Three Classes of Closed Sets of Monomials

We consider three classes of monomials: unary, binary with at least one linear literal, and idempotent binary. A functionally closed set containing a unary monomial may or may not contain identity, and it can be generated by a singleton or by an arbitrary set of monomials. This induces four different classes of functionally closed sets of unary monomials. These classes are ordered by set inclusion and the emphasis is put on minimal, maximal, least and greatest elements. For binary monomials with at least one linear literal, we describe the structure of the set of clones generated by singletons. Finally, for idempotent binary monomials, we determine the least and the greatest element.

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