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Topic: nonstandard ringoid structures on Z (Read 411 times) |
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baudolino
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nonstandard ringoid structures on Z
« on: Oct 9th, 2004, 2:30pm » |
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Consider this definition of a ringoid.
A ringoid is a set R with two binary operators, addition [oplus] and multiplication [otimes], where both [oplus] and [otimes] are commutative and associative, and [otimes] distributes over [oplus] left and right, (i.e. a [otimes] (b [oplus] c) = (a [otimes] b) [oplus] (a [otimes] c) and
(b [oplus] c) [otimes] a = (b [otimes] a) [oplus] (c [otimes] a)).
Consider the set Z of integers and the binary operation [otimes] given by a [otimes] b = a + b, where + is the standard addition in Z. Find all binary operations [oplus] on Z such that (Z, [oplus], [otimes]) is a ringoid.
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Barukh
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Re: nonstandard ringoid structures on Z
« Reply #1 on: Oct 11th, 2004, 10:31am » |
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Here’s one example.
[smiley=blacksquare.gif]
a [oplus] b = max(a,b), and the symmetric min function.
[smiley=blacksquare.gif]
Here’s a couple of wrong functions that came to my mind:
[smiley=square.gif]
1. a [oplus] b = (a+b)/2. This satisfies the distributive and commutative laws, but is not associative.
2. a [oplus] b = a. This satisfies the distributive and associative laws, but is not commutative.
[smiley=square.gif]
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