Unique inclusions of maximal C-clones in maximal clones

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.


Introduction
Clones are sets of operations on a fixed domain that are closed under composition and contain all projections. The clones on a finite set D are precisely the Galois closed sets of operations ( [7], translated in [5,6], independently [9]) with respect to the well-known Galois connection Pol D − Inv D induced by the relation "an operation f preserves a relation " (see also [10,11]). In other words, every clone F on D can be described by F = Pol D Q for some set Q of relations (cf. Section 2 for the notation).
In this paper, which is mainly based on [4], we continue the investigations from [2] and [16] concerning clones on a finite set D described by relations from a special set C R D . They are named clausal relations and were originally introduced in [8]. A clausal relation is the set of all tuples over D satisfying disjunctions of inequalities of the form x ≥ d and x ≤ d, where x, d belong to the finite set D = {0, 1, . . . , n − 1}.
We are interested in understanding the structure of clones that are determined by sets of clausal relations, so-called C-clones. Their lattice has been delineated completely in Theorem 2.14 of [16] for the case that |D| = 2. When |D| ≥ 3, the structure and even the cardinality of this lattice is largely unknown. In this paper we study the co-atoms in the lattice of all C-clones, the maximal C-clones, for an arbitrary finite set D. Since every clone on D either equals O D (the set of all finitary operations on D) or is contained in some maximal clone (co-atom of the lattice of all clones) (see, e.g., [12, Hauptsatz 3.1.5, p. 80; Vollständigkeitskriterium 5.1.6, p. 123] or [15, Proposition 1.15, p. 27]), our aim is to investigate which maximal C-clones are contained in which maximal clones. We achieve a complete description in Theorem 8.2 and thereby answer the question that was left open in the pre-print [3].
Using Rosenberg's theorem (see Theorem 2.4 below), all maximal clones on D can be classified into six types. In [3] it was already established that a few of them, e.g., centralisers of prime permutations, polymorphism sets of an affine, of a central relation of arity at least three or of an h-regular relation, do not contain any maximal C-clone. We shall see that this phenomenon extends to maximal clones of monotone functions with regard to some bounded partial order whenever |D| ≥ 3.
To our surprise, it turns out that every maximal C-clone is contained in a unique maximal clone, either given as polymorphism set of a non-trivial equivalence relation or a unary or binary central relation (vide infra for a definition of such relations). The respective details can be seen from our main result, Theorem 8.2. As a corollary we also deduce a new completeness criterion for C-clones.
We start by introducing our notation, recalling some fundamental facts about the Galois theory for clones, the characterisation of maximal clones and C-clones, respectively, and providing two basic lemmas in Section 2. After that we recollect the relevant results from the pre-print [3]. Then we devote one section each to examine possible inclusions of maximal C-clones in maximal clones of the form Pol D , where is a non-trivial unary relation, a bounded partial order relation, a non-trivial equivalence relation or an at least binary central relation. Finally, in Section 8, we deduce our main theorem from the previous results.

