Inma P. Cabrera, Pablo Cordero, Manuel Ojeda-Aciego, Relation-based Galois connections: towards the residual of a relation, CMMSE 2017: Proceedings of the 17th International Conference on Mathematical Methods in Science and Engineering ( ISBN: 978-84-617-8694-7) , pp. 469--475We explore a suitable generalization of the notion of Galois connection in which their components are binary relations. Many different approaches are possible depending both on the (pre-)order relation between subsets in the underlying powerdomain and the chosen type of relational composition.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Cabrera, Inma P.

Cordero-Ortega, Pablo

Ojeda-Aciego, Manuel

English

Repositorio Institucional Universidad de Málaga

Relation-based Galois-connections:towards the residual of a relationInma P. Cabrera, Pablo Cordero, Manuel Ojeda-AciegoSeptember 28, 20171 IntroductionGalois connections can be identified in lots of situations, and they have shownto be an interesting tool both for theory and for applications [6, 9]. Recentapplications can still be found in different topics, specially in the realm of FormalConcept Analysis and the foundations of Fuzzy Set Theory, see for instance [1,7].In all the different notions of generalized Galois connection the problem of itsconstruction is of capital importance: specifically, given a mapping f : A → Bhow can one obtain its residual (aka right adjoint). Freyd’s adjoint theoremcharacterizes when such a residual exists when both A and B have the samestructure. One of our recent research topics has been to study what happens ifA and B are differently structured, and a number of results have been obtainedconsidering different underlying settings. Namely, in [10] we worked with crispfunctions between a poset (resp. preordered set) and an unstructured set; later,in [2] we entered in the fuzzy arena, considering the case in which A is a fuzzypreposet; then, in [3], we extended the previous results by allowing fuzzy equiv-alence relations as an adequate substitute to equality. In [4] we introduced thenotion of relational fuzzy Galois connection, in which the components of theconnection are not fuzzy functions but fuzzy relations satisfying certain prop-erties. Before proceeding to further generalizations, it is worth to start againfrom the crisp case and consider a more adequate notion of Galois connectionwhose components are crisp relations, and that is the topic of the present paper.2 Preliminary definitionsA binary relation R between two sets A and B is a subset of the Cartesianproduct A×B and it can be also seen as a multivalued function R from the setA to the powerset P(B). For an element (a, b) ∈ R, it is said that a is relatedto b and denoted aRb.Definition 1 Given a binary relation R ⊆ A×B,• the afterset aR of an element a ∈ A is defined as {b ∈ B : aRb}1• the foreset Rb of an element b ∈ B is defined as {a ∈ A : aRb}• the domain of R is the set Dom(R) = {a ∈ A : aR 6= ∅}• the range of R is the set Rng(R) = {b ∈ B : Rb 6= ∅}Definition 2 Given a binary relation R ⊆ A×B and a subset X ⊆ A,• the direct image of X under R isR(X) = XR = {b ∈ B : there exists x ∈ X such that xRb} =⋃x∈XxRThe direct image of a subset X is the set of those elements of B that arerelated to at least one element of X.