Galois theory for semiclones

We present a Galois theory connecting finitary operations with pairs of finitary relations one of which is contained in the other. The Galois closed sets on both sides are characterised as locally closed subuniverses of the full iterative function algebra (semiclones) and relation pair clones, respectively. Moreover, we describe the modified closure operators if only functions and relation pairs of a certain bounded arity, respectively, are considered.


Introduction
Clones of operations, i.e. composition closed sets of operations containing all projections (cf. [30,34,24,15]), play an important role in universal algebra as they encode structural properties independently of the similarity type of the algebra. It is well-known (see [9,14], translations available in [7,8]) that on finite carrier sets clones are in a one-to-one correspondence with structures called relational clones. This is established via the Galois correspondence Pol -Inv, which is induced by the relation of "functions preserving relations". In general, i.e. including in particular the case of infinite sets, so-called local closure operators come into play (see [14,29,28,23,3]), and also the notion of relational clone as known from finite domains needs to be generalised (cf. ibid.). In this way the Galois connection singles out certain locally closed clones from the lattice of all clones on a given set. These clones can also be seen as those which are topologically closed w.r.t. the topology that one gets by endowing each set A A n , n ∈ N, with the product topology arising from A initially carrying the discrete topology (see e.g. [5,4]).
By equipping the set of all finitary functions on a fixed set A with a finite number of operations (including permutation of variables, identification of variables, introduction of fictitious variables, a certain binary composition operation and a projection as a constant; we present more details later on), one obtains the full function algebra of finitary functions on A. It is known (cf. e.g. [30,24]) that the clones on A are exactly the carrier sets of subalgebras of this structure. This relationship is a special case of the one between the full iterative function algebra, also known as iterative Post algebra (introduced by Maľcev in [26]), and its subuniverses (called Post algebras in [9]), which have often only been referred to as closed classes of functions in the Russian literature (e.g. [20,21]). These are similar in spirit to clones, but they do not need to contain the projections (selectors in the terminology of [26]) as the iterative Post algebra omits the projection constant in its signature compared to the full function algebra.
In analogy to the Pol -Inv Galois connection, a Galois correspondence Polp -Invp has been developed in [16] (see also [17,18,19]) based on the notion of functions preserving pairs (̺, ̺ ′ ) of relations ̺ ′ ⊆ ̺. For finite carrier sets the Galois closed sets have been characterised to be precisely the subuniverses of the full iterative Post algebra and the subuniverses of a suitably defined relation pair algebra, respectively. To the best knowledge of the author, a generalisation of this result to arbitrary base sets has not yet appeared in the literature. In particular the general (and thus infinite) case is also missing in Table 1 of [13, p. 296] summarising related Galois connections and characterisations of their closure operators.
In this article it is our aim to fill in this gap. We first coin the notion of a semiclone, which relates to transformation semigroups in the same way as clones relate to transformation monoids. It is not hard to figure out that semiclones and subuniverses of the full iterative Post algebra coincide. However, not only is the name shorter, but also do we feel that the way how a semiclone is defined is much more similar to the usual definition of a clone and easier to grasp than that of a subalgebra of the full iterative function algebra; hence the proposition of the new terminology of semiclones. Unfortunately, our semiclones are different from those appearing in [32], which are closed w.r.t. a different form of composition, do have to contain the identity operation, but are not necessarily closed under variable substitutions.
In a similar fashion as one needed to generalise the notion of relational clone to accommodate the closed sets of Inv A Pol A for infinite carrier sets A, it will be necessary to modify the relation pair algebra proposed by Harnau in [16]. We shall refer to the corresponding (new) subuniverses as relation pair clones.
Using the same local closure operator as introduced in [14,28,29] for sets of functions (the topological closure), and appropriately modifying the local closure on the side of relation pairs, we shall prove the following two main results: the Galois closed sets of operations w.r.t. Polp A Invp A are exactly the locally closed semiclones. Dually, the closed sets of Invp A Polp A are precisely the locally closed relation pair clones.
Because it fits nicely in this context, we shall more specifically study and characterise what it means that a semiclone can be described in the form Polp A Q for some set Q of at most s-ary relation pairs, and that a relation pair clone is given by Invp A F using a set F of at most s-ary operations. As in [29] this involves certain s-local closure operators, and, in general, the reader may find that quite a few results in our text are analogous to those in [29], where similar questions have been studied w.r.t. Pol -Inv.
We mention that a related, in some sense more general, Galois connection has been studied in [27] (finite case) and [12,11]. There, for fixed sets A and B, functions f : A n −→ B have been related to pairs of relations R ⊆ A m , S ⊆ B m for some m ∈ N + , called relational constraints. In this situation the Galois closed sets on the functional side are also closed w.r.t. variable substitutions (as our semiclones), but already for syntactic reasons cannot be closed w.r.t. compositions. So even if one considers the special case that B = A, the results from [12] and [11] describe similar but differently closed sets of functions due to other objects on the dual side (there is no containment condition for the relations as in our setting since for general A and B there cannot be one).
We acknowledge that, perhaps, it could be possible to derive our results by restricting the relational side of the Galois correspondence studied in [12,11], but we think that the way of describing the closed objects on the dual side used there is (and has to be) more complicated (using so-called conjunctive minors), in fact, too technical for our situation. Besides, our strategy of proof exhibits more similarities with the classical arguments known from clones and relational clones. Also the local closures developed for relational constraints in [12,11] necessarily need to be modified (see Remark 2.16) to be used with our relation pairs due to the inclusion requirement in their definition.
Still a different weakening of the notion of clone and an associated Galois theory for arbitrary domains has been considered in [25]: there sets of functions that contain (as clones do) all projections, are closed under substitution of one function into the first place of another one, permutation of positions and addition of fictitious variables but are not necessarily closed under variable identification (as semiclones are) have been characterised in terms of closed sets of so-called clusters. For the classes of functions characterised in [25] contain all projections, these results explore a separate direction and cannot be exploited either to obtain the missing general (infinite) case for semiclones.
Acknowledgements. The author expresses his gratitude to Erhard Aichinger for an invitation to the Institute for Algebra at Johannes Kepler University Linz, which enabled fruitful discussions on some aspects of the topic with members of the institute including Erhard Aichinger, Peter Mayr, Keith Kearnes and Ágnes Szendrei. The author wishes to thank them, too, for their valuable comments and contributions.

