Centralising groups of semiprojections and near unanimity operations

Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

“…For the co-atom M = F * (1) this means M = F * (1) = { f } * (1) * * (1) = { f } * (1) by virtue of the Galois connection given by K. So M is singly witnessed by a non-trivial operation f ; however, even more is known about these witnesses: they can always be chosen as a generator of a minimal clone of minimum arity, a so-called minimal function. Proposition 3 (Theorem 3.2 in [21], Proposition 2.2 in [27]). For any maximal centralising monoid M ⊆ O…”

“…In this paper, we address centralising monoids witnessed by a minimal function of types (I)-(III) with a special focus on the set {0, 1, 2, 3}; those relating to type (IV) have already been considered in [27], the ones of type (V) are part of ongoing research.…”

“…In Lemma 4.11 of [27] (see also Lemma 4.10 of [37]) we provided a characterisation of when a finitary operation f ∈ O A on a finite set A commutes with a permutation s ∈ Sym A, based on examining orbits of the permutation group generated by s. Under the assumption that f is a function the given condition was sufficient to ensure that f ⊥ s, but blindly fulfilling the orbit condition could also lead to some value assignments for f that would contradict the assumption of f being a function. Thus, some care had to be taken when working with Lemma 4.11 from [27], but this was never much of an issue when studying majority operations f on small sets as in [27]. Here we improve the mentioned characterisation by placing an additional necessary condition on the choice of the function values such that no contradictions can occur.…”

“…This research is part of the project, begun in [27], to take the results of [23] to the next level, that is, to determine all maximal centralising monoids on four-element carrier sets. According to Proposition 3 it is necessary for this to find all centralising monoids on {0, 1, 2, 3} induced by single functions of each of the five types of Rosenberg's Theorem and to determine the ones among them which are maximal proper transformation monoids under set inclusion.…”

“…As an initial step centralising groups of majority operations and semiprojections were studied in [28]. Extending the methods of [23], the case of majority operations on {0, 1, 2, 3} was already completed in [27], and further investigated under a different aspect in [29]. In this article we tackle the cases of unary operations (permutations of prime order and non-identical retractive operations), binary idempotent operations and ternary minority operations arising as Mal'cev operations corresponding to Boolean groups.…”