||jdSt S)N)rQrOrMrrDrSrGs r__sub__zModuleHomomorphism.__sub__NsK ??3   E99S__TY->->r-B-BCCDD DrcN|S)a Return True if ``self`` is injective. That is, check if the elements of the domain are mapped to the same codomain element. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_injective() False >>> h.quotient_domain(h.kernel()).is_injective() True )rr:rs r is_injectivezModuleHomomorphism.is_injectiveSs*{{}}$$&&&rc<||jkS)a Return True if ``self`` is surjective. That is, check if every element of the codomain has at least one preimage. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_surjective() False >>> h.restrict_codomain(h.image()).is_surjective() True )rrrs r is_surjectivez ModuleHomomorphism.is_surjectivejs*zz||t},,rcR|o|S)a~ Return True if ``self`` is an isomorphism. That is, check if every element of the codomain has precisely one preimage. Equivalently, ``self`` is both injective and surjective. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h = h.restrict_codomain(h.image()) >>> h.is_isomorphism() False >>> h.quotient_domain(h.kernel()).is_isomorphism() True )r^r`rs ris_isomorphismz!ModuleHomomorphism.is_isomorphisms',  "";t'9'9';';;rcN|S)aN Return True if ``self`` is a zero morphism. That is, check if every element of the domain is mapped to zero under self. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_zero() False >>> h.restrict_domain(F.submodule()).is_zero() True >>> h.quotient_codomain(h.image()).is_zero() True )rr:rs rr:zModuleHomomorphism.is_zeros.zz||##%%%rcT ||z S#t$rYdSwxYw)NF)r:r rGs r__eq__zModuleHomomorphism.__eq__s? 3J'')) )   55 s  ''c||k SrCrGs r__ne__zModuleHomomorphism.__ne__sCK  rN)"__name__ __module__ __qualname____doc__rrrrrr(r+r.r1r5r8r;r=rArErIrMrOrQrT__rmul__rWrYr\r^r`rbr:rerhrgrrr r s##J   ,,""""""""""""""""""%)%)%)N+++@)))>$+$+$+L"""MMM"""""""""KKK """H"""   '''.---.<<<0&&&2 !!!!!rr cNeZdZdZdZdZdZdZdZdZ dZ d Z d Z d Z d S) MatrixHomomorphisma Helper class for all homomoprhisms which are expressed via a matrix. That is, for such homomorphisms ``domain`` is contained in a module generated by finitely many elements `e_1, \ldots, e_n`, so that the homomorphism is determined uniquely by its action on the `e_i`. It can thus be represented as a vector of elements of the codomain module, or potentially a supermodule of the codomain module (and hence conventionally as a matrix, if there is a similar interpretation for elements of the codomain module). Note that this class does *not* assume that the `e_i` freely generate a submodule, nor that ``domain`` is even all of this submodule. It exists only to unify the interface. Do not instantiate. Attributes: - matrix - the list of images determining the homomorphism. NOTE: the elements of matrix belong to either self.codomain or self.codomain.container Still non-implemented methods: - kernel - _apply czt|||t||jkr't d|jdt||jjt|jttfr|jj jtfd|D|_ dS)NzNeed to provide z elements, got c3.K|]}|VdSrCrg.0x converters r z.MatrixHomomorphism.