+ iURt^RIHtHtHtHtHt^RIHt!RR4t !RR] 4t !RR] 4t !R R ] 4t R t R #) 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)CoercionFailedca]tRt^toRtRtRtRtRtRt Rt Rt R t R t R tR tR tRtRtRtRtRtRtRtRt]tRtRtRtRtRtRtRt Rt!Rt"Rt#Vt$R #)!ModuleHomomorphisma Abstract base class for module homomoprhisms. Do not instantiate. Instead, use the ``homomorphism`` function: >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [0, 1]]) Matrix([ [1, 0], : QQ[x]**2 -> QQ[x]**2 [0, 1]]) Attributes: - ring - the ring over which we are considering modules - domain - the domain module - codomain - the codomain module - _ker - cached kernel - _img - cached image Non-implemented methods: - _kernel - _image - _restrict_domain - _restrict_codomain - _quotient_domain - _quotient_codomain - _apply - _mul_scalar - _compose - _add cX\V\4'g\RV,4h\V\4'g\RV,4hVPVP8wd\ RV: RV: 24hWnW nVPVnRVnRVnR#)zSource must be a module, got %szTarget must be a module, got %sz0Source and codomain must be over same ring, got z != N) isinstancer TypeErrorring ValueErrordomaincodomain_ker_img)selfrrs&&&\/var/www/html/photoedit/myenv/lib/python3.14/site-packages/sympy/polys/agca/homomorphisms.py__init__ModuleHomomorphism.__init__8s&&))=FG G(F++=HI I ;;(-- '/5xAB B  KK   c `VPfVP4VnVP#)a Compute the kernel of ``self``. That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute `ker(\phi) = \{x \in M | \phi(x) = 0\}`. This is a submodule of `M`. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [x, 0]]).kernel() <[x, -1]> )r_kernelrs&rkernelModuleHomomorphism.kernelFs%$ 99  DIyyrc `VPfVP4VnVP#)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&rimageModuleHomomorphism.image\s%$ 99  DIyyrc \h)zCompute the kernel of ``self``.NotImplementedErrorrs&rrModuleHomomorphism._kernelr!!rc \h)zCompute the image of ``self``.r"rs&rrModuleHomomorphism._imagevr%rc \hz%Implementation of domain restriction.r"rsms&&r_restrict_domain#ModuleHomomorphism._restrict_domainzr%rc \hz'Implementation of codomain restriction.r"r*s&&r_restrict_codomain%ModuleHomomorphism._restrict_codomain~r%rc \hz"Implementation of domain quotient.r"r*s&&r_quotient_domain#ModuleHomomorphism._quotient_domainr%rc \hz$Implementation of codomain quotient.r"r*s&&r_quotient_codomain%ModuleHomomorphism._quotient_codomainr%rc VPPV4'g\RVP: RV: 24hWP8XdV#VPV4#)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_domain"ModuleHomomorphism.restrict_domainsU@{{''++ $ R12 2  K$$R((rc VPVP44'g"\RVP4: RV: 24hWP8XdV#VP V4#)a Return ``self``, with codomain restricted to to ``sm``. Here ``sm`` has to be a submodule of ``self.codomain`` containing the image. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.restrict_codomain(F.submodule([1, 0])) Matrix([ [1, x], : QQ[x]**2 -> <[1, 0]> [0, 0]]) z the image  must contain sm, got )r<rrrr0r*s&&rrestrict_codomain$ModuleHomomorphism.restrict_codomainsV2tzz|,, $ b23 3  K&&r**rc VP4PV4'g"\RVP4: RV: 24hVP4'dV#VP V4#)a Return ``self`` with domain replaced by ``domain/sm``. Here ``sm`` must be a submodule of ``self.kernel()``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.quotient_domain(F.submodule([-x, 