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I have an infinite monoid [itex]A[/itex] and a submonoid [itex]K[/itex].

let's assume I pick up an element [itex]x\in A-K[/itex],

now I consider the

*principal ideal*of [itex]K[/itex] generated by [itex]x[/itex], that is [itex]xK=\{xk|k\in K\}[/itex].

The question is:

if I consider another element [itex]x'[/itex] such that [itex]x'\in A-K[/itex] and [itex]x'\notin xK[/itex], is it possible to prove that [itex]xK\cap x'K=0[/itex] ?

If that statement is not generally true, is there an additional hypothesis that I could make to force [itex]xK\cap x'K=0[/itex] hold?

PS: I clicked too early and now I cannot change the title into something better.