GENERALISATIONS OF THE ALGEBRAIC THEORYOF CONNECTIVE SEGMENTATION
Seidon Alsaody1
Project Report for the Degree of Master of Science ... Supervisor: Prof. Em. Jean Serra, ESIEE Paris Examiner: Prof. Volodymyr Mazorchuk, Uppsala University ... Spring Semester 2010
1(Visiting) Universite´ Paris-Est, Laboratoire d’Informatique Gaspard-Monge, Equipe AS3I, ESIEE Paris (2, boulevard Blaise Pascal, Cite´ Descartes, BP 99, 93162 Noisy le Grand Cedex, France). Abstract
We begin by defining the set-up and the framework of connective segmentation. Then we start from a theorem based on connective criteria established for the lattice of subsets of an arbitrary set and formulate and prove its analogue for more general lattices. This also requires generalising the underlying definitions. Secondly, we consider partial partitions and partial connections. We recall the definitions, and quote a result that gives a characterisation of (partial) connections. We link this approach to the above one by means of a commutative diagram; then, as a continuation of the work in the first part, we state generalised versions of the concepts and results that underlie this char- acterisation, thereafter giving the characterisation in the generalised setting as well. Finally we generalise the commutative diagram to the general setting in two manners. Contents
1 Introduction 2 1.1 Outline of the Paper ...... 2 1.2 Origin of the Problems ...... 3 1.3 Note on Terminology and Notation ...... 3 1.4 Fundamental Conventions ...... 3 1.5 Note on the References ...... 3 1.6 The Author’s Acknowledgements ...... 4 1.7 Basic Definitions ...... 4 1.7.1 Some Examples ...... 6 1.8 Fundamentals of Connective Segmentation ...... 7
2 The Characterisation Theorem of Connective Segmentation 9 2.1 Serra’s Theorem ...... 9 2.2 First Generalisation: Viscous Lattices ...... 9 2.2.1 Further Definitions ...... 9 2.2.2 Adapting the Set-Up ...... 12 2.2.3 Generalising the Theorem to Viscous Lattices ...... 13 2.3 Main Generalisation ...... 15 2.3.1 Adapting the Set-Up ...... 16 2.3.2 Step I: Atomistic Lattices ...... 17 2.3.3 Step II: (Generally) Non-Atomistic Lattices ...... 19
3 Partial Connections & Partial Partitions 21 3.1 Main Definitions ...... 22 3.2 Openings on Partial Partitions and Ronse’s Corollary ...... 23 3.3 Relating Ronse’s Approach to Serra’s ...... 24
4 Generalising Ronse’s Corollary 25 4.1 Basic Concepts ...... 26 4.2 Underlying Results ...... 27 4.3 The Corollary Generalised ...... 33
5 Relating Serra’s & Ronse’s Approaches in General Lattices 34 5.1 Maintaining the Non-Partial Setting ...... 35 5.2 Using Partiality ...... 36
6 Concluding Remarks 37
A Appendix: On Partitions, Partial Partitions and Partial Connections 39
B Appendix: On an Augmentation to the Theory 41 B.1 The Non-Partial Setting ...... 41 B.2 The Partial Setting ...... 43
1 1 Introduction
1.1 Outline of the Paper
This paper lies within the field of complete lattices in the theory of partially ordered sets. We start by recalling some basic properties, and then consider the collection of all subsets of a given set, and recognise its proprties of a complete lattice. We then pass to the theory of math- ematical morphology1. Two themes, namely the notion of connectivity and that of a partition, are central in this work, and their definitions are quoted. We also quote the definition of a criterion, and thus arrive by the end of Section 1 at the apparatus of connective segmentation. This theory has been introduced and studied thoroughly by Serra and, among others, Ronse. Having done this, we generalise in Section 2 a theorem of Serra that characterises seg- mentation in terms of so called connective criteria, in that we replace the subset lattice (to be understood as the lattice of all subsets of a given set) by an arbitrary complete lattice. The first step is the case of viscous lattices, which are images of the subset lattice under a certain class of maps. The importance of this case is that such lattices lack several properties that hold for the subset lattice. To this end, we generalise the set-up and prove an analogue of the original theorem, thus establishing that it is, in this sense, independent of these properties. We then take the final step of generalising the set-up, now to the case of arbitrary complete lattices, and state the theorem. As the reader will see, the theorem holds in its original sense in a large sub-class of such lattices, namely atomistic lattices (defined below); however, for the most general case, we prove a weaker version, as well as present an example showing that the original version of the theorem does not generalise as far. In Section 3, we present an approach to segmentation that uses a broader notion, namely so called partial connectivity instead of connectivity. This comprises the introduction of partial partitions; indeed, it has been shown that partial connections are characterised by a class of operators on partial partitions. An important aim of Section 3 is to demonstrate, through a commutative diagram, the link between this result (obtained by Ronse) and the theorem of Serra from Section 2. This section is entirely concerned with the case of the subset lattice. Motivated by the generalisation in Section 2 and the link of Section 3, section 4 is dedicated to generalising the set-up of partial connections and partial partitions to arbitrary complete lattices. Ronse’s result is generalised as well; it holds in full strength even in the most general case, due to the notion of partial partitions. Having done this, we return in Section 5 to the commutative diagram of Section 3, aim- ing at generalising it as well. The treatment of this section consists of two parts. In the first, we generalise the diagram as faithfully as possible, whereas in the second, we use the adap- tation of Serra’s theorem into partial connections and partial partitions [4]; as we will have seen in Section 4, this adaptation generalises completely to arbitrary complete lattices. Thus we reformulate it in terms of connective criteria, and produce the commutative diagram in question. The first Appendix is dedicated to the proofs that the sets of partitions, partial partitions, and partial connections, as defined in the general setting, are all complete lattices. These
1See section 1.2
2 results underly various arguments in different sections of this work. The second appendix generalises an augmentation made in [4] to the theory of connective segmentation.
