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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 | )abbrev category NARNG NonAssociativeRng
++ Author: J. Grabmeier, R. Wisbauer
++ Date Created: 01 March 1991
++ Date Last Updated: 03 July 1991
++ Reference:
++ Scha10 An Introduction to Nonassociative Algebras
++ Brem08 Nonassociative Algebras
++ Description:
++ NonAssociativeRng is a basic ring-type structure, not necessarily
++ commutative or associative, and not necessarily with unit.\br
++ Axioms\br
++ \tab{5}x*(y+z) = x*y + x*z\br
++ \tab{5}(x+y)*z = x*z + y*z\br
++
++ Common Additional Axioms\br
++ \tab{5}noZeroDivisors\tab{5} ab = 0 => a=0 or b=0
NonAssociativeRng() : Category == SIG where
SIG ==> Join(AbelianGroup,Monad) with
associator : (%,%,%) -> %
++ associator(a,b,c) returns \spad{(a*b)*c-a*(b*c)}.
commutator : (%,%) -> %
++ commutator(a,b) returns \spad{a*b-b*a}.
antiCommutator : (%,%) -> %
++ antiCommutator(a,b) returns \spad{a*b+b*a}.
add
associator(x,y,z) == (x*y)*z - x*(y*z)
commutator(x,y) == x*y - y*x
antiCommutator(x,y) == x*y + y*x
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