Main notions and preliminaries
Throughout the text, D will denote the finite non-empty set Furthermore, for a binary relation ⊆ D 2 we denote its inverse relation by We want to study clones that are determined by sets of clausal relations. Even though, for almost all results, we shall need only binary clausal relations, we define them here in full generality (the publication [8] allowed the parameters p and q to be chosen in N such that p + q > 0; for compatibility with [2,16,17] we are slightly more restrictive here, by additionally requiring that p · q > 0).
In this expression ≤ denotes the canonical linear order on D and ≥ its dual.
For k ∈ N + we denote by O Next, we consider a Galois connection between sets of operations and relations that is based on the so-called preservation relation. It is the most important tool for our investigations. it follows that also f applied to these tuples belongs to , i.e., f • (r 1 , . . . , r k ) := (f (a 11 , . . . , a 1k ), . . . , f(a m1 , . . . , a mk )) ∈ .
Given F ⊆ O D , we denote by Inv D F the set of all relations that are invariant for all operations f ∈ F , i.e., Inv } denotes the set of polymorphisms of Q. Furthermore, for k ∈ N + we abbreviate Pol On a finite set D the Galois closed sets of relations [7,5,6,9] with respect to Pol D − Inv D are exactly the so-called relational clones. These can be characterised as those sets of finitary relations on D that are closed under primitive positively definable relations, i.e., those arising as interpretations of first order formulae where only predicate symbols corresponding to relations from Q, falsity, variable identifications, finite conjunctions and finite existential quantification are allowed. For a set Q ⊆ R D of relations, we denote by [Q] R D the closure of Q with regard to such formulae, which equals the least relational clone generated by Q, i.e., by the above, we have for some set Q of clausal relations. All C-clones on D, ordered by set inclusion, form a complete lattice, whose co-atoms are called maximal C-clones.
From [17] we have a description of all maximal C-clones on finite sets as polymorphism sets of clausal relations R (a) Likewise, the following characterisation of maximal clones on finite sets is well known.
where is a non-trivial relation belonging to one of the following classes: (1) The set of all partial orders with least and greatest element. For some sorts of relations from Theorem 2.4 we give a brief explanation. If s ∈ Sym(D) is a permutation, by its graph we mean the binary relation graph s : The permutation is called prime if, for some prime p, it has only cycles of length p. In particular such a function s cannot have cycles of length one, i.e., it has to be fixed point free.
For a prime p a group G = G; +, −, o is called an elementary Abelian p-group, if G is a commutative group and satisfies the law x + · · · + x ≈ o where the variable symbol x occurs p times in the sum. The latter means that every element in G \ {o} has order p. If G is finite, then, by the fundamental theorem of finitely generated Abelian groups, G must be isomorphic to a finite direct power of the cyclic group of order p, so in particular the cardinality of G must be a power of p.
For any (not necessarily commutative) group G = G; +, −, o , we define the corresponding affine relation G : In case G is Abelian, G is given by permuting the middle two variables of the graph of the Maľcev operation (x, y, u) → x − y + u, hence its name.
A central relation is a totally symmetric, totally reflexive relation having a central element and not being a diagonal relation. Total symmetry means closure under all permutations of entries of tuples; total reflexivity requires that every tuple having two identical entries has to belong to the relation. An element c ∈ D is central for if any tuple containing c as an entry is a member of .
The only unary diagonal relations are ∅ and D, the binary ones are Δ and D × D. Therefore, unary central relations are precisely all proper subsets ∅ D. Binary central relations can be described as follows. Note that for binary relations the notions of total symmetry and total reflexivity coincide with ordinary symmetry and reflexivity, respectively. For c ∈ D define the relations c := Δ∪({c}×D)∪(D×{c}) and A c : c , and it is easy to see that all of them arise in this way. Note that for n = |D| = 3 we always have S c = ∅ as A c contains only one pair.
Supposing |D| ≥ 3, the goal of the following sections is to understand completely, for which parameters a ∈ D \ {0}, b ∈ D \ {n − 1} and which relations from Theorem 2.4 we have the inclusion Pol D R To do this, we may want to use unary functions f ∈ Pol where Pol D is a maximal clone. The following lemma gives a simple sufficient condition for functions f ∈ O When constructing unary functions f ∈ Pol where Pol D is a maximal clone, it is helpful to know how much choice we have for f . We cannot achieve a converse to Lemma 2.5, but the following result seems to be as good as we can get in this respect.  ( Proof. Statement (c) follows from (a) as the condition im(f ) ⊆ {a, . . . , n − 1} implies f {a, . . . , n − 1}. The proof of statement (b) is dual to that of (a), so we only deal with the latter one. If f {0, . . . , b}, then there exists some