• the subdirect image of X under R isR/(X) = X/R = {b ∈ B : xRb for all x ∈ X} =⋂x∈XxRThe subdirect image of a subset X is the set of those elements of B thatare related to all the elements of X.Observe that if X = {a}, the direct image and the subdirect image of Xcoincides with the afterset, thus aR coincides with R(a) should we interpret Ras a multivalued mapping.In the realm of ordered structures, Ore [12] introduced in 1944 the so-calledGalois connections as a pair of antitone mappings for which both possible com-positions are inflationary. In order to extend this concept to binary relationsit is necessary to fix the meaning of antitone and inflationary relation. Onepossibility is to consider the relation as a multivalued function and extend itnaturally, but both the preordered structure in the powerset and the composi-tion of relations admit different approaches.Definition 3 Given P an arbitrary set and ≤ a preorder (reflexive and transi-tive relation) defined over P , it is possible to lift the preorder structure to thepowerset P(P ) by definingX  Y ⇐⇒ for all x ∈ X there exists y ∈ Y such that x ≤ y (1)X b Y ⇐⇒ for all y ∈ Y there exists x ∈ X such that x ≤ y (2)X n Y ⇐⇒ for all x ∈ X there exists y ∈ Y such that x ≤ y and (3)for all y ∈ Y there exists x ∈ X such that x ≤ yNote that the relations defined above are the preorder relations used in theconstruction of the, respectively, Hoare, Smyth, and Plotkin powerdomains.The concept of antitone multivalued function between two preordered setsdepends on the preorder relation considered over the powerset. We explore threepossibilities, being one approach more restrictive than the others:2Definition 4 Let A = (A,≤A),B = (B,≤B) be two preordered sets and R ⊆A×B a binary relation between A and B.• R is said to be h-antitone if a1 ≤A a2 implies a2R a1R, for all a1, a2 ∈A.• R is said to be s-antitone if a1 ≤A a2 implies a2R b a1R, for all a1, a2 ∈A.• R is said to be p-antitone if a1 ≤A a2 implies a2Rna1R, for all a1, a2 ∈ A,equivalently, if it is h-antitone and s-antitone.Analogously, the definition of inflationary multivalued function also admitsthree possibilities.Definition 5 Let A = (A,≤A) be a preordered set and R ⊆ A × A a binaryrelation on A.• R is said to be h-inflationary if {a}  aR, for all a ∈ A, that is, thereexists (at least) x ∈ aR such that a ≤ x.• R is said to be s-inflationary if {a} b aR, for all a ∈ A, that is, a ≤ x forall x ∈ aR.• R is said to be p-inflationary if it is h-inflationary and s-inflationary.Remark 1 Notice that if a relation is s-inflationary, then it is also h-inflationary,therefore p-inflationary and s-inflationary are equivalent notions.The condition of the composition of two multivalued mappings being infla-tionary in some sense requires also to fix which definition of composition will beused.Definition 6 Let R be a binary relation between A and B and S be a binaryrelation between B and C.• The classical composition of R and S is defined as followsR ◦ S = {(x, z) ∈ A× C : there exists b ∈ B such that xRb and bSz}= {(x, z) ∈ A× C : xR ∩ Sz 6= ∅}• The / composition of R and S is defined as followsR / S = {(x, z) ∈ A× C : for all b ∈ B such that xRb it holds that bSz}= {(x, z) ∈ A× C : xR ⊆ Sz}Observe that for an element a ∈ A, the afterset a[R ◦ S] coincides with thedirect image of the afterset aR under S, that isa[R ◦ S] = (aR)S =⋃b∈aRbS3Analogously, for an element a ∈ A, the afterset a[R / S] coincides with thesubdirect image of the afterset aR under S, that isa[R / S] = (aR)/S =⋂b∈aRbS3 Two types of relational Galois connectionsDefinition 7 Let A = (A,≤A),B = (B,≤B) be two posets, R ⊆ A×B a binaryrelation between A and B and S ⊆ B ×A a binary relation between B and A.The pair (R,S) is said to be an s-Galois connection between A and B ifi) R and S are s-antitone.ii) R ◦ S and S ◦R are s-inflationary.Proposition 1 Let A = (A,≤A),B = (B,≤B) be two posets and (R,S) be ans-Galois connection between A and B. Then if b ∈ Rng(R) then bS is at most asingleton, so, the restriction of S to Rng(R) is a (partial) single-valued function.Proof: If b ∈ Rng(R)r Dom(S), then there is nothing to prove; therefore, letus assume that b ∈ Rng(R) ∩Dom(S).1. As b ∈ Rng(R), there exists a ∈ A such that b ∈ aR and, as b ∈ Dom(S),we have that bS is nonempty. We will now see that, b ∈ xR for all x ∈ bS.Since {a} b a[R ◦ S], taking into account that b ∈ aR, we have a ≤ x forall x ∈ bS. Now, as R is s-antitone, there exists b′ ∈ xR such that b′ ≤ b.The other inequality b ≤ b′ follows because of {b} b b[S ◦R]. As a result,b = b′ ∈ xR.2. Consider two elements x, x? ∈ bS, by definition of composition and theprevious item, we have x? ∈ x[R ◦S]; since, by hypothesis, we have {x} bx[R ◦ S], it turns out that x ≤ x?. Applying that R is s-antitone, thereexists b? ∈ x?R such that b? ≤ b. Again, the hypothesis {b} b b[S ◦ R]implies b ≤ b?, and we obtain b = b? ∈ x?R. This, together with x ∈ bS,proves that x ∈ x?[R ◦ S]; finally, applying once again x? b x?[R ◦ S], weobtain x? ≤ x and, therefore x? = x and bS is a singleton. Notice that the previous result shows that the definition of s-Galois connec-tion necessarily collapses the relations R and S to be (partial) functions in thecase of posets. If we drop the antisymmetry and consider the more general caseof preordered sets, we obtain a similar result in that the images of the relationsare clusters.As a result, it seems more convenient to consider alternative approacheseither by changing the ordering between subsets and/or slightly modifying thenotions of antitone or inflationary relation.A promising definition seems to be the following:4Definition 8 Let A = (A,≤A),B = (B,≤B) be two preordered sets, R ⊆ A×Ba binary relation between A and B and S ⊆ B ×A a binary relation between Band A. The pair (R,S) is said to be an h-Galois connection between A and B ifi) R and S are h-antitone.ii) For all a ∈ A and all b ∈ B the following conditions hold:{a}  yS for all y ∈ aR and {b}  xR for all x ∈ bS. (4)It is not difficult to check that condition (4) above is a consequence of theproperty of R / S and S / R being h-inflationary but, in general, are not equiv-alent.The following result shows a necessary condition for a pair (R,S) to be anh-Galois connection.Lemma 1 If the pair (R,S) is an h-Galois connection between A and B, thenthe following inclusions hold: Rng(R) ⊆ Dom(S) and Rng(S) ⊆ Dom(R).Proof: Given b ∈ Rng(R), there exists a ∈ A such that b ∈ aR. Now, bycondition (4) above, we obtain that bS 6= ∅ and, therefore, b ∈ Dom(S).The other inclusion can be proved similarly. Notice that, in fact, the proof of the previous lemma does not use the anti-tonicity of either R or S.With the condition of Lemma 1 in mind, we obtain an equivalence with theusual notion of Galois connection, as stated below:Theorem 1 Let A = (A,≤A),B = (B,≤B) be two preordered sets, R ⊆ A×Ba binary relation between A and B and S ⊆ B ×A a binary relation between Band A. The pair (R,S) is an h-Galois connection between A and B if and onlyif the following holds:Rng(R) ⊆ Dom(S) and Rng(S) ⊆ Dom(R) (5){a}  bS ⇐⇒ {b}  aR (6)Proof: Given (R,S) an h-Galois connection between A and B, condition (5)follows by Lemma 1. Now, for condition (6), assume that {a}  bS. Then,there exists x ∈ bS such that a ≤ x and, by R h-antitone, we obtain xR aR.On the other hand, by condition (4) we have {b}  xR. Now, by transitivityof , we obtain that {b}  aR. The proof that {b}  aR implies {a}  bS issimilar.Conversely, assume that equivalence (6) holds, and let us prove that (R,S)is an h-Galois connection.Firstly, we will show condition (4): Given a ∈ A and y ∈ aR, since y ≤ y,then {y}  aR which by (6) implies that {a}  yS for all y ∈ aR. The otherpart is similar.Now, consider a1 ≤ a2 in A. Then, since {a2}  yS for all y ∈ a2R, it alsoholds that {a1}  yS for all y ∈ a2R. Hence, by (6), we have {y}  a1R,for all y ∈ a2R which means that a2R  a1R. The antitonicity of S followsanalogously. 