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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 44 45 46 47 48 49 50 51 52 53 | )abbrev category GRMOD GradedModule
++ Author: Stephen M. Watt
++ Date Created: May 20, 1991
++ Date Last Updated: May 20, 1991
++ References:
++ Saun79 Algebra 2d Edition, MacLane and Birkhoff, MacMillan 1979
++ Description:
++ GradedModule(R,E) denotes ``E-graded R-module'', that is, collection of
++ R-modules indexed by an abelian monoid E.
++ An element \spad{g} of \spad{G[s]} for some specific \spad{s} in \spad{E}
++ is said to be an element of \spad{G} with degree \spad{s}.
++ Sums are defined in each module \spad{G[s]} so two elements of \spad{G}
++ have a sum if they have the same degree.
++
++ Morphisms can be defined and composed by degree to give the
++ mathematical category of graded modules.
GradedModule(R,E): Category == SIG where
R : CommutativeRing
E : AbelianMonoid
SIG ==> SetCategory with
degree : % -> E
++ degree(g) names the degree of g. The set of all elements
++ of a given degree form an R-module.
0 : constant -> %
++ \spad{0} denotes the zero of degree 0.
_* : (R, %) -> %
++ r*g is left module multiplication.
_* : (%, R) -> %
++ g*r is right module multiplication.
_- : % -> %
++ -g is the additive inverse of g in the module of elements
++ of the same grade as g.
_+ : (%, %) -> %
++ g+h is the sum of g and h in the module of elements of
++ the same degree as g and h. Error: if g and h
++ have different degrees.
_- : (%, %) -> %
++ g-h is the difference of g and h in the module of elements of
++ the same degree as g and h. Error: if g and h
++ have different degrees.
add
(x: %) - (y: %) == x+(-y)
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