Selfdual and quasilinear functions, and such preserving central or h-regular relations
In this section we recall from [3] that Pol D R  Proof. It is clear that c a1 ∈ Pol D R a b . Moreover, by the above characterisation, c a1 is s-selfdual if and only if a 1 = c a1 (x) = s −1 (c a1 (s(x))) = s −1 (a 1 ) holds (for all x ∈ D), that is, if a 1 is fixed by s. Thus, by assumption, we have c a1 / ∈ Pol D graph s.
To deal with affine and at least ternary central and h-regular relations, we need the following observation. Proof. Let (x 1 , y 1 Central and h-regular relations share the common properties of total symmetry and total reflexivity; hence they can be dealt with in one lemma. Proof. Since is non-full, we have D m , and hence there exists some tuple x := (x 1 , . . . , x m ) ∈ D m \ . By total reflexivity, the entries x 1 , . . . , x m are pairwise distinct. Choose the unique i ∈ {1, . . . , m} such that x i is the least element among x 1 , . . . , x m with respect to ≤ D and pick indices j, ∈ {1, . . . , m} such that |{i, j, }| = 3. This is possible due to m ≥ 3. Define y, z ∈ D m by y k := x i for k = j and y k := x k else; z k := x k for k = j and z k : It remains to discuss the case of affine relations. In fact, for any elementary Abelian p-group G = G; +, −, o the clone Pol G G contains precisely all quasilinear (sometimes affine linear ) maps with regard to the canonical affine space induced by G over the field GF(p). Such clones never contain maximal C-clones as the following lemma proves, even under slightly more general assumptions. , however, we shall demonstrate below that ∨ D / ∈ Pol D G , which makes an inclusion impossible. Indeed, it is obvious that (0, −0, 1, −1), (1, 0, 1, 0) ∈ G . If the maximum operation preserved G , we would get, due to 0 being the least element with respect to ≤, that (1, −0, 1, −1) By definition of G this would imply 1 + (−0) = 1 + (−1) = o, i.e., 1 = 0, an evident contradiction.
In the next section, we will attack possible inclusions Pol D R

Non-trivial unary relations
The following lemma gives sufficient conditions for binary operations to belong to a given maximal C-clone.
(b) and we are done. Else, by the assumption on f we must have . The proof of the second claim is by dualisation.
We can use this type of functions to witness non-inclusions of maximal C-clones in maximal clones given by a non-trivial unary relation whenever there exists some x ∈ respecting b < x < a.

Corollary 4.2. Consider a, b ∈ D and suppose
D contains an element x ∈ such that b < x < a. Every binary function f ∈ O  In the next step we derive a necessary condition concerning the form of the unary relation that has to hold if Pol D R  As a partial converse the next result establishes a sufficient condition for an inclusion of a maximal C-clone in a maximal clone given by a non-trivial unary relation.
Proof. The second equality stated in the lemma will follow by variable identification from R  , x). If y ≥ a, thus, (x, y) ∈ {a, . . . , n − 1} 2 . Otherwise, we have x < a, such that y ≤ b < a due to (x, y) ∈ R The following lemma solves the task for non-trivial unary relations.

The case of bounded order relations
A bounded (partial) order relation is an order relation having both, a largest (top) element , and a least (bottom) element ⊥. If ⊆ D 2 is an order relation on D, considered to be clear from the context, and a, b ∈ D are any two elements, we occasionally use the notation [a, b] := {x ∈ D | a x b} and call it the interval from a to b. Clearly, if a b, then [a In the first step we construct binary functions witnessing non-inclusions of certain maximal C-clones in maximal clones described by non-trivial binary reflexive relations. Moreover, let D 2 be reflexive, (x, y) ∈ \Δ, (u, v) ∈ D 2 \ , b < z < a, and suppose, in addition to the above, that g(x, z) = u and g(y, z) = v. Then and x 2 ≥ a, then g(x 1 , x 2 ) ≥ a. Otherwise, we have x 2 < a and y 2 ≤ b, which implies g(y 1 , y 2 ) ≤ b. In both cases we have (g(x 1 , x 2 ), g(y 1 , y 2 )) ∈ R (a) (b) . Furthermore, we have (x, y), (z, z) ∈ , but (g(x, z), g(y, z)) = (u, v) / ∈ , proving g .
If a−b ≥ 2, the many requirements on the binary function in the previous lemma are actually satisfiable. For all a, b ∈ D such that a − b ≥ 2 and every non-trivial binary reflexive relation Δ D 2 , we have Pol Since a − b ≥ 2, functions g fulfilling the assumptions of Lemma 5.1 are indeed constructible. Choosing pairs (x, y) ∈ \ Δ and (u, v) ∈ D 2 \ , we may, for instance, define g(w, z) := 0 ≤ b for z ≤ b, g(w, z) := n − 1 ≥ a for z ≥ a, g(w, z) := u for b < z < a and w = x, and g(w, z) := v else, i.e., for all (w, z) ∈ D 2 satisfying b < z < a and w = x. Since y = x, this ensures that g(y, z) = v for all b < z < a, and thus g fulfils the conditions of Lemma 5.1.

