Suppose A is a finite set. For every clone C over A, the family C(1) of all unary functions in C is a monoid of transformations of the set A. We study how the lattice of clones is partitioned into intervals, where two clones belong to the same partition iff they have the same monoids of unary functions. The problem of Szendrei concerning the power of such intervals is investigated. We give new examples of intervals which are continual, one-element, and finite but not one-element. Moreover, it is proved that every lattice that is not more than a direct product of countably many finite chains is isomorphic to some interval in the lattice of clones, establishing, in passing, the number of E-minimal algebras on a finite set.
KeywordsMathematical Logic Direct Product Unary Function Finite Chain Monoid Interval
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