__init__..s+99QIIaLL999999r) r rlenrankrrrDr rr containertuplematrix)rrrr{rus @rrzMatrixHomomorphism.__init__s##D&(;;; v;;&+ % %* & S[[[:;; ;M) dmi1B%C D D 8 /7I9999&99999 rcddlm}dtjtt frd|fdjDjS)z=Helper function which returns a SymPy matrix ``self.matrix``.r)Matrixc|SrCrgrts rz2MatrixHomomorphism._sympy_matrix..sarc|jSrC)datars rrz2MatrixHomomorphism._sympy_matrix..s!&rc>g|]}fd|DS)cDg|]}j|Srg)rto_sympy)rsyrs r z?MatrixHomomorphism._sympy_matrix...s)<<.s6RRR<<<  rV)reprrsplitrrrwrangejoin)rlinestsnis r__repr__zMatrixHomomorphism.__repr__sT''))**0066![[[$-- 8 AJ JJqAv  A !HHHMHHHH a1f  q!tax##  A !HHHMHHHHyyrc8t||j|jSr%)SubModuleHomomorphismrr{r&s rr(z#MatrixHomomorphism._restrict_domains$R DDDrcD||j||jSr*) __class__rr{r&s rr+z%MatrixHomomorphism._restrict_codomains~~dk2t{;;;rcT||j|z |j|jSr-rrrr{r&s rr.z#MatrixHomomorphism._quotient_domains"~~dk"ndmT[IIIrc|j|z }|jt|jtr |jj||j|j|z fd|jDS)r0c&g|] }|Srgrgrrs rrz9MatrixHomomorphism._quotient_codomain.. s! / / /aYYq\\ / / /r)rrDr rryrrr{)rr'Qrus @rr1z%MatrixHomomorphism._quotient_codomainsp M" I dmY / / , +I~~dk4=+; / / / /4; / / /11 1rc ||j|jdt|j|jDS)Ncg|] \}}||z Srgrg)rsrtrs rrz+MatrixHomomorphism._add..s NNNAq1uNNNr)rrrzipr{rGs rrOzMatrixHomomorphism._addsB~~dk4=NNT[#*1M1MNNNPP Prch||j|jfd|jDS)Ncg|]}|zSrgrgrsrtrLs rrz2MatrixHomomorphism._mul_scalar..s:T:T:T11Q3:T:T:TrrrKs `rrMzMatrixHomomorphism._mul_scalars4~~dk4=:T:T:T:T :T:T:TUUUrch||jjfd|jDS)Nc&g|] }|Srgrg)rsrtrHs rrz/MatrixHomomorphism._compose..s!9V9V9VQ##a&&9V9V9VrrrGs `rrIzMatrixHomomorphism._composes4~~dk3<9V9V9V9V$+9V9V9VWWWrN)rirjrkrlrrrr(r+r.r1rOrMrIrgrrroros: : : :VVV    EEE<<<JJJ111PPPVVVXXXXXrroc$eZdZdZdZdZdZdS)FreeModuleHomomorphisma Concrete class for homomorphisms with domain a free module or a quotient thereof. Do not instantiate; the constructor does not check that your data is well defined. Use the ``homomorphism`` function instead: >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [0, 1]]) Matrix([ [1, 0], : QQ[x]**2 -> QQ[x]**2 [0, 1]]) ct|jtr|j}t dt ||jDS)Nc3&K|] \}}||zV dSrCrgrsrtes rrvz0FreeModuleHomomorphism._apply../*< 2 2 9D<>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> M = QQ.old_poly_ring(x).free_module(2)*x >>> homomorphism(M, M, [[1, 0], [0, 1]]) Matrix([ [1, 0], : <[x, 0], [0, x]> -> <[x, 0], [0, x]> [0, 1]]) ct|jtr|j}t dt ||jDS)Nc3&K|] \}}||zV dSrCrgrs rrvz/SubModuleHomomorphism._apply..Trr)r rrrrrr{r?s rrAzSubModuleHomomorphism._applyQsG dk#4 5 5 9D<.Ws!(K(K(KQa(K(K(Kr)rrrrrs`rrzSubModuleHomomorphism._imageVs/&t}&(K(K(K(K$+:J(K(K(KLLrc}jjfd|jDS)Nc rg|]3}tdt|jjD4S)c3&K|] \}}||zV dSrCrg)rsxigis rrvz;SubModuleHomomorphism._kernel...