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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | )abbrev category GRALG GradedAlgebra
++ Author: Stephen M. Watt
++ Date Created: May 20, 1991
++ Date Last Updated: May 20, 1991
++ References:
++ Iyan60 Encyclopedic Dictionary of Mathematics, MIT Press, 1977
++ Description:
++ GradedAlgebra(R,E) denotes ``E-graded R-algebra''.
++ A graded algebra is a graded module together with a degree preserving
++ R-linear map, called the product.
++
++ The name ``product'' is written out in full so inner and outer products
++ with the same mapping type can be distinguished by name.
GradedAlgebra(R,E) : Category == SIG where
R : CommutativeRing
E : AbelianMonoid
SIG ==> Join(GradedModule(R, E),RetractableTo(R)) with
1 : constant -> %
++ \spad{1} is the identity for \spad{product}.
product : (%, %) -> %
++ product(a,b) is the degree-preserving R-linear product:
++
++ \spad{degree product(a,b) = degree a + degree b}
++ \spad{product(a1+a2,b) = product(a1,b) + product(a2,b)}
++ \spad{product(a,b1+b2) = product(a,b1) + product(a,b2)}
++ \spad{product(r*a,b) = product(a,r*b) = r*product(a,b)}
++ \spad{product(a,product(b,c)) = product(product(a,b),c)}
add
if not (R is %) then
0: % == (0$R)::%
1: % == 1$R::%
(r: R)*(x: %) == product(r::%, x)
(x: %)*(r: R) == product(x, r::%)
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