1.2 Origin of the Problems
The notion of connective segmentation, introduced in Definitions 5–9 below, is a subfield of the field of mathematical morphology, which is, briefly, the study and analysis of images, using a lattice-theoretic framework. Image segmentation consists of subdividing, or segment- ing, an image into its different components, e.g. foreground and background. This is the practical application of the concepts studied below, and hopefully explains the terminology. The principle of mathematical morphology is to express the problems of image analysis in mathematical terms, using this theoretical set-up.
It should be stressed that this is a paper in mathematics and hence is entirely concerned with the theoretical apparatus of mathematical morphology, in particular its algebraic parts. The applications into image analysis are outside the scope of this paper, and the interested reader is recommended to consult the appropriate literature listed in the bibliography.
1.3 Note on Terminology and Notation
In the field under study, it sometimes occurs that different terminologies are used, and many notions have more than one name. The choices made in this work aim at maximising legibility and accuracy. Whenever possible, alternative terminology will be provided, in brackets, in the definitions. For more details on differences of terminology, Ronse [5] has made an extensive account.
1.4 Fundamental Conventions
Throughout this paper, all sets considered, that are not explicitely stated to be subsets of other sets (i.e. all “ground sets”, usually denoted by E) are assumed to be non-empty, unless otherwise stated. Analogously, all complete lattices (to be defined below) that do not arise from a construction on other complete lattices (i.e. all “ground complete lattices”, usually denoted by L) are assumed to contain at least two different elements.
1.5 Note on the References
At the formulation of Definitions 19, 24 and 26, the author was unaware of the existence of the articles [2] and [6]. Thus, those definitions were made independently of these sources, which explains the parantheses at each occurrence. The articles are however listed as references since they were, to the author’s knowledge, the first to introduce the notions of Definitions 24 and 26.
3 1.6 The Author’s Acknowledgements
I am most grateful to Professor Emeritus Jean Serra for his invitation and efforts in order for this Master Thesis project to take place, as well as for being attentive to my research interests, and supervising and nourishing the work in progress with valuable remarks and ideas, and I am moreover much indebted for his warm welcoming and friendliness during the progress of the work. I would further like to express my deep gratitude to Doctor Hugues Talbot for arranging the financial support without which this work could not have been done, and for steadily and patiently being willing to help. I also wish to express my thankfulness to the entire team at ESIEE Paris among whose members I have spent a very enriching period of time. Moreover, I would like to thank Professor Christian Ronse for bringing the sources [2] and [6] to my attention.
I am sincerely grateful to Professor Volodymyr Mazorchuk for undertaking the responsi- bility of examining this Master Thesis, and for offering me the possibility to give a presenta- tion at the Algebra and Geometry Seminar at Uppsala University. My profound thankfulness is finally expressed toward Professor Emeritus Christer Kisel- man for once establishing the personal link that has given birth to this work, and for always being supporting, trusting, and encouraging.
1.7 Basic Definitions
The definitions in this section are standard and can be found in textbooks in lattice theory, such as [1]. We assume that the reader is familiar with the definition of a partially ordered set (poset). The partial order on any poset P that occurs in the text will be denoted by ≤ when the poset is clear from the context, and otherwise by ≤P . Two elements p, q in a poset P are said to be comparable if any of p ≤ q or q ≤ p holds; we then write p - q. It is easy to show that the relation - on P is reflexive and symmetric, but in general not transitive. A poset where any two elements are compareble is called a chain. Moreover, we assume familiarity with the operations of supremum (i.e smallest upper bound) and infimum (i.e. greatest lower bound). (Other names are join and meet, respectively, which terminology we will however not use.) As binary operations, these will here be denoted by ∨ and ∧ respectively. A poset L in which these operations are well defined, in the sense that for each x, y ∈ L, there exist unique z, w ∈ L such that z = x ∨ y and w = x ∧ y, is called a lattice with respect to the given partial order. The binary supremum and infimum can be extended naturally to larger sets. If for each W (possibly infinite) family {xi}i∈I ⊆ L of elements in the lattice L the supremum i∈I xi and V the infimum i∈I xi exist and belong to L, then L is said to be complete. Given a subset A ⊆ L of a complete lattice L, the supremum and infimum of A will be denoted by W A and V A, respectively. It follows that a complete lattice has a smallest element, denoted by 0, which is the supremum of the empty set (also called the empty supremum). Likewise, there is a greatest element 1, which is the infimum of the empty set (also called the empty infimum). A
4 subset A of a complete lattice is said to be closed under the supremum (resp. infimum) if for each subset B ⊆ A, it holds that W B ∈ A (resp. V B ∈ A). Unless otherwise stated, the smallest and greatest elements of all complete lattices will be denoted by 0 and 1 respectively. We will moreover switch freely between the notion of a subset of a set A, and that of a family of elements in A indexed by a parameter, whichever is more convenient for notation and legibility.
