This musing requires the prerequisites of musing 1, and the knowledge that a band is an idempotent (that is, ) semigroup (that is, it is associative as well).
In the literature on subvarieties of the variety of bands, there is a series of identities based on equivalences between certain recursively-defined words.
In particular, the -words are defined as , with * indicating the dual - that is, reversal - and the base case , and the -words are defined as , with the base case . Let’s call bands satisfying as -bands.
In particular, consider -bands. Let’s build this up one-by-one:
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Using Mace4, we can observe that the (unique) smallest band which is not a -band is of order two, the (unique) smallest band which is not a -band is of order 5, and the (unique) smallest band which is not a -band is of order 9. What is the order of the smallest band which is not a -band, and is it unique?
Also, are these smallest counterexamples possibly free algebra? The concept of a free algebra is recalled in musing 2.