1])) Matrix([ [1, x], : QQ[x]**2/<[-x, 1]> -> QQ[x]**2 [0, 0]]) zkernel r@)rr<ris_zeror4r*s&&rquotient_domain"ModuleHomomorphism.quotient_domainsY0{{}))"--"kkmR12 2 ::<<K$$R((rc VPPV4'g\RVP: RV: 24hVP4'dV#VP V4#)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 r;)rr<rrDr8r*s&&rquotient_codomain$ModuleHomomorphism.quotient_codomainsU>}}))"-- $ r34 4 ::<<K&&r**rc \h)zApply ``self`` to ``elem``.r"relems&&r_applyModuleHomomorphism._applyr%rcVPPVPVPPV444#N)rconvertrMrrKs&&r__call__ModuleHomomorphism.__call__s/}}$$T[[1D1DT1J%KLLrc \h)z Compose ``self`` with ``oth``, that is, return the homomorphism obtained by first applying then ``self``, then ``oth``. (This method is private since in this syntax, it is non-obvious which homomorphism is executed first.) r"roths&&r_composeModuleHomomorphism._composes "!rc \h)z8Scalar multiplication. ``c`` is guaranteed in self.ring.r")rcs&&r _mul_scalarModuleHomomorphism._mul_scalar'r%rc \h)z^ Homomorphism addition. ``oth`` is guaranteed to be a homomorphism with same domain/codomain. r"rUs&&r_addModuleHomomorphism._add+s "!rc \V\4'gR#VPVP8H;'dVPVP8H#)zEHelper to check that oth is a homomorphism with same domain/codomain.F)r r rrrUs&&r _check_homModuleHomomorphism._check_hom2s;#122zzT[[(JJS\\T]]-JJrc \V\4'd-VPVP8XdVP V4#VP VP PV44# \d \u#i;irP) r r rrrWr[r rQrNotImplementedrUs&&r__mul__ModuleHomomorphism.__mul__8sh c- . .4;;#,,3N<<% % "##DII$5$5c$:; ; "! ! "s)A..BBcVP^VPPV4, 4# \d \u#i;i))r[r rQrrdrUs&&r __truediv__ModuleHomomorphism.__truediv__CsA "##Adii&7&7&<$<= = "! ! "s03AAc^VPV4'dVPV4#\#rP)rar^rdrUs&&r__add__ModuleHomomorphism.__add__Is% ??3  99S> !rcVPV4'd:VPVPVPP R444#\ #)rh)rar^r[r rQrdrUs&&r__sub__ModuleHomomorphism.__sub__Ns> ??3  99S__TYY->->r-BCD Drc >VP4P4#)a Return True if ``self`` is injective. That is, check if the elements of the domain are mapped to the same codomain element. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_injective() False >>> h.quotient_domain(h.kernel()).is_injective() True )rrDrs&r is_injectiveModuleHomomorphism.is_injectiveSs*{{}$$&&rc <VP4VP8H#)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_surjective ModuleHomomorphism.is_surjectivejs*zz|t}},,rc RVP4;'dVP4#)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 )rsrvrs&ris_isomorphism!ModuleHomomorphism.is_isomorphisms$,  ";;t'9'9';;rc >VP4P4#)a Return True if ``self`` is a zero morphism. That is, check if every element of the domain is mapped to zero under self. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_zero() False >>> h.restrict_domain(F.submodule()).is_zero() True >>> h.quotient_codomain(h.image()).is_zero() True )rrDrs&rrDModuleHomomorphism.is_zeros.zz|##%%rcTW, P4# \dR#i;i)F)rDr rUs&&r__eq__ModuleHomomorphism.__eq__s* J'') )  s  ''c W8X*#rPrUs&&r__ne__ModuleHomomorphism.__ne__s   r)rrrrr N)%__name__ __module__ __qualname____firstlineno____doc__rrrrrr,r0r4r8r=rArErHrMrRrWr[r^rare__rmul__rirlrprsrvryrDr~r__static_attributes____classdictcell__ __classdict__s@rr r s#J ,,""""""%)N+@)>$+L"M"""K "H"   '.-.