The following definition describes how complete lattices can be generated.
Definition 1. A subset S ⊆ L of a complete lattice L is said to be sup-generating (or a sup-basis) if for each l ∈ L, one has
_ l = {s ∈ S; s ≤ l}, (1) in which case L is said to be sup-generated by S.
The next definition reminds of that of irreducible and prime elements in a domain (in the sense of ring theory), partially ordered by divisability.
Definition 2. Let L be a complete lattice.
1. An element l ∈ L \ {0} is an atom if ∀m ∈ L \ {0}; m ≤ l ⇒ m = l.
2. An element l ∈ L \ {0} is called co-prime if ∀m, n ∈ L; l ≤ m ∨ n ⇒ (l ≤ m or l ≤ n).
If L is sup-generated by a set of atoms, it is said to be atomistic, and if L is sup-generated by a set of co-prime elements, it is said to be co-primary.
The next definition presents two notions that will be important in what follows.
Definition 3. Let L be a complete lattice (with smallest element 0 and greatest element 1).
1. A complete sublattice of L is a subset M ⊆ L such that M is a complete lattice with respect to the partial order on L, and whose least and greatest elements are 0 and 1, respectively.
2. A dual Moore family in L is a subset N ⊆ L such that
∀l ∈ L, ∃nl ∈ N ; nl ≤ l and ∀n ∈ N ; n ≤ l ⇒ n ≤ nl. (2)
In item 1, the requirement of the coincidence of greatest and smallest elements is analo- gous to the requirement that the identity element of e.g. a subgroup of a group coincides with the identity element of the group. Item 2 states, in other words, that a dual Moore family is a subset that, for each element in the lattice, contains a maximal element smaller than or equal to it. It holds [7] that a subset N of a complete lattice L is a dual Moore family if and only if it is closed arbitrary suprema. As the reader may have guessed, there is the notion of a Moore family, in which the order relations are the opposite to those in Definition 3.2, and the supremum is replaced by the infimum. The general notion of duality with respect to a partial order will not be further dealt with here; a treatment is given in e.g. [3].
Finally, the following properties are defined. 5 Definition 4. 1. A lattice L is modular if for each l, m ∈ L, l ≤ m ⇒ (l ∨ n) ∧ m = l ∨ (n ∧ m) for all n ∈ L.
2. A lattice L is distributive if for each l, m, n ∈ L, the following distrivutive laws hold:
l ∧ (m ∨ n) = (l ∧ m) ∨ (l ∧ n), (3)
l ∨ (m ∧ n) = (l ∨ m) ∧ (l ∨ n). (4)
3. Let L be a complete lattice and l ∈ L. Any element m ∈ L that satisfies l ∧ m = 0 and l ∨ m = 1 is said to be a complement of l. If for each element of L there exists a complement, L is said to be complemented.
One can show that the distributive laws (3) and (4) are equivalent, and that distributiv- ity implies modularity. It is moreover worth noting that the complement is not necessarily unique.
1.7.1 Some Examples
We give here some standard examples of lattices, which are rather different one from another. The second illustrates the supremum and infimum in terms of set theory and is a helpful tool for visualising these operations. Example 1. The closed interval [0, 1] ⊆ R is a complete lattice with respect to the usual order on R, in which the zero and one elements coincide with the real numbers 0 and 1. The open interval (0, 1) as well as the whole real line are lattices, though not complete due to the lack of greatest and smallest elements. The extended real line R = R ∪ {∞} ∪ {−∞} is a complete lattice. In all these cases, the lattices are totally ordered, and completeness, whenever it holds, is implied by completeness in the sense of topology (i.e. the convergence of any Cauchy se- quence). Note that the set of rational numbers in R and [0, 1] is sup-generating as it is dense in these sets. None of these two complete lattices is atomistic, but both are co-primary. Example 2. Given any set E, the set 2E of all subsets of E, ordered by inclusion, is a complete lattice, called the subset lattice of E. The supremum and infimum of a collection of subsets are the union and intersection, respectively, of its members; the greatest and smallest elements are respectively E and ∅. This lattice is atomistic, its atoms being the singleton (one-element) subsets, and complemented as in the sense of set theory, with each element admitting a unique complement. By set-theoretic distributivity, it is also distributive. Example 3. Given two complete lattices L and M, the set of all maps φ : L → M, henceforth denoted (L, M), is itself a complete lattice under the elementwise order on their images in M, i.e. for any two maps φ, ψ : L → M,
φ ≤(L,M) ψ ⇔ ∀l ∈ L, φ(l) ≤M ψ(l). (5)
The lattice structure is thus implied by that of M, and the smallest and greatest elements are respectively the map l 7→ 1 and the map l 7→ 0 for all l ∈ L. Furthermore, if M = L, then the set (L, L) (whose elements are called operators on L) also carries the structure of a monoid
6 under the composition of operators. The identity of the monoid is of course the identity operator id : l 7→ l for all l ∈ L. Remark. Given a complete lattice L and an operator φ on L, the notation Inv(φ) stands for the invariance set of φ, i.e. Inv(φ) = {l ∈ L; φ(l) = l}. Example 4. Given a complete lattice L and an element m ∈ L, the set Lm = {l ∈ L; l ≤ m} is m a complete lattice under the order induced by the order on L (i.e. for x, y ∈ L , x ≤Lm y ⇔ m x ≤L y). The smallest element in L coincides with that of L, and the largest element is m; hence, Lm is not a complete sublattice of L.