Notation, functions and relations.
In this article the symbol N will denote the set of all natural numbers (including zero), and N + will be used for N \ {0}. Moreover, we shall make use of the standard set theoretic representation of natural numbers by John von Neumann, i.e. n = { i ∈ N | i < n} for n ∈ N. The power set of a set S will be denoted by P (S).
When discussing semiclones, relation pair clones and their Galois theory we shall make no further assumptions on the carrier set, which we usually represent by A. Any finite (including 0) or infinite cardinality is allowed for A.
For sets A and B we write A B for the set of all mappings from B to A. The order of composition employed in this article is from right to left, i.e. g • f ∈ C A for f ∈ B A and g ∈ C B . That is, g • f maps elements a ∈ A to g(f (a)). For any index set I, sets A and (B i ) i∈I and maps (f i : is the i-th projection map belonging to the Cartesian product j∈I B j . As any ambiguity can usually be resolved from the context, we denote the tupling h by (f i ) i∈I , in the same way as the tuple (f i ) i∈I .
The notion of tupling is, of course, meaningful (by definition) in any category having suitable products, and hence the following simple lemma about composition of tuplings can be proven in such a general context. We recall here just its instance for the category of sets (cf. [3, Lemma 2.5]): Lemma 2.1. Let I and J be arbitrary index sets, k, m, n ∈ N natural numbers, and A, B, D, X and B i (i ∈ I), C j (j ∈ J) be sets. Furthermore, suppose that we are given mappings r : then we have the superassociativity law for finitary operations on X.
As in our modelling natural numbers are sets, we consequently interpret tuples as maps, too: if B = n ∈ N is a natural number, then A B = A n is the set of all n-tuples x = (x(i)) i<n . We shall often write x i for the entry x(i) (i ∈ n), and, whenever convenient, we shall also refer to the entries of tuples by different indexing, e.g. x = (x 1 , . . . , x n ). Note that the sole element of A 0 = A ∅ is the empty mapping (tuple), whose graph is the empty relation. It will consistently be denoted by ∅. As tuples are functions we may compose them with other functions: for instance, if x ∈ A n and α : m −→ n, (m, n ∈ N), then x • α is the tuple in A m whose entries are Any mapping f ∈ A A n (n ∈ N) is called an n-ary operation on A, and the number n is referred to as its arity, denoted by ar (f ). The set of all finitary operations on A is O A := k∈N A A k . Note that we explicitly include nullary operations here, which is slightly uncommon in standard clone theory. For a set of operations F ⊆ O A we denote its n-ary part by F (n) := F ∩ A A n . We extend this notation to operators yielding subsets of operations: if OP : is an operator on a set S, then we define OP (n) : S −→ P O (n) A by the restriction OP (n) (s) := (OP(s)) (n) for s ∈ S. Based on this, we put moreover OP (n1,...,n k ) (s) := k i=1 OP (ni) (s) for s ∈ S and a finite list of arities n 1 , . . . , n k , k > 0. We also abbreviate OP (0,...,n) as OP (≤n) , and for s ∈ S we let OP (>0) (s) := OP(s) \ OP (0) (s).
The projection operations belonging to the finite Cartesian powers of the carrier set play a special role. For n ∈ N and i ∈ n, we denote by e the n-ary projection on the i-th coordinate. Evidently, there do not exist any nullary projections. Therefore, the set of all projections on A, denoted by J A , equals n∈N+ e (n) i 0 ≤ i < n . For the identity operation e (1) 0 we occasionally also use the notation id A .
Knowing about re-indexing tuples, we can recollect the notion of polymer. If m, n ∈ N are arities, α : n −→ m is any indexing map and f ∈ O A by some map α : n −→ m, m ∈ N, is said to be a polymer of f . Clearly any polymer of f can be obtained by composition with a suitable tupling of projections: Besides operations we shall also need relations: for m ∈ N any subset ̺ ⊆ A m of m-tuples is an m-ary relation on A. Thus P (A m ) is the set of all m-ary relations, and, again allowing arity equal to null, the set of all finitary relations is defined by    for each m ∈ N, which is given by set inclusion in both components. That is, we write (σ,  We say that a relation pair (σ, σ ′ ) ∈ Rp A is a relaxation of some other pair (̺, ̺ ′ ) ∈ Rp A (cf. [12, p. 153   (m ∈ N) on a set A, we say that f preserves (̺, ̺ ′ ) and write f ⊲ (̺, ̺ ′ ) if the following equivalent conditions hold: (i) For every tuple r ∈ ̺ n , the composition of f with the tupling (r) of the tuples in r belongs to the smaller relation: f • (r) ∈ ̺ ′ .
(ii) For every (m × n)-matrix X ∈ A m×n the columns X −,j (j ∈ n) of which are tuples in ̺, the tuple (f (X i,− )) i∈m obtained by row-wise application of f to X yields a tuple of ̺ ′ .
Note in this respect that for any tuple r = (r j ) 0≤j<n ∈ (A m ) n where for 0 ≤ j < n each tuple is given as r j = (r ij ) 0≤i<m , the definition of tupling precisely yields that f • (r) = f (r ij ) 0≤j<n 0≤i<m , i.e. the result of applying f row-wise to the matrix (r ij ) (i,j)∈m×n ∈ A m×n . Note furthermore, that for ̺ ∈ R A and f ∈ O A the condition f ⊲ (̺, ̺) coincides with the usual preservation condition for functions and relations (cf. [3,Definition 2.3] for the framework involving nullary operations).
Based on the preservation condition we introduce a Galois correspondence in the usual way: for a set F ⊆ O A we denote by the set of its invariant relation pairs, and, dually, for Q ⊆ R A , the set If we restrict the latter just to relation pairs (̺, ̺ ′ ) where ̺ = ̺ ′ , then we get the standard Galois connection Pol -Inv : The name polymorphism attributed to the functions in Pol A Q for sets of relations Q ⊆ R A comes from the fact that an operation f ∈ O A belongs to Pol A Q if and only if it is a homomorphism from the power  The proof is a straightforward rewriting of the definitions and is therefore omitted.
It is an evident consequence of the definition of preservation that sets of the form Invp A F , F ⊆ O A , are closed w.r.t. relaxation (cf. [18,Lemma 8,p. 16]).
Proof. Although the statement easily follows from the definition, we present here another argument, based on the local closure LOC A of sets of relation pairs (see Definition 2.8). For any Q ⊆ Rp A we have Q ⊆ → Q ← , and we shall prove → Q ← ⊆ LOC A Q in Corollary 2.10. Hence, we get the inclusions The following result provides a simple reformulation of the previous lemma.
Some relation pairs are preserved by no operation. If they are part of a set Q ⊆ Rp A or, more generally, part of Invp A Polp A Q, then Polp A Q is forced to be empty. The next lemma characterises when this happens (cf. [16, p. 15] and [18, p. 12]).
Proof. If ̺ ∈ Rp A is non-empty, then also ̺ n = ∅ for any possible n ∈ N. Therefore, the condition in Definition 2.
which clearly contains the relation pair A 0 , ∅ . The nullary relation A 0 is never empty, even for A = ∅, so the exhibited example is of the right form.
Remark 2.7. The previous lemma demonstrates the necessity to include nullary relations in the framework, caused by our wish not to impose any restriction on the carrier set A. Namely, for A = ∅, we have A m = ∅ for all m ∈ N + , and thus R Both sets are evidently semiclones (subalgebras of the iterative Post algebra), on any carrier set A, so, in view of our overall objective, it is more than desirable to be able to model them with our Galois correspondence. Restricting to relations of positive arity, this would clearly be impossible for A = ∅. In fact, in order to characterise Galois closures of sets of at most s-ary operations / relations (s ∈ N) we define more specific variants of s-local closure operators. Note that apart from extending the scope of the definition to s = 0 and nullary operations, the operators s-Loc A and Loc A we define coincide with those from [29, 1.9, p. 15] (see also [28, 1.5 and call these s-local and local closure operators, respectively.
It is easy to check that s-Loc A , Loc A , s-LOC A and LOC A are indeed closure operators on the sets of finitary operations and relation pairs, respectively. Likewise, it is not hard to see that for every s, n ∈ N we have s-Loc To make a technical remark: if we had not insisted on using the disjoint union for the definition of Rp A , then for any n ∈ N we would have 0-LOC A Q (n) = Rp A whenever (∅, ∅) ∈ Q (as in this case (∅, ∅) ∈ Q (n) (m) were true for all m ∈ N), and this would obviously violate the equality mentioned above:

