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        riddles >> putnam exam (pure math) >> nonstandard ringoid structures on Z
        
(Message started by: baudolino on Oct 9^th, 2004, 2:30pm)

Title: nonstandard ringoid structures on Z
Post by baudolino on Oct 9^th, 2004, 2:30pm

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.

Title: Re: nonstandard ringoid structures on Z
Post by Barukh on Oct 11^th, 2004, 10:31am

Here’s one example.
[smiley=blacksquare.gif][hide]
a [oplus] b = max(a,b), and the symmetric min function.
[/hide][smiley=blacksquare.gif]

Here’s a couple of wrong functions that came to my mind:
[smiley=square.gif][hide]
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.
[/hide][smiley=square.gif]