So the preceding result shows that inclusions Pol
impossible whenever a − b ≥ 2 and is a non-trivial equivalence, bounded order relation or binary central relation. In order to exclude more inclusions, we shall use the following trivial observation.
then there exists some f ∈ Pol Proof. In each case one can explicitly define a unary operation f ∈ O   The second main case is when a ≤ ⊥ ≤ b. If there is some a ≤ x ≤ b such that x = ⊥, then we use (x, ⊥). Otherwise, [a, b] ⊆ {⊥}, and so a = ⊥ = b. Due to n ≥ 3, we have again = ⊥ = a = b. Let us consider the situation < a. If there exists some x < a, x = , then we may use (x, ), else every x < a equals , so = 0 < a = 1 = b = ⊥. This possibility is treated in Lemma 5.4(e). The opposite situation is that > a = b. If there exists some x > b, x = , then we use (x, ), otherwise every x > b equals , and so = n − 1 > b = n − 2 = a = ⊥, which is solved in case (f) of Lemma 5.4. Third, let us deal with the possibility that ⊥ > b. If there exists some x > b, x = ⊥, then we can use the transposition (x, ⊥). Otherwise, every x > b equals ⊥, so ⊥ = n − 1 > b = n − 2. Due to n ≥ 3, we have = ⊥ = n − 1, i.e., ≤ n − 2 = b. The first subcase is that < a. If there exists some x < a, x = , we use the transposition (x, ). Else, all x < a satisfy x = , so we obtain = 0 < a = 1, b = n−2 < ⊥ = n−1, which is treated in Lemma 5.4(c).
which has been dealt with in Lemma 5.4(d).
So in the case that a − b ≤ 1, we have always found a transposition or a unary operation as constructed in Lemma 5.4 that preserves R

The case of non-trivialequivalence relations
Throughout this section, we shall employ the notation Eq D for the set of all equivalence relations on D. It is our aim to show that maximal C-clones Pol D R (a) (b) are contained in a maximal clone given by a non-trivial equivalence relation if and only if a = b+1. In this case the equivalence relation is uniquely determined.
As our first result, we provide a simple sufficient condition for an inclusion in a maximal clone described by an equivalence relation. In the remainder of this section we shall prove that the situation described in Lemma 6.1 is the only one, where a maximal C-clone can be contained in a maximal clone given by a non-trivial equivalence relation.
As a first step, we establish a few necessary conditions. (c) For all x, y, z ∈ D where (x, y) ∈ θ \ Δ, we have the implication  Obviously, (z, x) ∈ θ, but (f (z), f(x)) = (z, y) / ∈ θ, as otherwise (x, z) ∈ θ and transitivity would imply (x, y) ∈ θ. Thus, f θ. Moreover, as x, y ∈ I, we have f ∈ Pol D S, which implies that f R (c): Let x, y, z ∈ D where (x, y) ∈ θ and x = y. Moreover, the assumption of the implication is that we can find w ∈ {x, y} such that w, z ≥ a or w, z ≤ b. We define f ∈ O So we get (f (x), f(y)) ∈ θ from (x, y) ∈ θ. If w = x, this means (x, z) ∈ θ. Else, if w = y, we obtain (z, y) ∈ θ, which together with (x, y) ∈ θ yields (x, z) ∈ θ.