54 Conclusions and further workThe problem of considering relations within the notion of Galois connection isnot new, since it can be dated back to [8], nor outdated, since one can still findrecent references dealing with different aspects of the integration of relationsand Galois connections, see for instance [5, 11,13].We have obtained some prospective results on the notion of relational-basedGalois connection, in which the components of the connection are relations be-tween posets. There are several possibilities depending both on the (pre-)orderrelation between subsets in the underlying powerdomain and the chosen type ofrelational composition. We have just scratched the surface of the problem, andshown that one of the most reasonable approaches collapses in that the involvedrelations R and S should actually be functions. The second proposed definitionuses a different approach in that, apart from considering an alternative orderingin the underlying powerdomain for the definition of antitonicity, it also gener-alizes the notion of being inflationary. This way, we have obtained a promisingresult in the form of Theorem 1.As future work, we are planning to continue the line initiated in [2, 3] andattempt the construction of the residual, in the sense of relation-based (fuzzy)Galois connections, to a given mapping between differently structured domainand codomain, as stated in the introduction.References[1] L. Antoni, S. Krajcˇi, and O. Kr´ıdlo, Representation of fuzzysubsets by Galois connections. Fuzzy Sets and Systems, 2017.https://doi.org/10.1016/j.fss.2017.05.020.[2] I.P. Cabrera, P. Cordero, F. Garc´ıa-Pardo, M. Ojeda-Aciego, andB. De Baets. On the construction of adjunctions between a fuzzy preposetand an unstructured set, Fuzzy Sets and Systems, 320:81–92, 2017.[3] ——, Adjunctions between a fuzzy preposet and an unstructured set withunderlying fuzzy equivalence relations, 2017, submitted.[4] I.P. Cabrera, P. Cordero, and M. Ojeda-Aciego. Relational fuzzy Galoisconnections. In Proc. of the 17th World Congress of Intl Fuzzy SystemsAssociation (IFSA-SCIS’17), 2017. To appear.[5] M. Couceiro. Galois connections for generalized functions and relationalconstraints. In Contributions to General Algebra, volume 16, pages 35–54.Verlag Johannes Heyn, 2005.[6] K. Denecke, M. Erne´, and S. Wismath. Galois connections and applications,ser. Mathematics and its Applications. Springer, 2004, vol. 565.6[7] J.T. Denniston, A. Melton, S.E. Rodabaugh. Formal Contexts, FormalConcept Analysis, and Galois Connections. Electr. Proc. on TheoreticalComputer Science 129: 105–120, 2013.[8] I. Fleischer and I. Rosenberg. The Galois connection between partial func-tions and relations. Pacific Journal of Mathematics, 79(1):93–97, 1978.[9] F. Garc´ıa-Pardo, I.P. Cabrera, P. Cordero, and M. Ojeda Aciego. On Galoisconnections and soft computing, Lect. Notes in Computer Science, vol.7903, pp. 224–235, 2013.[10] F. Garc´ıa-Pardo, I.P. Cabrera, P. Cordero, M. Ojeda Aciego, andF. Rodr´ıguez. On the definition of suitable orderings to generate adjunc-tions over an unstructured codomain, Information Sciences, vol. 286, pp.173–187, 2014.[11] E. Jerˇa´bek. Galois connection for multiple-output operations.https://arxiv.org/abs/1612.04353v1, dec 2016.[12] O. Ore. Galois Connexions. Transactions of the American MathematicalSociety, 55(3):493–513, 1944.[13] A´. Sza´z. A particular Galois connection between relations and set functions.Acta Univ. Sapientiae, Mathematica, 6(1):73–91, 2014.7

Relation-based Galois connections: towards the residual of a relation

https://riuma.uma.es/xmlui/bitstream/10630/14552/3/para_riuma.pdf

Relation-based Galois connections: towards the residual of a relation

Abstract

Similar works

Full text

Available Versions

Repositorio Institucional Universidad de Málaga