\s*??FB"R%??????r)rrrr)rsrrs rrz1SubModuleHomomorphism._kernel..\sO!!!??c!T[-=&>&>?????!!!rrrs` rrzSubModuleHomomorphism._kernelYsXjjll((**$t{$!!!!x!!!" "rNrrgrrrr>sN$=== MMM"""""rrc d}||\}}}}||\}} } t|| fd|D|| | |S)a> Create a homomorphism object. This function tries to build a homomorphism from ``domain`` to ``codomain`` via the matrix ``matrix``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> R = QQ.old_poly_ring(x) >>> T = R.free_module(2) If ``domain`` is a free module generated by `e_1, \ldots, e_n`, then ``matrix`` should be an n-element iterable `(b_1, \ldots, b_n)` where the `b_i` are elements of ``codomain``. The constructed homomorphism is the unique homomorphism sending `e_i` to `b_i`. >>> F = R.free_module(2) >>> h = homomorphism(F, T, [[1, x], [x**2, 0]]) >>> h Matrix([ [1, x**2], : QQ[x]**2 -> QQ[x]**2 [x, 0]]) >>> h([1, 0]) [1, x] >>> h([0, 1]) [x**2, 0] >>> h([1, 1]) [x**2 + 1, x] If ``domain`` is a submodule of a free module, them ``matrix`` determines a homomoprhism from the containing free module to ``codomain``, and the homomorphism returned is obtained by restriction to ``domain``. >>> S = F.submodule([1, 0], [0, x]) >>> homomorphism(S, T, [[1, x], [x**2, 0]]) Matrix([ [1, x**2], : <[1, 0], [0, x]> -> QQ[x]**2 [x, 0]]) If ``domain`` is a (sub)quotient `N/K`, then ``matrix`` determines a homomorphism from `N` to ``codomain``. If the kernel contains `K`, this homomorphism descends to ``domain`` and is returned; otherwise an exception is raised. >>> homomorphism(S/[(1, 0)], T, [0, [x**2, 0]]) Matrix([ [0, x**2], : <[1, 0] + <[1, 0]>, [0, x] + <[1, 0]>, [1, 0] + <[1, 0]>> -> QQ[x]**2 [0, 0]]) >>> homomorphism(S/[(0, x)], T, [0, [x**2, 0]]) Traceback (most recent call last): ... ValueError: kernel <[1, 0], [0, 0]> must contain sm, got <[0,x]> cbttrfdfSttrjjjfdfStt rjjjjfdfSjfdfS)z Return a tuple ``(F, S, Q, c)`` where ``F`` is a free module, ``S`` is a submodule of ``F``, and ``Q`` a submodule of ``S``, such that ``module = S/Q``, and ``c`` is a conversion function. c.|SrC)rDrtmodules rrz0homomorphism..freepres..sPQARARrc8|jSrC)rDrrs rrz0homomorphism..freepres..sfnnQ//4rcBj|jSrC)ryrDrrs rrz0homomorphism..freepres..sf.66q99>rc8j|SrC)ryrDrs rrz0homomorphism..freepres..s&*22155r)r rrrbase killed_modulerry)rs`rfreepreszhomomorphism..freepress fj ) ) S66#3#3#5#57R7R7R7RR R fn - - 6Kf.B44446 6 f/ 0 0 @K)6;8L>>>>@ @ &&*:*:*<*<55557 7rc&g|] }|Srgrgrs rrz homomorphism..s!*@*@*@A11Q44*@*@*@r)rr5r8r=r;) rrr{rSFSSSQ_TFTSTQrLs @r homomorphismr`sx777$HV$$MBBHX&&MBB !"b*@*@*@*@*@*@*@  ?2  00   R !4!45rN)rlsympy.polys.agca.modulesrrrrrsympy.polys.polyerrorsrr rorrrrgrrrsJ""""""""""""""111111 g!g!g!g!g!g!g!g!T ZXZXZXZXZX+ZXZXZXz"0"0"0"0"0/"0"0"0J"""""."""DS5S5S5S5S5r