\hUSrSSKJrJrJrJrJr SSKJr "SS5r "SS\ 5r "SS \ 5r "S S \ 5r S r g )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)CoercionFailedc\rSrSrSrSrSrSrSrSr Sr S r S r S r S rS rSrSrSrSrSrSrSrSrSr\rSrSrSrSrSrSrSr Sr!Sr"S r#g!)"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@[U[5(d[SU-5e[U[5(d[SU-5eURUR:wa[ SU<SU<35eXlX lURUlSUlSUlg)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 V/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/agca/homomorphisms.py__init__ModuleHomomorphism.__init__8s&&))=FG G(F++=HI I ;;(-- '/5xAB B  KK   c^URcUR5UlUR$)a Compute the kernel of ``self``. That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute `ker(\phi) = \{x \in M | \phi(x) = 0\}`. This is a submodule of `M`. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [x, 0]]).kernel() <[x, -1]> )r_kernelrs rkernelModuleHomomorphism.kernelFs%$ 99  DIyyrc^URcUR5UlUR$)a Compute the image of ``self``. That is, if ``self`` is the homomorphism `\phi: M \to N`, then compute `im(\phi) = \{\phi(x) | x \in M \}`. This is a submodule of `N`. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [x, 0]]).image() == F.submodule([1, 0]) True )r_imagers rimageModuleHomomorphism.image\s%$ 99  DIyyrc[e)zCompute the kernel of ``self``.NotImplementedErrorrs rrModuleHomomorphism._kernelr!!rc[e)zCompute the image of ``self``.r$rs rr ModuleHomomorphism._imagevr'rc[ez%Implementation of domain restriction.r$rsms r_restrict_domain#ModuleHomomorphism._restrict_domainzr'rc[ez'Implementation of codomain restriction.r$r,s r_restrict_codomain%ModuleHomomorphism._restrict_codomain~r'rc[ez"Implementation of domain quotient.r$r,s r_quotient_domain#ModuleHomomorphism._quotient_domainr'rc[ez$Implementation of codomain quotient.r$r,s r_quotient_codomain%ModuleHomomorphism._quotient_codomainr'rcURRU5(d[SUR<SU<35eXR:XaU$URU5$)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$$R((rcURUR55(d![SUR5<SU<35eXR:XaU$UR U5$)a Return ``self``, with codomain restricted to to ``sm``. Here ``sm`` has to be a submodule of ``self.codomain`` containing the image. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.restrict_codomain(F.submodule([1, 0])) Matrix([ [1, x], : QQ[x]**2 -> <[1, 0]> [0, 0]]) z the image  must contain sm, got )r>r!rrr2r,s rrestrict_codomain$ModuleHomomorphism.restrict_codomainsV2tzz|,, $ b23 3  K&&r**rcUR5RU5(d![SUR5<SU<35eUR5(aU$UR U5$)a Return ``self`` with domain replaced by ``domain/sm``. Here ``sm`` must be a submodule of ``self.kernel()``. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h Matrix([ [1, x], : QQ[x]**2 -> QQ[x]**2 [0, 0]]) >>> h.quotient_domain(F.submodule([-x, 1])) Matrix([ [1, x], : QQ[x]**2/<[-x, 1]> -> QQ[x]**2 [0, 0]]) zkernel rB)rr>ris_zeror6r,s rquotient_domain"ModuleHomomorphism.quotient_domainsY0{{}))"--"kkmR12 2 ::<<K$$R((rcURRU5(d[SUR<SU<35eUR5(aU$UR U5$)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>rrFr:r,s rquotient_codomain$ModuleHomomorphism.quotient_codomainsU>}}))"-- $ r34 4 ::<<K&&r**rc[e)zApply ``self`` to ``elem``.r$relems r_applyModuleHomomorphism._applyr'rcURRURURRU555$N)rconvertrOrrMs r__call__ModuleHomomorphism.