Sets of integers form a monoid, where the product of two sets
and is defined as the set containing for all $a\in A$ and
$b\in B$. We give a characterization of when a family of finite
sets is a code in this monoid, that is when the sets do not satisfy
any nontrivial relation. We also extend this result for some
infinite sets, including all infinite rational sets.

Sets of integers form a monoid, where the product of two sets
and is defined as the set containing for all $a\in A$ and
$b\in B$. We give a characterization of when a family of finite
sets is a code in this monoid, that is when the sets do not satisfy
any nontrivial relation. We also extend this result for some
infinite sets, including all infinite rational sets.

A -abelian cube is a word , where the factors , , and are either pairwise equal, or have the same multiplicities for every one of their factors of length at most . Previously it has been shown that -abelian cubes are avoidable over a binary alphabet for ≥ 8. Here it is proved that this holds for ≥ 5.

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