We classify binary minimal clones into seven categories: affine algebras, rectangular bands, $p$-cyclic groupoids, spirals, non-Taylor partial semilattices, melds, and dispersive algebras. Each category has nice enough properties to guarantee that any clone of one of these types contains a minimal clone of the same type.

2023-01-30math.RA

Minimal clones with weakly abelian representations

We show that a minimal clone has a nontrivial weakly abelian representation iff it has a nontrivial abelian representation, and that in this case all representations are weakly abelian.

2011-02-10math.RA

Majority-closed minions of Boolean functions

The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all $(C_1,C_2)$-stable classes of Boolean functions are determined for an arbitrary clone $C_1$ and for any clone $C_2$ containing the clone of self-dual monotone functions.

2021-02-03math.COmath.RA

Almost associative operations generating a minimal clone

Characterizations of "almost associative" binary operations generating a minimal clone are given for two interpretations of the term "almost associative". One of them uses the associative spectrum, the other one uses the index of nonassociativity to measure how far an operation is from being associative.

2011-02-10math.RA

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The minimal clones above the permutations

We determine the atoms of the interval of the clone lattice consisting of those clones which contain all permutations, on an infinite base set. This is equivalent to the description of the atoms of the lattice of transformation monoids above the permutations.

2005-12-15math.LOmath.RA

Equivalence of operations with respect to discriminator clones

For each clone C on a set A there is an associated equivalence relation, called C-equivalence, on the set of all operations on A, which relates two operations iff each one is a substitution instance of the other using operations from C. In this paper we prove that if C is a discriminator clone on a finite set, then there are only finitely many C-equivalence classes. Moreover, we show that the smallest discriminator clone is minimal with respect to this finiteness property. For discriminator clones of Boolean functions we explicitly describe the associated equivalence relations.

2007-06-01math.RA

All centralising monoids with majority witnesses on a four-element set

We study unary parts of centraliser clones on the set $\{0,1,2,3\}$, so-called centralising monoids. We describe and count all centralising monoids on the set $\{0,1,2,3\}$ having majority operations as witnesses, and we list the inclusion maximal proper submonoids of the full transformation monoid among them. This extends previous work by Goldstern, Machida and Rosenberg.

2020-10-31math.LOmath.RA

Examples, counterexamples, and structure in bounded width algebras

We study bounded width algebras which are minimal in the sense that every proper reduct does not have bounded width. We show that minimal bounded width algebras can be arranged into a pseudovariety with one basic ternary operation. We classify minimal bounded width algebras which have size at most three, and prove a structure theorem for minimal bounded width algebras which have no majority subalgebra, which form a pseudovariety with a commutative binary operation. As a byproduct of our results, we also classify minimal clones which have a Taylor term.

2020-02-14cs.CCmath.LOmath.RA