The lattice of submonoids generated by idempotents and one- or two-sided units of a monoid is classified by a four-bit type, and the functors producing these submonoids generate a 15-element monoid.
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Idempotents and one-sided units: Lattice invariants and a semigroup of functors on the category of monoids
The lattice of submonoids generated by idempotents and one- or two-sided units of a monoid is classified by a four-bit type, and the functors producing these submonoids generate a 15-element monoid.