1.8 Fundamentals of Connective Segmentation
The notions to be dealt with below were originally defined by Serra for the lattice 2E of a set E. Recall that the motivation for this apparatus comes from situations where given a set, typically a subset of the Euclidean plane, the aim is to partition it in a certain way into subsets that satisfy some criterion. This is made precise by the following definitions. First is the definition of a criterion.
Definition 5 ([11]). Let E and T be two sets, and F a family of functions f : E → T .A criterion on F is a map σ : F × 2E → {0, 1} satisfying σ((f, ∅)) = 1. The criterion σ is said to be validated on (f, A) ∈ F × 2E whenever σ((f, A)) = 1; otherwise it is refuted on (f, A).
To avoid stacking parantheses, we will write σ(f, A) instead of σ((f, A)). Intuitively, the criterion shows whether or not a given function satisfies some condition on a given subset. When the function f is clear from context, we will simply say that σ is validated on a set A, meaning that σ(f, A) = 1. Next is the definition of connectivity.
Definition 6 ([8]). A subset C of 2E is a connection if it satisfies
1. ∅ ∈ C,
2. ∀x ∈ E; {x} ∈ C, and T S 3. if {Ci}i∈I ⊆ C for some index set I, and i∈I Ci 6= ∅, then i∈I Ci ∈ C.
The elements of C are said to be C-connected, or connected if the connection is clear from the context, and an element of C which is not a proper subset of another element of C is called a connected component. A connected component of a set A ⊆ E is an element C ∈ C such that there exists no D ∈ C that satisfies C $ D ⊆ A. The notion of connectedness is evidently inspired from topology, and indeed the above axioms hold for connectedness in the usual sense (in which a set is called connected if there are no two disjoint open sets each containing a non-empty subset of it), as well as in the sense of arc-wise connectedness. Note however that the definition above does not require any pre- defined topology and rather relies on the lattice structure of 2E. The two definitions above combine to give rise to the following
Definition 7 ([11]). A criterion σ on a family F of functions from a set E to a set T is connective whenever 7 1. ∀f ∈ F, ∀x ∈ E; σ(f, {x}) = 1, and
E T 2. for all f ∈ F and for each family {Ai}i∈I ⊆ 2 such that i∈I Ai 6= ∅ and σ(f, Ai) = 1 S for all i ∈ I, it holds that σ(f, i∈I Ai) = 1.
This means, in other words, that the collection of all subsets on which the criterion is validated for a given function is a connection. The following definition is rather classical and characterises a partition of a set as a map that assigns to each point of the set the class to which it belongs.
Definition 8. A partition of E is a map π : E → 2E that satisfies
1. ∀x ∈ E; x ∈ π(x), and
2. ∀x, y ∈ E such that π(x) ∩ π(y) 6= ∅ it holds that π(x) = π(y).
When confusion is improbable, we will denote the image of the partition π, i.e the col- lection of all its classes, by π as well. Thus the notation C ∈ π means that C is a class of π. The set of partitions of E, denoted Π(E), is ordered by refinement: π ≤ π0 ⇔ ∀x ∈ E; π(x) ⊆ π0(x). It has been shown that with respect to this order, the set Π(E) is a complete lattice, in which the infimum of a non-empty family of partitions is given by the classwise infimum (i.e. intersection), and the supremum is the smallest partition greater than or equal to all partitions in the family. The greatest element in this lattice is the partition 1; 1(x) = E for all x ∈ E, while the smallest partition is I; I(x) = {x} for all x ∈ E. This enables the following definition
Definition 9 ([11]). Given f ∈ F where F is a family of functions from a set E to a set T , and E given A ∈ 2 and a criterion σ on F, let Π(A, σf ) be the family of all partitions π on A such that σ is validated on each class of π. The criterion σ is said to segment f (over A) if
1. ∀x ∈ E, σ(f, {x}) = 1, and
2. the family Π(A, σf ) is closed under the supremum of partitions.
In that case, the supremum of Π(A, σf ) is the segmentation of f over A by σ.
Remark. The second item of the definition is equivalent to saying that Π(A, σf ) is a dual Moore family in Π(A). When the set A is not specified, the segmentation is understood to be over E (as is the case in the theorem below; segmentation on subsets of E is discusses in Appendix B). Hence the segmentation of a function by a criterion is the greatest partition on all of whose classes the criterion is validated for the function. In this sense, it is the optimal partition subject to the criterion.
8 2 The Characterisation Theorem of Connective Segmentation
2.1 Serra’s Theorem
The purpose of this part of the work is to generalise the following result, which relates seg- mentation to connectivity.
Theorem 10 ([11]). Given two sets E and T , let F be a family of functions f : E → T , and σ a criterion on F. The following statements are equivalent:
1. σ is connective.
E 2. For all f ∈ F, the collection Cf = {A ∈ 2 ; σ(f, A) = 1} is a connection.
3. σ segments all f ∈ F.
From [11] we know that in case any, and hece all, of the statements holds, then for each 2 f ∈ F, the segmentation obtained is the partition of E into its Cf -connected components.
2.2 First Generalisation: Viscous Lattices
2.2.1 Further Definitions
The first step in generalising the theorem above is to consider so called viscous lattices, for reasons mentioned in the Introduction (Section 1.1). The definition of this type of lattices relies on some particular maps and properties of maps of lattices, and we begin by quoting their definitions. Again, the definitions are standard and can be found in textbooks on lattice theory.
