sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 50 ## sage: P.<x,y,z> = A.g_algebra(relations={y*x:-x*y}, order = 'lex') ## line 51 ## sage: P ## line 53 ## Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -x*y} sage: y*x + 1/2 ## line 56 ## -x*y + 1/2 sage: A.<x,y,z> = FreeAlgebra(GF(17), 3) ## line 59 ## sage: P.<x,y,z> = A.g_algebra(relations={y*x:-x*y}, order = 'lex') ## line 60 ## sage: P ## line 61 ## Noncommutative Multivariate Polynomial Ring in x, y, z over Finite Field of size 17, nc-relations: {y*x: -x*y} sage: y*x + 7 ## line 64 ## -x*y + 7 sage: from sage.matrix.constructor import Matrix ## line 71 ## sage: c = Matrix(3) ## line 72 ## sage: c[0,1] = -2 ## line 73 ## sage: c[0,2] = 1 ## line 74 ## sage: c[1,2] = 1 ## line 75 ## sage: d = Matrix(3) ## line 77 ## sage: d[0, 1] = 17 ## line 78 ## sage: P = QQ['x','y','z'] ## line 79 ## sage: c = c.change_ring(P) ## line 80 ## sage: d = d.change_ring(P) ## line 81 ## sage: from sage.rings.polynomial.plural import NCPolynomialRing_plural ## line 83 ## sage: R.<x,y,z> = NCPolynomialRing_plural(QQ, c = c, d = d, order=TermOrder('lex',3),category=Algebras(QQ)) ## line 84 ## sage: R ## line 85 ## Noncommutative Multivariate Polynomial Ring in x, y, z over Rational Field, nc-relations: {y*x: -2*x*y + 17} sage: R.term_order() ## line 88 ## Lexicographic term order sage: a,b,c = R.gens() ## line 91 ## sage: f = 57 * a^2*b + 43 * c + 1; f ## line 92 ## 57*x^2*y + 43*z + 1 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 97 ## sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') ## line 98 ## sage: TestSuite(P).run() ## line 99 ## sage: loads(dumps(P)) is P ## line 100 ## True sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 103 ## sage: P = A.g_algebra(relations={y*x:-x*y}, order = 'lex') ## line 104 ## sage: P.is_commutative() ## line 105 ## False sage: R.<x,y,z> = FreeAlgebra(QQ, 3) ## line 108 ## sage: P = R.g_algebra(relations={}, order='lex') ## line 109 ## sage: P.is_commutative() ## line 110 ## True sage: sig_on_count() # check sig_on/off pairings (virtual doctest) ## line 112 ## 0 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 157 ## sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) ## line 158 ## sage: H is A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) # indirect doctest ## line 159 ## True sage: sig_on_count() # check sig_on/off pairings (virtual doctest) ## line 161 ## 0 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 175 ## sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) ## line 176 ## sage: sorted(H.relations().items(), key=str) ## line 177 ## [(y*x, x*y - z), (z*x, x*z + 2*x), (z*y, y*z - 2*y)] sage: sig_on_count() # check sig_on/off pairings (virtual doctest) ## line 179 ## 0 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 204 ## sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) ## line 205 ## sage: H is A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) # indirect doctest ## line 206 ## True sage: P = A.g_algebra(relations={}, order='lex') ## line 209 ## sage: P.category() ## line 210 ## Category of commutative algebras over Rational Field sage: sig_on_count() # check sig_on/off pairings (virtual doctest) ## line 212 ## 0 sage: A.<x,y,z> = FreeAlgebra(QQ, 3) ## line 253 ## sage: H = A.g_algebra({y*x:x*y-z, z*x:x*z+2*x, z*y:y*z-2*y}) ## line 254 ## sage: H._is_category_initialized() ## line 255 ## True sage: H.category() ## line 257 ## Category of algebras over Rational Field sage: TestSuite(H).run() ## line 259 ## sage: H.<x,y,z> = A.g_algebra({z*x:x*z+2*x, z*y:y*z-2*y}) ## line 264 ## sage: x*y == y*x ## line 265 ## True sage: sig_on_count() # check sig_on/off pairings (virtual doctest) ## line 267 ## 0 sage: from sage.matrix.constructor import Matrix ## line 294 ## sage: c0 = Matrix(3) ## line 295 ## sage: c0[0,1] = -1 ## line 296 ## sage: c0[0,2] = 1 ## line 297 ## sage: c0[1,2] = 1 ## line 298 ## sage: d0 = Matrix(3) ## line 300 ## sage: d0[0, 1] = 17 ## line 301 ## sage: P = QQ['x','y','z'] ## line 302 ## sage: c = c0.change_ring(P) ## line 303 ## sage: d = d0.change_ring(P) ##

slow doctest:: Test ran for 5.58s cpu, 4.51s wall Check ran for 0.00s cpu, 0.00s wall

Variable 'a' referenced here was set only in doctest marked '# needs sage.libs.linbox sage.rings.finite_rings'; '# needs sage.libs.ntl sage.rings.finite_rings'; '# needs sage.libs.pari sage.rings.finite_rings'