Moreover, it follows directly from the definition that t-Loc
It follows from these relations that Note that our definition of s-local closure of relation pairs for s ∈ N + entails the corresponding one for relations given in [29, 1.9, p. 16] in the following way: for

s-Loc A t-Loc
The local closure LOC A of sets of non-nullary relations can be handled in a similar way.
Furthermore, the following characterisation is also simple to verify.
From this result it easily follows that closure w.r.t. relaxation is just a special case of the local closure of relation pairs.
The following consequence is now evident.
In particular, in case of finite carrier sets, the inclusion in Corollary 2.10 is always an equality.

Corollary 2.13. For finite
The following closure property will become important regarding the characterisation of the closure operator Invp A Polp Clearly, this condition is equivalent to T being non-empty and that for all (X i ) i∈s ∈ T s and r ∈ i∈s X i there exists We prove now that sets of the form s-LOC A Q, where Q ⊆ Rp A , are closed w.r.t. unions of s-directed systems of relation pairs of the same arity.
Proof. Clearly, we have σ ′ : A . In order to prove that it belongs to s-LOC A Q, we consider any subset For m ∈ N and we say that a set T ⊆ Rp This means we require the condition presented before Lemma 2.14 to hold for any finite sequence of relations and tuples.
We call a set T ⊆ Rp This is equivalent to saying that for any finite subset F ⊆ T there is an pair (̺, ̺ ′ ) ∈ T such that (µ,µ ′ )∈F µ ⊆ ̺, wherefore directedness clearly implies ℵ 0 -directedness. As a consequence of this implication we get that locally closed sets of relation pairs are closed under directed unions of sets of pairs of identical arity.
Under additional assumptions on the set of relation pairs Q we shall extend Lemma 2.14 and Corollary 2.15 to characterisations of local and s-local closedness. We conclude this subsection with remarks on the relationship of our local closure operators to others defined in the more general setting treated in [11].
Remark 2.16. The local closure operators (and s-local closure operators for s ∈ N + ) defined here cannot directly be derived as special cases of the corresponding closure operators from [11]. As the case of local closures is similar, we shall only argue for s-local closures. Specialising the framework in the mentioned article for a pair of carrier sets (A, B) where B = A, we may apply the As this set contains pairs (R, S) that are not relation pairs, i.e. failing the condition R ⊇ S, the canonical modification would be to simply intersect the result with Rp A , leading to This set equals Q on any set A (in fact, the second part of the union is empty, whenever A = ∅, as for C = ∅ and T = A m = ∅ the condition (C, T ) ∈ Q is never satisfied). So the original definition of LO s (or its canonical modification) is not helpful at all in our setting.
Suppose, in the union over m ∈ N + , we change the condition describing when a relation pair (R, S) is added to the s-local closure of Q as follows: among all relational constraints (C, T ) relaxing (R, S) and verifying |C| ≤ s only those are required to be in Q that are indeed relation pairs. Then we get This set still differs from s-LOC A Q as defined above. For instance for any s ∈ N + and Q = ∅ we have s-LOC A Q = ∅, while the previously displayed collection contains all relation pairs (R, S) ∈ Rp A where |S| > s.
We do not see an obvious way how to translate LO s into s-LOC A or vice versa.

