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) ##

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

slow doctest:: Test ran for 5.48s cpu, 4.11s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 24.71s cpu, 24.78s wall Check ran for 0.00s cpu, 0.00s wall

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

slow doctest:: Test ran for 21.65s cpu, 29.16s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 16.77s cpu, 29.25s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 14.71s cpu, 25.10s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 5.99s cpu, 10.65s wall Check ran for 0.00s cpu, 0.02s wall

slow doctest:: Test ran for 9.31s cpu, 17.94s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 5.73s cpu, 8.48s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 9.50s cpu, 15.84s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 11.63s cpu, 21.78s wall Check ran for 0.00s cpu, 0.00s wall

The 'defaults' channel might have been added implicitly. If this is intentional, add 'defaults' to the 'channels' list. Otherwise, consider setting 'conda-remove-defaults' to 'true'.

slow doctest:: Test ran for 5.50s cpu, 7.97s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 6.39s cpu, 4.90s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 7.99s cpu, 5.98s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 14.73s cpu, 31.62s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 5.21s cpu, 9.94s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 6.16s cpu, 9.19s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 8.60s cpu, 14.89s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 5.79s cpu, 11.97s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 10.86s cpu, 18.95s wall Check ran for 0.00s cpu, 0.00s wall

slow doctest:: Test ran for 11.09s cpu, 19.36s wall Check ran for 0.00s cpu, 0.00s wall