Definition 11. Given two lattices L and M, a map φ : L → M is said to be isotone (or order preserving) if ∀x, y ∈ L; x ≤ y ⇒ φ(x) ≤ φ(y), and antitone (or order reversing) if ∀x, y ∈ L; x ≤ y ⇒ φ(y) ≤ φ(x). An operator ψ : L → L is said to be extensive if ∀x ∈ L; x ≤ φ(x), and intensive (or anti-extensive) if ∀x ∈ L; φ(x) ≤ x.
Note that the terms increasing and decreasing will not be used due to their ambiguity, since in the literature they may refer to both isotonity/antitonity and extensiveness/intensiveness. The following definitions introduce some special maps on complete lattices.
Definition 12. [7] Let L and M be complete lattices.
1. An extensive operator κ : L → L is a closure (or a closing) if
∀x, y ∈ L; x ≤ κ(y) ⇒ κ(x) ≤ κ(y). (6)
2. An intensive operator γ : L → L is an opening (or a dual closure) if
∀x, y ∈ L; γ(x) ≤ y ⇒ γ(x) ≤ γ(y). (7)
2This will indeed be the case in every generalisation of the theorem, as the reader will see in the coming sections.
9 3. A map δ : L → M is a dilation (or a complete join-morphism) if it commutes with the supremum, i.e. if
_ _ ∀K ⊆ L; δ K = {δ(k); k ∈ K}. (8)
4. A map : L → M is an erosion (or a complete meet-morphism) if it commutes with the infimum, i.e. if
^ ^ ∀K ⊆ L; K = {(k); k ∈ K}. (9)
5. A map φ : L → M is a complete morphism if it is a dilation and an erosion.
6. A map ψ : L → M is an isomorphism of complete lattices if ψ is bijective and
∀x, y ∈ L; x ≤ y ⇔ ψ(x) ≤ ψ(y). (10)
If such a map exists, L is said to be isomorphic to M.
Remark. It follows from the definitions that closures and openings are idempotent and isotone. Conversely, an extensive (resp. intensive) operator satisfies (6) (resp. (7)) if it is idempotent and isotone; thus, there are two equivalent definitions of closures and of openings. Moreover, it is seen from the definition that the identity operator on a lattice, the inverse map of a lattice isomorphism, and the composition of two lattice isomorphisms are all lattice isomorphisms, whereby isomorphism of lattices is an equivalence relation. Remark. There is, by [7], a one-to-one correspondence between openings on a complete lattice L, and dual Moore families in L, as follows. To each opening γ, the set Inv(γ) is a dual W Moore family, and to each dual Moore family D, the operator γD : L → L; l 7→ {d ∈
D; d ≤ l} is an opening. One can check, using the definition of an opening, that Inv(γD) = D for any dual Moore family D, and that any opening γ satisfies γ = γInv(γ) This establishes the correspondence. (Of course, there is an analogous correspondence between closures and Moore families.) Finally, we have the following
Definition 13. Let L and M be complete lattices, and φ : L → M, ψ : M → L two maps. If φ, ψ satisfy the following Galois connection
∀x ∈ L, ∀y ∈ M; φ(x) ≤ y ⇔ x ≤ ψ(y) (11) then the ordered pair (ψ, φ) is said to be an adjunction (or a residuation).
Observe that some textbooks define Galois connection dually with respect to the order.
Given two complete lattices L and M, a result in [7] states that to each dilation δ : L → M there exists a unique erosion : M → L such that (, δ) is an adjunction, and, conversly, to each erosion 0 : M → L there exists a unique dilation δ0 : L → M such that (0, δ0) is an adjunction. In addition, whenever an ordered pair of maps (ψ, φ) is an adjunction, then 10 ψ is an erosion and φ a dilation, and moreover ψφ is a closure and φψ is an opening. By uniqueness of adjunctions, any dilation and any erosion gives rise to a unique closure and a unique opening through such compositions. We will refer to such a closure (resp. opening) as the closure (resp. opening) induced by the dilation or erosion in question. Example 5. [7] a. For the lattice 2E, where E is a topological space, the topological closure in E is a closure on 2E in the sense of Definition 12, whereas the topological interior is an opening. n b. Let E = R , n ≥ 1 equipped with the canonical topology, and Br(x), x ∈ E be the n-dimensional (topologically) open ball of (fixed) radius r and centre x. One can show that the operator
E E [ δ : 2 → 2 ,A 7→ Br(x) (12) x∈A is an (extensive) dilation that “adds” to each point to the ball centred at it. Moreover, the operator E E : 2 → 2 ,A 7→ {x ∈ E; Br(x) ⊆ A)} (13)
is an (intensive) erosion, namely the erosion adjoint to δ. The open ball in the example can be replaced by any subset B(x) ∈ E containing x for each x ∈ E. Further examples are given in [10]. Now the viscous lattice is introduced in the following proposition.
Proposition 14 ([10]). Given a set E, and an extensive dilation δ : 2E → 2E, the collection {δ(A); A ∈ 2E} is a complete lattice under the inclusion order, where the supremum of a family of subsets is the union of its members, and the infimum is the opening induced by δ of their intersection.