Semiclones and the full iterative Post algebra
The following definition is very similar to that of a clone of operations. The only difference is that a clone F ⊆ O A is additionally required to contain the set J A of projections as a subset.
The closure property stated in Definition 3.1 is formulated in terms of partial composition operations on O A as the functions making up the tupling all have to be of identical arity. However, it is possible to extend these operations in a conservative way to totally defined operations on O A such that semiclones are exactly the subuniverses of a certain universal algebra on the carrier set O A : for each n, m ∈ N and each subset I ⊆ n and any tuple (g i ) i∈I ∈ J (m) A of m-ary projections we define an (|n \ I| + 1)-ary operation on O A , which maps f, A of unary transformations, abbreviate its generated transformation semigroup by S : Proof. S is obtained from G by closure w.r.t. composition of unary operations, which is part of the requirement in Definition 3.1. Thus, we have S ⊆ [G] OA , but now the closure property clearly yields that [G] OA must contain the whole set on the right-hand side as a subset.
Conversely, it is easy to check that the latter collection, first of all, contains S and therefore G, and second, actually forms a semiclone. Thus, it must be a superset of the least semiclone containing G, which is Proof. By definition, the restriction F (1) of any semiclone F ∈ S A forms a transformation semigroup. The converse inclusion follows from Lemma 3.2 as we have [S] (1) As mentioned in the introduction, semiclones are not a new invention. They are just the subuniverses ("closed classes of functions") of the full iterative Post algebra. In order to see this we need a few definitions.
In this way, we obtain an algebra O A := O A ; ζ, τ, ∆, ∇, * of arity type (1, 1, 1, 1, 2) that we call full iterative Post algebra. It is easy to see that A is a subuniverse, and the corresponding subalgebra is the one that has been introduced under precisely the same name in [26]. The difference in terminology is just of technical nature and shows up because we wish to accommodate all nullary constants in our framework.
The algebra O A obviously is less prodigal of its fundamental operations than O A ; Φ introduced above. The following lemma proves that both actually do the same job. Proof. We saw earlier that any polymer can be expressed as a composition with a tupling of projections under which semiclones are closed by definition. Thus any semiclone is closed w.r.t. the unary operations ζ, τ , ∆ and ∇. By construction of * , it is also closed w.r.t. * .
It is a tedious, but well-known exercise (known from the proof that clones are exactly the subuniverses of function algebras, which differ from iterative algebras by just adding an additional constant representing a projection) that the converse also holds: for arities m, n ∈ N and operations f ∈ F (n) and . . , g n−1 ) can be expressed as the result of a term operation of O A applied to (f, g 0 , . . . , g n−1 ). For F ∈ Sub (O A ) this means that any such composition also has to belong to F .
The following facts on the relationship of semiclones and clones are wellknown (see [16, p. 5 et seq.] or [17, p. 8 et seq.], Lemmata 3 and 4, and Satz 1). In this context, we recollect that F OA denotes the least clone containing some set F ⊆ O A , i.e. the clone generated by F . The symbol L A stands for the set of all clones on A.
Lemma 3.5. For any set F ⊆ O A and any 0 ≤ i < n, n ∈ N, the following assertions are true: Proof. Fix any set F ⊆ O A and any projection e The set of projections is a clone, and hence a semiclone. We only need to check that e (n) i generates any other projection. First, we note that follows from (a), and the inclu- ⊆ F OA holds as each clone is a semiclone. Finally, [F ] OA ∪ J A contains F , and it is easy to check that it is indeed a clone. Therefore, it has to contain F OA as a subset. (c) If F is a semiclone and F ∩ J A = ∅, then for some arity n ∈ N and some As a consequence of the previous lemma, we can describe those semiclones whose unary parts yield proper transformation semigroups, i.e. those which are no monoids.

Corollary 3.6. On any set A we have
The Galois correspondence Polp -Invp gives us plenty of examples of semiclones (cf. [18,Lemma 2,p. 12] for the situation without nullary operations).

Lemma 3.7. Any polymorphism set Polp
n . We prove that the The latter tuple is a member of The following facts can be routinely proven using Lemma 3.7.
Proof. If Q ⊆ Rp A only consists of identical pairs, then we saw already in Sub- This set always is a clone. On the other hand, if Polp A Q is a clone, then we have id A ∈ Polp A Q, which implies ̺ ⊆ ̺ ′ and thus ̺ = ̺ ′ for every (̺, ̺ ′ ) ∈ Q.
Note that (along with an appropriate generalisation of preservation) the three previous statements remain true if one considers relation pairs of arbitrary, possibly infinite arity. That is to say, pairs (R, S), where S ⊆ R ⊆ A K for some fixed set K.
The following two results are in close analogy to Proposition 1.11(a),(b) from [29, p. 17].
Proof. Consider n ∈ N and g ∈ s-Loc For this consider any n-tuple r = (r j ) 0≤j<n ∈ ̺ n of tuples from ̺ and define , and the latter tuple belongs to ̺ ′ as f ⊲ (̺, ̺ ′ ) ∈ Q. Lemma 3.10. As the pair (̺, ̺ ′ ) ∈ Q was arbitrarily chosen, we obtain g ∈ Polp A Q.