<0&2 !!rr c`a]tRtRtoRtRtRtRtRtRt Rt R t R t R t R tR tVtR#)MatrixHomomorphismia 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 ca\PWV4\V4VP8wd'\ RVP: R\V4: 24hVP P o\VP \\34'd!VP PP o\;QJd.V3RlV4FNK 5Vn R#!V3RlV44Vn R#)zNeed to provide z elements, got c34<"TF pS!V4xK R#5irPr).0x converters& r .MatrixHomomorphism.__init__..s9&QIaLL&sN) r rlenrankrrrQr rr containertuplematrix)rrrrrs&&&&@rrMatrixHomomorphism.__init__s##D(; v;&++ % & S[:; ;MM)) dmmi1B%C D D //77Ie9&9e e9&99 rc  2^RIHpRp\VP\\ 34'dRpT!VP UUu.uF2q2!V4Uu.uFq@PPV4NK upNK4 upp4P#uupiuuppi)z=Helper function which returns a SymPy matrix ``self.matrix``.)MatrixcV#rPrrs&r2MatrixHomomorphism._sympy_matrix..sarcVP#rP)datars&rrrs!&&r) sympy.matricesrr rrrrr to_sympyT)rrrZrys& r _sympy_matrix MatrixHomomorphism._sympy_matrixsp)  dmmn6G%H I I AdkkRkqt  )reprrsplitrrrrangejoin)rlinestsnis& r__repr__MatrixHomomorphism.__repr__sT'')*006![[$-- 8 AJ JqAvA HMH 1f  q!tax#A HMH$yyrc B\WPVP4#r))SubModuleHomomorphismrrr*s&&rr,#MatrixHomomorphism._restrict_domains$R DDrc NVPVPWP4#r/) __class__rrr*s&&rr0%MatrixHomomorphism._restrict_codomains~~dkk2{{;;rc rVPVPV, VPVP4#r3rrrrr*s&&rr4#MatrixHomomorphism._quotient_domains%~~dkk"ndmmT[[IIrc  PVPV, pVPp\VP\4'dVPPpTP VP VPV, VPUu.uF qC!V4NK up4#uupir7)rrQr rrrrr)rr+Qrrs&& rr8%MatrixHomomorphism._quotient_codomainsx MM" II dmmY / / ++I~~dkk4==+;#';; /;aYq\; /1 1 /s B# c TPVPVP\VPVP4UUu.uF wr#W#,NK upp4#uuppirP)rrrzipr)rrVrrs&& rr^MatrixHomomorphism._addsM~~dkk4==14T[[#**1MN1Mquu1MNP PNsA c TPVPVPVPUu.uF q!V,NK up4#uupirPr)rrZrs&& rr[MatrixHomomorphism._mul_scalars7~~dkk4== :T 1Q33 :TUU:TsA c TPVPVPVPUu.uF q!!V4NK up4#uupirPr)rrVrs&& rrWMatrixHomomorphism._composes7~~dkk3<<$++9V+Q#a&+9VWW9VsA )rN)rrrrrrrrr,r0r4r8r^r[rWrrrs@rrrsH: :V  E<J1PVXXrrc6a]tRtRtoRtRtRtRtRtVt R#)FreeModuleHomomorphismia 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]]) c\VP\4'd VPp\ R\ WP 444#)c36"TFwrW,xK R#5irPrrres& rr0FreeModuleHomomorphism._apply../<%;TQ155%;)r rrrsumrrrKs&&rrMFreeModuleHomomorphism._apply,s6 dkk> 2 299Da Concrete class for homomorphism with domain a submodule of 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 >>> 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]]) c\VP\4'd VPp\ R\ WP 444#)c36"TFwrW,xK R#5irPrrs& rr/SubModuleHomomorphism._apply..Trr)r rrrrrrrKs&&rrMSubModuleHomomorphism._applyQs7 dkk#4 5 599D.\s?&>FB"%%&>r)rrrrrrr)rrrs& rrSubModuleHomomorphism._kernelYskjjl((*{{$$xx!!?c![[-=-=&>??!" "!s8BrNrrs@rrr>s$= M""rrc RpV!V4wrErgV!V4wrr\YHVU u.uF q!V 4NK up 4PV4PV 4PV 4P V4#uup i)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]> ca\S\4'dSSSP4V3Rl3#\S\4'd)SPSPSP V3Rl3#\S\ 4'd3SPPSPSP V3Rl3#SPSSP4V3Rl3#)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&<SPV4#rP)rQrmodules&rr0homomorphism..freepres..sPQARrc:<SPV4P#rP)rQrrs&rrrsfnnQ/44rcN<SPPV4P#rP)rrQrrs&rrrsf..66q9>>rc:<SPPV4#rP)rrQrs&rrrs&**2215r)r rrrbase killed_modulerr)rsfrfreepreshomomorphism..freepress fj ) )66#3#3#57RR R fn - -KKf.B.B46 6 f/ 0 0KK))6;;8L8L>@ @  &&*:*:*<57 7r)rr=rArHrE) rrrrSFSSSQ_TFTSTQrZrs &&& r homomorphismr`sxx7$V$MBBX&MBB !"*@A1Q4*@ ?2 00  R !45*@sA7 N)rsympy.polys.agca.modulesrrrrrsympy.polys.polyerrorsrr rrrrrrrrsX""1 g!g!T ZX+ZXz"0/"0J"."DS5r