This lattice is called the viscous lattice of δ, and will be denoted V(E,δ) or, when the dilation
δ is clear from the context, VE. The dilation in Example 5 gives a viscous lattice where each element is the union of open balls of radius r. This lattice is hence atomistic, since one sees that any two different such balls have empty infimum. In general, to guarantee that a viscous lattice is atomistic, the following condition is posed.
Definition 15 ([10]). A dilation on 2E for a set E is said to be separated if it satisfies the follow- ing equivalence for any x, y ∈ E:
δ({x}) ⊆ δ({y}) ⇔ x = y. (14)
The name is motivated by the role of the condition as a separation axiom (in the sense E used in topology). To simplify notation, if for a set E the lattice VE is obtained from 2 by a separated dilation, we say that VE is separated.
It has been shown [10] that V(E,δ) is atomistic for any set E and separated extensive dila- tion δ, the set of atoms being {δ({x}); x ∈ E}, but that there exist separated extensive dilations such that the corresponding viscous lattice is neither modular, nor co-primary, nor admits a unique complement to each of its elements.
11 2.2.2 Adapting the Set-Up
We now generalise the definitions of Section 1 to the case of the viscous lattice of an extensive dilation δ and a set E. To begin with, in order to define criteria, one needs to replace the set E in Definition 5 by a set of atoms that sup-generates VE, in order to give the notion a meaning similar to that in the case of 2E. The set {δ({x}); x ∈ E} is obviously sup-generating, and to enusure that its members are atoms, we assume δ to be separated. We avoid stacking parantheses as in the note following Definition 5.
Definition 16. Let E and T be two sets, and F a family of functions f : {δ({x}); x ∈ E} → T , E where δ is a separated extensive dilation on 2 .A criterion on F is a map σ : F × V(E,δ) →
{0, 1} satisfying σ(f, ∅) = 1. The criterion σ is said to be validated on (f, A) ∈ F × V(E,δ) whenever σ(f, A) = 1; otherwise it is refuted on (f, A).
It is then straight-forward to define connections.
Definition 17 ([10]). Let E be a set, and δ a separated extensive dilation on 2E. A subset C of
V(E,δ) is a connection if it satisfies
1. ∅ ∈ C,
2. ∀x ∈ E; δ({x}) ∈ C, and V S 3. if {Ci}i∈I ⊆ C for some index set I, and i∈I Ci 6= ∅, then i∈I Ci ∈ C.
The notion of a connected component, introduced after Definition 6, generalises to the case of the viscous lattice in a straight-forward manner. The definition of a connective criterion becomes as follows.
Definition 18. Let E and T be sets, and δ a separated extensive dilation on 2E. A criterion σ on F, where F is a family of functions f : {δ({x}); x ∈ E} → T is connective whenever
1. ∀f ∈ F, ∀x ∈ E; σ(f, δ({x})) = 1, and V 2. for all f ∈ F and for each family {Ai}i∈I ⊆ V(E,δ) such that i∈I Ai 6= ∅ and σ(f, Ai) = 1 S for all i ∈ I, it holds that σ(f, i∈I Ai) = 1.
To be able to introduce the notion of segmentation, we begin by defining partitions. These have been defined for 2E for a set E in [11], and for complete lattices in general in [9] and, as the author has later learnt, in [2]. We define them here for viscous lattices.
E Definition 19. Let E be a set, and δ a separated extensive dilation on 2 .A partition on V(E,δ) is a map π : {δ({x}); x ∈ E} → VE that satisfies
1. ∀x ∈ E; δ({x}) ⊆ π(δ({x})), and
2. ∀x, y ∈ E such that π(δ({x})) ∧ π(δ({y})) 6= ∅ it holds that π(δ({x})) = π(δ({y})).
12 Note that the classes of a partition in general overlap, but their infimum, i.e the image of the overlap under the opening induced by δ, is empty. This is an example of the fact that in
VE, the infimum being empty does not imply that the intersection is as well.
0 The set Π(V(E,δ)) of partitions on V(E,δ) is ordered by refinement: π ≤ π ⇔ ∀x ∈ E; π(δ({x})) ⊆ 0 π (δ({x})). With respect to this order, the set Π(V(E,δ)) is a complete lattice (see the appendix for a proof of a stronger statement). This enables the following definition.
Definition 20. Given two sets E and T and a separated extensive dilation δ, let F be a family of functions f : {δ({x}); x ∈ E} → T , and σ a criterion on F. Given f ∈ F, let Π(V(E,δ), σf ) be the collection of all partitions π on V(E,δ) such that σ is validated on each class of π. The criterion σ is said to segment f if
1. ∀x ∈ E, σ(f, δ({x})) = 1, and
2. the family Π(V(E,δ), σf ) is closed under the supremum of partitions.
In that case, the supremum of Π(V(E,δ), σf ) is the segmentation of f under σ.
Note that for this definition, we did not use partitions of subsets of E as in Definition 9, since the original theorem of Serra does not rely on them (the segmentation there is over the whole set E); hence, neither will the generalisation in this section (see, however, Appendix B).
2.2.3 Generalising the Theorem to Viscous Lattices
We have now arrived at the point of generalising the theorem of Serra. As the next theorem shows, the result holds indeed, mutatis mutandis, for the viscous lattice as well. Of course, a new proof is requiered which differs from that of the original result, especially in the case of the last implication. Altogether, we have the following
Proposition 21. Given two sets E and T and a separated extensive dilation δ, let F be a family of functions f : {δ({x}); x ∈ E} → T , and σ a criterion on F. The following statements are equivalent:
1. σ is connective.