Relation pair clones
In this section we first recollect the so-called general superposition of relations ([29, Definition 3.4(R4), p. 27], see also [28, Definition 2.2(ii), p. 258] and [3]), which comes into play when generalising the notion of relational clone from finite carrier sets to arbitrary ones. It is not surprising that it will be important for the generalisation of relation pair algebras as introduced in [16, p. 21] (see also [18, p. 16]) to carrier sets of arbitrary cardinality, as well. , i ∈ I, be given. The general superposition of these relations w.r.t. the given data is defined to be the m-ary relation β (αi) i∈I We mention in passing that, in general, a relational clone can be defined as any set Q ⊆ R A that is closed w.r.t. general superposition. That is, whenever data as in Definition 4.1 is given and all relations ̺ i , i ∈ I, belong to Q, then also β (αi) i∈I (̺ i ) i∈I has to be an element of Q (if nullary relations are disregarded, then one restricts the integers m and (m i ) i∈I to positive ones only). Depending on the carrier set A, one can work out cardinality bounds on the sets I and µ involved in this closure property, but this is not our concern here. Different specialisations of the general superposition yield operations known from the closure property corresponding to relational clones on finite carrier sets: variable permutation, projection onto arbitrary subsets of coordinates, variable identification, addition of fictitious coordinates, all diagonal relations as nullary constants, and (even arbitrary) intersection of relations of the same arity.
We now straightforwardly extend the general superposition from relations to relation pairs.
we define their general superposition to be β (αi) i∈I It is easy to see that this definition is well-defined, i.e. that we really have in the situation described in Definition 4.2. This allows us to define relation pair clones as such sets of relation pairs that are closed under general superposition.
. One can routinely check that for a given carrier set A the collection of all relation pair clones on A is a closure system. We denote the corresponding closure operator by Q → [Q] Rp A for Q ⊆ Rp A and refer to [Q] Rp A as the relation pair clone generated by Q.
Note that for finite carrier sets A = ∅, and provided that (∅, ∅) ∈ Q (m) for all m ∈ N, our concept of locally closed relation pair clone, by taking Q \ Rp A , subsumes that of subuniverses of the full relation pair algebra defined in [16, p. 21] (see also [18, p. 16]).
There are two issues here: the necessity to add local closure and the requirement that pairs of empty relations have to belong to relation pair algebras in Harnau's sense. We noted above in Corollary 2.13 that for finite carrier sets closure under relaxation coincides with our local closure of relation pairs. Moreover, we shall prove in Corollary 5.14 that the closed sets w.r.t. Invp A Polp A are precisely the locally closed relation pair clones, which implies for finite carrier sets that they are exactly those relation pair clones that are closed w.r.t. relaxations. In [16] and [18] this additional closure property (with the goal of characterising the Galois closures) has been incorporated into the definition of the full relation pair algebra via multioperations d v and d h ; however, it has been noted that these operators are of a different nature than the other fundamental operations of relation pair algebras. Comparing to the situation known from clones and relational clones on arbitrary domains (see [29,28,33]) and looking from the perspective of infinite carrier sets, which requires local closures anyway, it is justified to modify Harnau's definition by separating closure properties related to concrete constructions involving relations from local interpolation properties. We mention that for finite A the constructive part can be expressed via interpretations of primitive positive formulae in both components. In fact, it was noted by Ágnes Szendrei that given a set Q ⊆ Rp A , one may consider the relational structures If σ and σ ′ are both non-empty, then one may obtain the relation pair (σ, σ ′ ) defined by ϕ as projections ofσ. If one of them is the empty set, thenσ = ∅ and therefore both projections will be empty. Thus only taking projections ofσ (i.e. of pp-definable relations in the product ✿✿ A × ✿✿ A ′ ) will never produce relation pairs (σ, σ ′ ) where σ ′ = ∅ σ, which is certainly needed, e.g., to model intersection in both components. However, collecting all pairs (σ, σ ′ ) arising from primitive positive formulae ϕ correctly describes the closure [Q] Rp A in the case of finite carrier sets.
The second issue pointed out above is related to nullary operations. In the literature these are often neglected, which makes it necessary for relation pair algebras ( [18]) and for relational clones (relation algebras, [29]) to contain the empty pair (∅, ∅) and the empty relation, respectively, in order to be in accordance with the corresponding Galois theory.
If nullary operations are given their proper place, this absurdity vanishes (see [3] for clones and relational clones); then empty relations (pairs) get a true function, indicating by their presence the absence of nullary operations on the dual side (see Lemma 4.8 below). This is also the reason why we cannot and do not add the empty pairs of all arities as nullary constants to the closure condition of relation pair clones.
Relational clones (as given in [3, Definition 2.2, p. 8]) relate to relation pair clones in the following way: For this let f ∈ F and put n := ar (f ). To verify that f ⊲ (̺, ̺ ′ ), let us take any r ∈ ̺ n . By definition of ̺ = β (αi) i∈I (̺ i ) i∈I , for each 0 ≤ j < n there exists a j ∈ A µ such that r (j) = a j • β and a j • α i ∈ ̺ i for all i ∈ I. By putting a := (a j ) j∈n ∈ (A µ ) n , we hence obtain r = (a j • β) j∈n = a • β (cp. Lemma 2.1(a)). Therefore, by associativity, we get f As a direct consequence we get the following compulsory corollary.

Corollary 4.7. For any set Q ⊆ Rp
Next, we quickly address how nullary operations affect the associated relation pair algebras.  A Invp A F . Consider any f ∈ F , then n := ar (f ) necessarily fulfils n ≤ s. Therefore, if we consider any r = (r j ) 0≤j<n in σ n and put B := { r j | 0 ≤ j < n} ⊆ σ ⊆ A m , we clearly have a finite subset B ⊆ σ of cardinality at most n ≤ s.

Corollary 4.10. The equality LOC A Invp
As the function f ∈ F was arbitrarily chosen, we obtain (σ, σ ′ ) ∈ Invp A F .

Characterisation of closures related to Polp -Invp
In this section we characterise, for any parameter s ∈ N, the operators Polp A Invp Note that the lemma also shows that the relations ̺, ̺ ′ ∈ R (m) A do not depend on the order of the entries of the tuple b. Furthermore, instead of the finite cardinal m, any cardinal or, in fact, any indexing set K can be used, provided the notion of preservation is straightforwardly extended to relation pairs of arbitrary arity, i.e. pairs (R, S) such that S ⊆ R ⊆ A K .
A F . Proof. The claim follows from Lemma 5.1 by observing that the equality We are now prepared to prove our first theorem, characterising the closure Polp A Invp (≤s) A for s ∈ N.

Theorem 5.3. For s ∈ N and any set of operations F ⊆ O A we have the equality Polp
For the converse inclusion take g ∈ Polp F for any n ∈ N; we want to prove that g ∈ s-Loc To do so, we consider any finite X ⊆ A n where k := |X| ≤ s and an arbitrary bijection β : k −→ X. Now Corollary 5.2 yields that ̺ X,n , ̺ ′ X,n ∈ Invp The following simple observation is not unexpected.

Lemma 5.4. Any relation pair clone Q ⊆ Rp A on a non-empty carrier set
Hence, we can prove the first corollary to our theorem.