__call__s/}}$$T[[1D1DT1J%KLLrc[e)z Compose ``self`` with ``oth``, that is, return the homomorphism obtained by first applying then ``self``, then ``oth``. (This method is private since in this syntax, it is non-obvious which homomorphism is executed first.) r$roths r_composeModuleHomomorphism._composes "!rc[e)z8Scalar multiplication. ``c`` is guaranteed in self.ring.r$)rcs r _mul_scalarModuleHomomorphism._mul_scalar'r'rc[e)z^ Homomorphism addition. ``oth`` is guaranteed to be a homomorphism with same domain/codomain. r$rWs r_addModuleHomomorphism._add+s "!rc[U[5(dgURUR:H=(a URUR:H$)zEHelper to check that oth is a homomorphism with same domain/codomain.F)r r rrrWs r _check_homModuleHomomorphism._check_hom2s8#122zzT[[(JS\\T]]-JJrc[U[5(a+URUR:XaUR U5$UR UR RU55$![a [s$f=frR) r r rrrYr]rrSrNotImplementedrWs r__mul__ModuleHomomorphism.__mul__8sh c- . .4;;#,,3N<<% % "##DII$5$5c$:; ; "! ! "s)A,,A?>A?cURSURRU5- 5$![a [s$f=f)N)r]rrSrrfrWs r __truediv__ModuleHomomorphism.__truediv__CsA "##Adii&7&7&<$<= = "! ! "s,/AAc\URU5(aURU5$[$rR)rcr`rfrWs r__add__ModuleHomomorphism.__add__Is% ??3  99S> !rcURU5(a9URURURR S555$[ $)N)rcr`r]rrSrfrWs r__sub__ModuleHomomorphism.__sub__Ns> ??3  99S__TYY->->r-BCD Drc>UR5R5$)a Return True if ``self`` is injective. That is, check if the elements of the domain are mapped to the same codomain element. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_injective() False >>> h.quotient_domain(h.kernel()).is_injective() True )rrFrs r is_injectiveModuleHomomorphism.is_injectiveSs*{{}$$&&rc<UR5UR:H$)a Return True if ``self`` is surjective. That is, check if every element of the codomain has at least one preimage. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_surjective() False >>> h.restrict_codomain(h.image()).is_surjective() True )r!rrs r is_surjective ModuleHomomorphism.is_surjectivejs*zz|t}},,rcPUR5=(a UR5$)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 )rurxrs ris_isomorphism!ModuleHomomorphism.is_isomorphisms!,  ";t'9'9';;rc>UR5R5$)a Return True if ``self`` is a zero morphism. That is, check if every element of the domain is mapped to zero under self. Examples ======== >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> h = homomorphism(F, F, [[1, 0], [x, 0]]) >>> h.is_zero() False >>> h.restrict_domain(F.submodule()).is_zero() True >>> h.quotient_codomain(h.image()).is_zero() True )r!rFrs rrFModuleHomomorphism.is_zeros.zz|##%%rcHX- R5$![a gf=f)NF)rFrrWs r__eq__ModuleHomomorphism.__eq__s* J'') )  s  !!c X:X+$rRrWs r__ne__ModuleHomomorphism.__ne__s   r)rrrrrN)$__name__ __module__ __qualname____firstlineno____doc__rrr!rr r.r2r6r:r?rCrGrJrOrTrYr]r`rcrg__rmul__rkrnrrrurxr{rFrr__static_attributes__rrrr r s#J ,,""""""%)N+@)>$+L"M"""K "H"   '.-.<0&2 !rr cT\rSrSrSrSrSrSrSrSr Sr S r S r S r S rS rg)MatrixHomomorphismia Helper class for all homomoprhisms which are expressed via a matrix. That is, for such homomorphisms ``domain`` is contained in a module generated by finitely many elements `e_1, \ldots, e_n`, so that the homomorphism is determined uniquely by its action on the `e_i`. It can thus be represented as a vector of elements of the codomain module, or potentially a supermodule of the codomain module (and hence conventionally as a matrix, if there is a similar interpretation for elements of the codomain module). Note that this class does *not* assume that the `e_i` freely generate a submodule, nor that ``domain`` is even all of this submodule. It exists only to unify the interface. Do not instantiate. Attributes: - matrix - the list of images determining the homomorphism. NOTE: the elements of matrix belong to either self.codomain or self.codomain.container Still non-implemented methods: - kernel - _apply c^[RXU5 [U5UR:wa&[ SUR<S[U5<35eUR R m[UR [[45(a UR RR m[U4SjU55Ul g)NzNeed to provide z elements, got c34># UH nT"U5v M g7frRr).0x converters r .MatrixHomomorphism.