We show how to reconstruct the topology on the monoid of endomorphisms of the rational numbers under the strict or reflexive order relation, and the polymorphism clone of the rational numbers under the reflexive relation. In addition we show how automatic homeomorphicity results can be lifted to polymorphism clones generated by monoids.

Abstract. C-clones are polymorphism sets of so-called clausal relations, a special type of relations on a finite domain, which first appeared in connection with constraint satisfaction problems in work by Creignou et al. from 2008. We completely describe the relationship regarding set inclusion between maximal C-clones and maximal clones. As a main result we obtain that for every maximal C-clone there exists exactly one maximal clone in which it is contained. A precise description of this unique maximal clone, as well as a corresponding completeness criterion for C-clones is given.Mathematics Subject Classification. 08A40, 08A02, 08A99.

As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Maľcev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.

In this paper, we define the action of M , the monoid of embeddings of (Q, ≤), on Q, in the monoid (M, •). That is we show that Q itself can be interpreted in (M, •), and in addition, so can the action of M on Q. This is extended to the monoid E of all endomorphisms of (Q, ≤).

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