Corollary 5.5. For s ∈ N and any set of operations F ⊆ O
Proof. By Lemma 4.6, the set Invp A F is a relation pair clone, so Lemma 5.4 yields Invp A F for all m ≤ s, and hence, we have where the last equality holds by Theorem 5.3.
The claim for A = ∅ follows by similar transformations.
The second corollary characterises the closure Polp A Invp A .
Proof. Using the definition of the operator Loc A and Theorem 5.3, we can write These two facts can be seen as generalisations of Lemma 2.5(ii),(iii) in [29, p. 22], where similar results have been proven for clones.

Corollary 5.7. For s ∈ N a set F ⊆ O A of operations is an s-locally (locally) closed semiclone if and only if s-Loc
Proof. (a) By Corollaries 5.5 and 5.7, and since m ≤ s, we have A once more on both sides.
Next, we turn to the characterisation of the other part of the Galois connection.
5.2. The side of relation pairs. We start by preparing the proof of our theorem with a lemma.

Theorem 5.10. For s ∈ N and any set Q ⊆ Rp A of relation pairs we have
For the converse inclusion let us consider m ∈ N and an arbitrary m-ary pair (σ, σ ′ ) ∈ Invp OA due to superassociativity. Hence, we obtain F (n) ⊆ F (s) OA . For n = 0 < s the previous lemma (and its proof) fail. This is why in the following corollary to Theorem 5.10 arities s and 0 are required.

Corollary 5.12. For s ∈ N and any set Q ⊆ R A of relation pairs we have the equality Invp
Proof. As, by Lemma 3.7, the set Polp A Q is a semiclone, Lemma 5.11 is applicable and yields Polp where the last equality is true by Theorem 5.10.
In case that the relation pairs contain an empty pair, the nullary polymorphisms in Corollary 5.12 vanish.

Corollary 5.13. For s, m ∈ N and any set Q ⊆ Rp
Proof. Since a pair of empty relations belongs to Q, and thus to Invp A Polp A Q, A Q = ∅. Therefore, the claim follows from Corollary 5.12. Next, we characterise the closure Invp A Polp A .

Corollary 5.14. We have Invp
Proof. From the definition of the operator LOC A and Theorem 5.10, we obtain Moreover, we can infer that a set Q ⊆ Rp A is closed w.r.t

. [ ] Rp A and s-LOC A if (and clearly only if) it is closed w.r.t. to the operator s-LOC A [ ] Rp
A . An analogous result holds, of course, for the operators [ ] Rp A , LOC A and LOC A [ ] Rp A . These two facts can be seen to generalise Proposition 3.8(ii),(iii) in [29, p. 30], where similar statements have been proven for relational clones.

Proof. Suppose that [Q] Rp A = Q and s-LOC
Proof. (a) It suffices to prove that Polp true for all n ≤ s. Upon application of Corollary 5.12, we can infer that Proof. To prove the inclusion "⊇", let us consider any m ∈ N and a pair (σ, σ ′ ) ∈ Rp (m) A satisfying the lengthy condition in the proposition. Its first part says that there is an s-directed system T ⊆ Rp (m) A whose union equals (σ, σ ′′ ) for some m-ary relation σ ′′ ⊆ σ ′ . The remaining part states that for every pair (̺, ̺ ′ ) ∈ T there is an m-ary relation̺ ⊇ ̺ such that (̺, ̺ ′ ) ∈ Q (m) .
Since the set s-LOC A Q is s-locally closed, Corollary 2.11 implies that it is also closed under relaxation. Hence, we have T ⊆ → s-LOC A Q ← = s-LOC A Q. Now as T is an s-directed system, Lemma 2.14 yields that (σ, σ ′′ ) ∈ s-LOC A Q. Thus, from For the converse inclusion, take any (σ, σ ′ ) ∈ s-LOC (m) A Q, m ∈ N. Then for any B ⊆ σ such that |B| ≤ s, the set Σ B := (̺, ̺ ′ ) ∈ Q (m) B ⊆ ̺ ∧ ̺ ′ ⊆ σ ′ is non-empty; using the axiom of choice, one can fix some pair ( , the collection T satisfies the second part of the condition we need to verify. We shall check that T is s-directed farther below; first we deal with the union (µ, µ ′ ) := T (meaning union in both components). Since for every subset B ⊆ σ, |B| ≤ s, we have ̺ ′ B ⊆ σ ′ and ̺ B ⊆ σ, it follows that also µ ′ ⊆ σ ′ and µ ⊆ σ. Due to s > 0, we have that σ = B⊆σ,|B|≤s B ⊆ B⊆σ,|B|≤s ̺ B = µ, wherefore µ = σ. This shows that (σ, σ ′ ) has the right form to fit into the set on the right-hand side, provided we establish that the non-empty set T is s-directed.
For this goal, we consider t ≤ s subsets B 0 , . . . , B t−1 ⊆ σ subject to the condition |B i | ≤ s for each 0 ≤ i < t and tuples r i ∈ ̺ Bi ⊆ σ. Let us define C := { r i | 0 ≤ i < t} ⊆ σ. As |C| ≤ s, the pair (̺ C , ̺ ′ C ) belongs to T by definition. Thus C ⊆ ̺ C demonstrates s-directedness, concluding the proof.
Remark 5.18. The inclusion "⊆" in Proposition 5.17 fails to hold for s = 0. Consider, for example, any pair of relations ̺ ′ ⊆ ̺ A m for some fixed m ∈ N.
then Q is certainly closed w.r.t. relaxation, and, moreover, it is not hard to see that it is also closed under arbitrary non-empty unions, i.e. 0-directed unions. Therefore, the set appearing on the right-hand side in Proposition 5.17 is contained in Q. Now, the set 0-LOC A Q = (σ, σ ′ ) ∈ Rp (m) A ∃ (µ, µ ′ ) ∈ Q (m) : σ ′ ⊇ µ ′ clearly contains (A m , ̺ ′ ), but this pair does not belong to the set on the right for it fails to belong to Q due to A m ⊆ ̺.
In generalisation of Proposition 1.13(i) of [29, p. 18] (see also [28, Proposition 1.6(i), p. 256]), a similar characterisation as above can be achieved for the local closure operator.