2. ∀f ∈ F the collection Cf = {A ∈ V(δ,E); σ(f, A) = 1} is a connection.
3. σ segments all f ∈ F.
To make the proof below more legible, we write δ(x) instead of δ({x}) for x ∈ E.
Proof. 1 ⇒ 2: By Definition 16, the empty set is in Cf . If σ is connective, then items 1 and 2 of
Definition 18 imply that Cf satisfies items 2 and 3 in Definition 17. Hence Cf is a connection.
2 ⇒ 1: Since Cf is a connection and σ is a criterion, Definition 17, items 2 and 3 imply that σ satisfies the conditions in Definition 18.
Throughout the rest of the proof, let f be any element of F, and let Π(V(δ,E), Cf ) be the col- lection of all partitions on V(δ,E) whose classes belong to Cf , i.e. all partitions for every class 13 A of which σ(f, A) = 1.
2 ⇒ 3: Item 1 of Definition 20 is satisfied by the definition of a connection. As for item 2, the empty supremum in Π(V(δ,E), Cf ) is the partition π˜;π ˜(δ(x)) = δ(x) for all x ∈ E, (which satisfies the axioms of a partition trivially), since the sets δ(x) are atoms. By definition π˜ ∈
Π(V(δ,E), Cf ).
Next, let {πi}i∈I , for some index set I, be a non-empty family in Π(V(δ,E), Cf ) with π = W i∈I πi. For each class B ∈ π, each δ(x) ⊆ B is included in a Cf -connected component of B (the union of all C ∈ Cf such that δ(x) ⊆ C ⊆ B), and the infimum of any collection of more than one component of B is empty by definition of a connection. Since moreover the classes of π have pairwise empty infimum, the collection of all connected components of all classes of π is itself a partition, say πC , such that
πC ≤ π (15)
by construction. Moreover, for all i ∈ I, each class of πi is contained in a class of πC since each class of πi is connected and included in a class of π. Hence πC is an upper bound of {πi}, and π being the least such, we have
π ≤ πC (16)
and altogether π = πC . Since by construction πC ∈ Π(V(δ,E), Cf ), we deduce that Π(V(δ,E), Cf ) is closed under non-empty suprema. Together with the above, and the definition of Cf , this implies that σ segments all f ∈ F.
3 ⇒ 2: By definition, σ is validated on the empty set and on the atoms for all f ∈ F. Take V any {Ci, i ∈ I} ⊆ Cf satisfying i∈I Ci 6= ∅. This implies that
^ ∃p ∈ E; δ(p) ⊆ Ci. (17) i∈I
Each Ci induces a partition πi defined by ( πi(δ(x)) = Ci if δ(x) ⊆ Ci, and (18) πi(δ(x)) = δ(x) otherwise,
(for which the properties of a partition are easily verified). It is clear that πi ∈ Π(V(δ,E), Cf ) W for all i ∈ I, and since σ segments all f ∈ F, it follows that π = i∈I πi ∈ Π(V(δ,E), Cf ). W S Define X = i∈I Ci = i∈I Ci. To complete the proof that Cf is a connection, we need show that X ∈ Cf , which will be done by proving that X is equal to a class of π, namely the one containing δ(p). To begin with, combining (17) and (18) we deduce that
[ ∀i ∈ I; πi(δ(p)) = Ci ⊆ Ci = X. (19) i∈I Now define the partition π0 by
14 ( π0(δ(x)) = X if δ(x) ⊆ X, and (20) π0(δ(x)) = δ(x) otherwise. The axioms of a partition are again easily verified. In particular we have π0(δ(p)) = X. In 0 the lattice of partitions, π is an upper bound of the family {πi}i∈I due to (19), together with the fact that all other classes of each πi are atoms. Since π, being the supremum, is the least upper bound, it necessarily follows that π ≤ π0, i.e.
π(δ(p)) ⊆ π0(δ(p)) = X. (21)
To prove the other inclusion we observe that from the definition of an upper bound of partitions, it holds for all i ∈ I that
Ci = πi(δ(p)) ⊆ π(δ(p)), (22) S for all i ∈ I, from which follows that X = i∈I Ci ⊆ π(δ(p)). Thus by mutual inclusions we have shown that
X = π(δ(p)), (23)
whereby X ∈ Cf since Π(V(δ,E), Cf ) is closed under suprema. This completes the proof.
The choice of the viscous lattice as a first step of the generalisation was made due to its role as a pathological example in the context of subset lattices. Since the viscous lattice is in gen- eral neither modular, nor co-primary, nor uniquely complemented, Proposition 21 shows that Serra’s theorem generalises independently of distributivity, co-primarity and the existense of a unique complement. However, we stress that the proof of Proposition 21 relies on the lattice being atomistic, as we take the next step in the generalisation.
2.3 Main Generalisation
In this section, we will generalise Theorem 10 to atomistic complete lattices, and formulate a weaker statement for even more general cases. We will consider arbitrary complete lattices L sup-generated by a set S ⊆ L. (Recall that if L is atomistic, then S can be chosen to be the set of all atoms.) Before stating the generalisation of the relevant concepts, and of the theorem, we give the following elementary lemma, which is formulated without reference to atomisticity. It highlights the role of the sup-generating set as a defining set of the entire lattice, and will be used in order to make future proofs less dependent on atomisticity.
