Anti-closure (?)
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Not sure if this is the right word, but is there a proper word describing an operation on a set that never gives back an element from the set?
i.e. you have some magma (N, *)
for any x, y in P
x * y is not in P (but is in N)
(not necessarily a binary operation)
e.g.
the primes under multiplication
characters under concatenation
forgot to say
*P is a subset of N
>>8204022
I don't think there is a word for it. Anti-closure sounds good.
As a side effect, if the operation has an identity then said identity cannot be contained in P in order for it to have this property.
>>8204022
>Asking an algebra question to /sci/
Most people here don't even know Calculus II.
Any function [math] P \times P \rightarrow N \cap P^\ast \backslash P[/math] where P* is the free magma
Challenge:
let S be a subset ot natural numbers such that for any a, b in S, a*b is not in S
what is maximum upper density of this set
>>8204022
An operation with disjoint image and preimage
>>8206743
Yes but is there a single word for it, like anti-closure
>>8206741
We already know the primes is a solution
-,2,3,-,5,-,7,-,-,-,11,-,13,-,-,-,17,-,19,-,...
If we delete 2, then we can add all numbers of the form 2*p where p is prime
-,-,3,4,5,6,7,-,-,10,11,-,13,14,-,-,17,-,19,-,...
If we delete 3, then we can add all numbers of the form 3*p where p is prime
and even 2*3*p like 12 or 18
-,-,-,4,5,6,7,-,9,10,11,12,13,14,15,-,17,18,19,-,...
having arrived at 4, we can in fact add all numbers 4n, with n<4, like 8
-,-,-,4,5,6,7,8,9,10,11,12,13,14,15,-,17,18,19,-,...
Deleting 4, we can add 16 and I'm running out of missing numbers
It's like a reverse prime filter where the set beyond a fixed m gets always denser
>>8206763
PNT*
>>8206758
that's still density zero