Corollary 5.19. For any set Q ⊆ Rp A of relation pairs, we have
Proof. The proof of Proposition 5.17 can be literally copied employing the following modifications: the use of Lemma 2.14 has to be substituted by Corollary 2.15; every occurrence of "s-directed", "s-locally closed" and the operator s-LOC A has to be replaced by "ℵ 0 -directed", "locally closed" and the operator LOC A , respectively; every restriction of the form |B| ≤ s, |B i | ≤ s and |C| ≤ s should be changed to |B| < ℵ 0 , |B i | ≤ ℵ 0 and |C| < ℵ 0 , respectively; and, finally, the phrase "Due to s > 0," is to be removed completely.
Placing an additional closure requirement on the sets Q ⊆ Rp A in Corollary 5.19, we can sharpen the statement by replacing ℵ 0 -directed unions by directed unions.
Corollary 5.20. For any set Q ⊆ Rp A of relation pairs that is closed under arbitrary intersections of pairs of identical arity, the following equality holds: Proof. The proof of the inclusion "⊇" stays the same as in Corollary 5.19, one just reads "directed" in place of "ℵ 0 -directed".
The dual inclusion requires a few more changes. Consider any relation pair (σ, σ ′ ) ∈ LOC (m) A Q, m ∈ N. Then for any finite subset B ⊆ σ, the set we can continue as in the proof of Corollary 5.19; only the final paragraph needs further modifications to demonstrate that T is directed and not only ℵ 0 -directed.
For this we consider any finite subset B ⊆ { B ⊆ σ | |B| < ℵ 0 }. Clearly, the union C := B is again a finite subset of σ. Moreover, for all B ∈ B, we have B ⊆ C and hence Σ C ⊆ Σ B , which implies̺ B ⊆̺ C and, consequently, ̺ B = σ ∩̺ B ⊆ σ ∩̺ C = ̺ C . Since this holds for all B ∈ B, we get B∈B ̺ B ⊆ ̺ C , proving directedness of T .
6. Special cases 6.1. Proper semiclones. Based on the results of the previous section, we may also characterise all s-locally closed semiclones that fail to be clones. If we had ̺ = ̺ ′ for all (̺, ̺ ′ ) ∈ Q, then it would follow id A ∈ Polp A Q = F , implying F ∈ L A according to Lemma 3.5(d). Hence, there is at least one In a very analogous fashion we may prove the following result.

Proposition 6.2. For all carriers A we have the equality
Proof. In the proof of Proposition 6.1 replace the use of Lemma 3.10 by Corollary 3.11, Theorem 5.3 by Corollary 5.6, the operators s-Loc A by Loc A , Invp  by Rp A , respectively, and "s-locally" by "locally".
Using the theory of the previous sections, we can also prove a decidability result regarding the question if a clone with projections removed yields a semiclone, or if the non-trivial functions generate the projections.
For this we need a more detailed analysis of the process generating Γ F (B) occurring in Lemma 5.1.
If R n = R n+1 , i.e. S n ⊆ R n , holds for some n ∈ N, then it is S m = S n and R m = R n for all m ≥ n. Therefore, for finite A and finite K, the condition R n = R n+1 is satisfied for some n ≤ A K .
Proof. First, we note that, by definition, R j ⊆ R j+1 , which implies S j ⊆ S j+1 , holds for all j ∈ N. Hence, the unions defining R and S are directed.
It is not difficult to see that (R, S) belongs to Q B . Namely, we have B = R 0 ⊆ R and S j ⊆ R j+1 ⊆ R for every j ∈ N, whence S ⊆ R. To prove that (R, S) is preserved by every n-ary f ∈ F , one considers an n-tuple of tuples (g 0 , . . . , g n−1 ) ∈ R n . Due to directedness of the union producing R and finiteness of n, there exists one j ∈ N such that (g 0 , . . . , g n−1 ) ∈ R j n , wherefore f • (g 0 , . . . , g n−1 ) ∈ S j ⊆ S. Consequently, (R, S) is preserved by every member of F and thus belongs to Q B .
Second, take any pair (̺, ̺ ′ ) ∈ Q B . By definition, we have R 0 = B ⊆ ̺. Moreover, supposing that R j ⊆ ̺, by the preservation condition, we get that S j ⊆ ̺ ′ ⊆ ̺ and hence R j+1 = R j ∪ S j ⊆ ̺, as well as S j ⊆ ̺ ′ . Thus, by induction we have shown R = j∈N R j ⊆ ̺ and S = j∈N S j ⊆ ̺ ′ . This proves that (R, S) ≤ (̺, ̺ ′ ), whence (R, S) is the ≤-least member of Q B .
It is easy to check by induction that R n = R n+1 entails S m = S n and R m = R n for all m ≥ n. Moreover, if R 0 , R 1 , . . . , R n are pairwise distinct (i.e. form a strictly increasing chain R 0 R 1 · · · R n ), then n ≤ |R n | ≤ A K . Therefore, for finite A and K, the condition R n = R n+1 must be satisfied for some n ≤ A K .
The following lemma goes back to an idea by Peter Mayr.
Proof. Using Lemma 3.5(b) we have With this result we can now prove the problem of whether a finitely generated clone on a finite set generates projections from its non-trivial members to be decidable.

Proposition 6.5. For both
Proof. Since F OA is a clone, and thus, in particular, a semiclone, containing OA are all equivalent. By Lemma 6.4 we get F OA \ J A OA , and using Corollary 5.2 for n = 1, X = A and some bijection β between A and its cardinality, we have a description of the invariant pair ̺ A,1 , ̺ ′ A,1 = Γ F \JA ({id A •β}). Via Lemma 6.3 this invariant relation pair generated by id A •β can be expressed as j∈N R j , j∈N S j and finiteness of A guarantees that R n = R n+1 happens for some n ≤ A A . This implies that f can be written as the finite union 0≤j≤|A A | S j , which due to finiteness of F can be straightforwardly calculated using the definitions of Lemma 6.3. Hence, one may check if id A belongs to [F \ J A ] , we obtain Polp A Q = s-Loc A Polp A Q by Lemma 3.10, wherefore we may express H as Note for the second reading that id A does not preserve relation pairs ̺ ′ ̺.
In an analogous way we may characterise the locally closed (proper) transformation semigroups.