__init__..s9&QIaLL&s) r rlenrankrrrSr rr containertuplematrix)rrrrrs @rrMatrixHomomorphism.__init__s##D(; v;&++ % & S[:; ;MM)) dmmi1B%C D D //77I9&99 rc 4SSKJn Sn[UR[[ 45(aSnU"UR VVs/sH3o2"U5Vs/sHo@RRU5PM snPM5 snn5R$s snfs snnf)z=Helper function which returns a SymPy matrix ``self.matrix``.r)MatrixcU$rRrrs r2MatrixHomomorphism._sympy_matrix..sarcUR$rR)datars rrrs!&&r) sympy.matricesrr rrrrrto_sympyT)rrr\rys r _sympy_matrix MatrixHomomorphism._sympy_matrixsp)  dmmn6G%H I I AdkkRkqt  rj)reprrsplitrrrrangejoin)rlinestsnis r__repr__MatrixHomomorphism.__repr__sT'')*006![[$-- 8 AJ JqAvA HMH 1f  q!tax#A HMH$yyrcB[XRUR5$r+)SubModuleHomomorphismrrr,s rr.#MatrixHomomorphism._restrict_domains$R DDrcNURURXR5$r1) __class__rrr,s rr2%MatrixHomomorphism._restrict_codomains~~dkk2{{;;rcjURURU- URUR5$r5rrrrr,s rr6#MatrixHomomorphism._quotient_domains%~~dkk"ndmmT[[IIrc @URU- nURn[UR[5(aURRnUR UR URU- URVs/sH oC"U5PM sn5$s snfr9)rrSr rrrrr)rr-Qrrs rr:%MatrixHomomorphism._quotient_codomainsx MM" II dmmY / / ++I~~dkk4==+;#';; /;aYq\; /1 1 /sB c URURUR[URUR5VVs/sH up#X#-PM snn5$s snnfrR)rrrzipr)rrXrrs rr`MatrixHomomorphism._addsK~~dkk4==14T[[#**1MN1Mqu1MNP PNsAc URURURURVs/sHo!U-PM sn5$s snfrRr)rr\rs rr]MatrixHomomorphism._mul_scalars5~~dkk4== :T 1Q3 :TUU:TsA c URURURURVs/sH o!"U5PM sn5$s snfrRr)rrXrs rrYMatrixHomomorphism._composes7~~dkk3<<$++9V+Q#a&+9VWW9VsA )rN)rrrrrrrrr.r2r6r:r`r]rYrrrrrrs?: :V  E<J1PVXrrc*\rSrSrSrSrSrSrSrg)FreeModuleHomomorphismia Concrete class for homomorphisms with domain a free module or a quotient thereof. Do not instantiate; the constructor does not check that your data is well defined. Use the ``homomorphism`` function instead: >>> from sympy import QQ >>> from sympy.abc import x >>> from sympy.polys.agca import homomorphism >>> F = QQ.old_poly_ring(x).free_module(2) >>> homomorphism(F, F, [[1, 0], [0, 1]]) Matrix([ [1, 0], : QQ[x]**2 -> QQ[x]**2 [0, 1]]) c[UR[5(a URn[ S[ XR 555$)Nc3.# UH upX-v M g7frRrrres rr0FreeModuleHomomorphism._apply../<%;TQ15%;)r rrrsumrrrMs rrOFreeModuleHomomorphism._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[UR[5(a URn[ S[ XR 555$)Nc3.# UH upX-v M g7frRrrs rr/SubModuleHomomorphism._apply..Trr)r rrrrrrrMs rrOSubModuleHomomorphism._applyQs7 dkk#4 5 599D.\s?&>FB"%&>r)r!rrrrrr)rrrs rrSubModuleHomomorphism._kernelYskjjl((*{{$$xx!!?c![[-=-=&>??!" "!s9BrNrrrrrr>s$= M"rrc SnU"U5upEpgU"U5upp[XHUV s/sH o"U 5PM sn 5RU5RU 5RU 5R U5$s sn f)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]> c^[T[5(aTTTR5U4Sj4$[T[5(a(TRTRTR U4Sj4$[T[ 5(a2TRRTRTR U4Sj4$TRTTR5U4Sj4$)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&>TRU5$rR)rSrmodules rr0homomorphism..freepres..sPQARrc:>TRU5R$rR)rSrrs rrrsfnnQ/44rcN>TRRU5R$rR)rrSrrs rrrsf..66q9>>rc:>TRRU5$rR)rrSrs rrrs&**2215r)r rrrbase killed_modulerr)rs`rfreepreshomomorphism..freepress fj ) )66#3#3#57RR R fn - -KKf.B.B46 6 f/ 0 0KK))6;;8L8L>@ @  &&*:*:*<57 7r)rr?rCrJrG) rrrrSFSSSQ_TFTSTQr\rs r homomorphismr`sxx7$V$MBBX&MBB !"*@A1Q4*@ ?2 00  R !45*@sA8 N)rsympy.polys.agca.modulesrrrrrsympy.polys.polyerrorsrr rrrrrrrrsX""1 g!g!T ZX+ZXz"0/"0J"."DS5r