Numbers with anticlosed sets of non-squares

[Edwin Clark]

On Tue, 17 Jan 2006 jens at voss-ahrensburg.de wrote:

> Let R be a ring. We will call a subset T of R *anticlosed* if for any
> elements t1 and t2 of T, neither the sum t1 + t2 nor the product t1 * t2
> lies in T.

I suggest that you call such a subset "sum and product free". There is a 
large literature on "sum-free sets in abelian groups". If G is an additive 
group, a subset S of G is "sum-free" if x,y in S implies x + y is not in 
S. The OEIS contains the sequence (note that in both cases apparently 
addition mod n is not used)

A007865(n) = subsets of {1,...,n} containing no solutions of x+y=z.

and

A085489(n) = subsets of {1,...,n} containing no solutions of x+y=z with x 
and y distinct.

[[I recall coming across this concept many years ago and didn't know how 
to find similar work. A year later I accidentally came across the survey 
paper by Ann Penfeld Street on the subject of sum-free sets in abelian 
groups. There has also been work on product-free subsets of not 
necessarily abelian groups. But a cursory search  by MathSciNet doesn't 
come up with sum and product free sets in rings.]]
 
--Edwin

---------------------------------------------------------
  W. Edwin Clark, Math Dept, University of South Florida
           http://www.math.usf.edu/~eclark/
---------------------------------------------------------