A of transformations the following facts are equivalent: (a) H is a locally closed transformation semigroup [and id
Proof. In the proof of Proposition 6.6 substitute "s-locally" by "locally", the operators Invp  By intersecting (in a similar way as outlined in this subsection) with other classes of functions, for example, the set of all permutations instead of all unary operations, one can obtain further characterisations of locally closed classes of functions in terms of relation pairs. Continuing the example of permutations, one may get a characterisation of all locally [s-locally] closed (proper) transformation semigroups that consist of permutations only. As on finite carrier sets every permutation has a finite order, such a result is necessarily more appealing on infinite domains. 6.3. Classical Pol -Inv Galois correspondence. Here we demonstrate that it is not difficult to derive the characterisations of the closure operators of the Galois connection given by polymorphisms and invariant relations from our theorems above. In this respect, we first consider the framework including nullary operations and relations as discussed in [3]; from there it will be a small step to obtain the variants known from [29,28].
First we recollect information concerning the relationship of the operators Pol A and Inv A w.r.t. Polp A and Invp A , which we already briefly discussed before Lemma 2.3.
Proof. The claim follows since a function f ∈ O A preserves a relation ̺ ∈ R A (w.r.t. Pol -Inv) if and only if f ⊲ (̺, ̺) (in the sense of Polp -Invp).
We shall also need to express the set Invp A F in terms of Inv A F for F ⊆ O A containing at least one projection.
Proof. Put Q := Invp A F . By Lemma 3.7, Polp A Q is a semiclone, and, as Polp A Q ⊇ F contains projections, it is even a clone (cp. Lemma 3.5(d)). Hence, Lemma 3.9 yields that Q ⊆ m∈N (̺, ̺) ̺ ∈ R (m) A , which implies In contrast to semiclones, nullary relations are never needed to discern locally closed clones. Even more generally, invariants of small arity may always be neglected.
Proof. As a consequence of Theorem 6.10 we have The following observation will be helpful in deriving the original formulations (without nullary operations) of the previously presented results.
A of positive arity and every s ∈ N the equality s-Loc Proof. Combining Lemma 6.13 with Corollary 6.12 (for m = 1) yields emptiness of the set Pol In a similar way, we may invoke Theorem 6.10 together with Lemma 6.13 to get Pol Using again Theorem 6.10 we can infer Pol In order to attack the relational side of the Pol -Inv Galois correspondence, we need to express generated relational clones, i.e. the closure [ ] R A of a set of relations under general superpositions, in terms of generated relation pair clones. This is prepared in the following lemma.

The equalities Loc
Proof. It is sufficient to prove for fixed m ∈ N that the m-ary part of the left set, s-LOC Due to s > 0 any relation pair (σ, σ ′ ) belonging to the previously displayed set satisfies σ = { B ⊆ σ | |B| ≤ s} ⊆ σ ′ ⊆ σ, i.e. σ = σ ′ . Therefore, we obtain s-LOC A (̺, ̺) ̺ ∈ Q (m) = (σ, σ) σ ∈ R (m) We may now characterise the closure operator Inv A Pol          Proof. By definition of the operator LOC A we have for Q ⊆ R A : The following evident observation will be needed for the next corollaries.
This is equivalent to Pol  describes the appropriate notion of generated relational clone (as employed e.g. in [29]) if one does neither consider nullary operations nor relations in connection with Pol -Inv. With the two stated equalities we have therefore established the two main results (see Theorem 4.2, p. 32, and Theorem 3.3, p. 260, respectively) of [29,28] regarding the relational side of the mentioned Galois connection.

Possible applications
In the literature the Pol -Inv Galois connection has been very successfully employed to discover the structure of the lattice of all clones (e.g. [31,22,37,36]), but it is also fundamentally involved in investigating other problems in algebra and theoretical computer science ( [10,2,1,6,35]). It is to be expected that the theory developed within this article will find similar applications w.r.t. semiclones in the future, especially regarding infinite carrier sets.
In this connection we briefly outline one possible idea, picking up again the topic of topologically closed (proper) transformation semigroups from the previous section. According to Proposition 6.7, for any set Q ⊆ Rp A of relation pairs, the set of all locally closed transformation semigroups S ⊆ O         A Q is a monoid, i.e. Q contains only relation pairs with identical components and Polp A Q is a real clone, then one may also be interested in the maximal locally closed proper transformation semigroups below it. This additional requirement enforces that the pair (̺, ̺ ′ ) one had to add above even has to be a proper relation pair, i.e. ̺ ′ ̺.
In a similar way all maximal locally closed (or s-locally closed) (possibly proper) semiclones (or s-locally closed transformation semigroups) below one specific structure of the respective sort can be described by preserving one additional relation pair. It is plausible that for certain sets Q a complete characterisation in analogy to [31] can be attempted. Furthermore, on infinite carrier sets, the machinery developed in this paper can also be useful to reveal counterexamples, e.g. structures having no maximal proper (locally or s-locally closed) substructures below them. It is, for example, not hard to prove for Q = ∅ that proper semigroups of the form Polp (1) A {(̺, ̺ ′ )} with ̺ ′ ̺ ⊆ A can never be maximal among all locally closed proper transformation semigroups on any at least two-element set A.
The author is, moreover, confident that a generalisation of the presented theory to categories with finite powers is possible along the lines of [23], where a similar project has been realised for clones and the Pol -Inv Galois connection (at the same time dualising the involved notions, which is not in our focus). Most of our results do not impose any restrictions on the carrier set, i.e. the particular object of the category of sets the Galois theory is based on. Therefore, the main theorems of this article could be a guideline and used to hint at what form of results to expect in the general setting. Once such a generalisation has been established, the corresponding results can be instantiated in any category of interest, as long as it has finite powers, for instance, in that of topological spaces. In this way, it may be possible to perform similar investigations as sketched above also for transformation semigroups consisting of continuous functions.