Modularity in the semilattice of \(\omega \)-words (Q2249955)

Summary: A partial ordering of \(\omega \)-words can be introduced with regard to whether an \(\omega \)-word can be transformed into another by a Mealy machine. It is known that the poset of \(\omega \)-words that is introduced by this ordering is a join-semilattice. The width of this join-semilattice has the power of continuum while the depth is at least \(\aleph_0\). We have created a technique for proving that power-characteristic \(\omega \)-words are incomparable. We use this technique to show that this join-semilattice is not modular.