Linear Equations in the Stone-čech Compactification of N
نویسندگان
چکیده
Let a and b be distinct positive integers. We show that the equation u + a · p = v + b · p has no solutions with u, v ∈ βN and p ∈ βN\N. More generally, we show that if (S,+) is any commutative cancellative semigroup and S has no nontrivial solutions to n · s = n · t for n ∈ N and s, t ∈ S, then the equation u + a · p = v + b · p has no solutions with u, v ∈ βS and p ∈ βS\S. We characterize completely the Abelian groups for which such an equation can be satisfied. We also show that if S can be embedded in the circle group T, then the equation a · p + u = b · p + v has no solutions with u, v ∈ βS and p ∈ βS\S. Finally, we investigate solutions to the equation a1 · p+ a2 · p+ . . .+ an · p = b1 · p+ b2 · p+ . . .+ bm · p where p ∈ βN\N and a1, a2, . . . , an, b1, b2, . . . , bm ∈ N.
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