Lemma 22. Let L be a lattice sup-generated by a subset S ⊆ L. If for two elements x, y ∈ L, it holds that
∀s ∈ S; s ≤ x ⇒ s ≤ y, (24)
then x ≤ y.
15 Proof. Since L is sup-generated by S, the sets Sx = {s ∈ S; s ≤ x} and Sy = {s ∈ S; s ≤ y} satisfy
_ _ x = Sx and y = Sy. (25) By the assumption of the lemma, we have Sx ⊆ Sy, whereby y is an upper bound of Sx. Since x is the least such upper bound, the result follows.
2.3.1 Adapting the Set-Up
We now take the ultimate step in generalising the definitions above. The definitions below apply to any lattice with a sup-generating subset S, and in general do not require atomisticity. Note that every complete lattice is sup-generated by itself; hence, it is of little interest to speak about “sup-generated lattices” as a class of complete lattices (in the same way that one does not talk about e.g “generated groups”). We present here the generalisation of the whole apparatus of connective segmentation, fixing for each lattice a sup-generating subset, and beginning by the definition of criteria. Again, we refrain from stacking parantheses.
Definition 23. Let L be a complete lattice sup-generated by a subset S ⊆ L, T an arbitrary set, and F a family of functions f : S → T .A criterion on F is a map σ : F × L → {0, 1} satisfying σ(f, 0) = 1. The criterion σ is said to be validated on (f, l) ∈ F × L whenever σ(f, l) = 1; otherwise it is refuted on (f, l).
For simplicity, when the function f is clear from context, we will say that σ is validated on l ∈ L, meaning that σ(f, l) = 1. The reader might find it peculiar that the domain of the functions is not the whole lattice; however, such a definition would not be coherent with the original one when applied to the special case of a subset lattice. Since in general there may exist more than one sup-generating subset of a lattice, the definition above fixes, once and for all, a sup-generating set for the criterion σ; hence, the definition of a connection in [9] must be modified into the following, more restrictive one. (It has later come to the authors knowledge that this restrictive definition appears in [6]. There, the term used is connection; we however use the term S-connection in order to stress the choice of a sup-generating set.)
Definition 24. Let L be a complete lattice sup-generated by a subset S ⊆ L. A subset C of L is an S-connection if it satisfies
1. 0 ∈ C,
2. S ⊆ C, and V W 3. if {ci}i∈I ⊆ C for some index set I, and i∈I ci 6= 0, then i∈I ci ∈ C.
Note that for any fixed sup-generating set S, an S-connection is a connection in the sense of [9], and thus the theory established there holds for S-connections as well. An S-connected component of an element l ∈ L greater than or equal to a given s ∈ S is an element c ∈ C such that s ≤ c ≤ l and there exists no d ∈ C such that c d ≤ l; due to item 3 of Definition 24, this is precisely W{c ∈ C; s ≤ c ≤ l}. Next is the following 16 Definition 25. Let L be a complete lattice sup-generated by a subset S ⊆ L. A criterion σ on a family F of functions from S to a set T is S-connective whenever
1. ∀f ∈ F, ∀s ∈ S; σ(f, s) = 1, and V 2. for all f ∈ F and for each family {li}i∈I ⊆ L such that i∈I li 6= 0 and σ(f, li) = 1 for all W i ∈ I, it holds that σ(f, i∈I li) = 1.
Next, we define the framework of segmentation. The following definition is a modification of the corresponding definition in [9]. (It has later come to the author’s knowledge that it appears in [2].)
Definition 26. Let L be a complete lattice sup-generated by a subset S ⊆ L. An S-partition on L is a map π : S → L that satisfies
1. ∀s ∈ S; s ≤ π(s), and
2. ∀s, t ∈ S such that π(s) ∧ π(t) 6= 0 it holds that π(s) = π(t).
Ordered by refinement, the set ΠS(L) of S-partitions on L is a complete lattice (see the appendix for a proof). As before, we will use the same notation for a partition and its image (which we continue to call the set of its classes, even though the term no longer bears its true meaning), and the prefix S will sometimes be dropped when the set S is clear from context. Segmentation is defined as follows.
Definition 27. Let L be a complete lattice sup-generated by a subset S ⊆ L. Given f ∈ F where F is a family of functions from S to a set T , and a criterion σ on F, let ΠS(L, σf ) be the family of all S-partitions π of L such that σ is validated on each class of π. The criterion σ is said to S-segment f if
1. ∀s ∈ S, σ(f, s) = 1, and
2. the family ΠS(L, σf ) is closed under the supremum of S-partitions.
In that case, the supremum of ΠS(L, σf ) is the S-segmentation of f under σ.
Again, we do not generalise the concept of segmentation on subsets here — this is done in Appendix B.
2.3.2 Step I: Atomistic Lattices
When the lattice L is in fact atomistic, the evident choice of sup-generating family is the set of all atoms of L. To fix terminology, whenever L is atomistic with A being the set of all atoms, the A-connections (A-connectivity, A-partitions, A-segmentations, etc.) will be called canonical connections (canonical connectivity, etc.). Serra’s theorem then generalises as follows:
Theorem 28. Let L be a complete, atomistic lattice and denote the set of all atoms by A. Let F be a family of functions f : A → T where T is an arbitrary set, and let σ be a criterion on F. The following statements are equivalent:
17 1. σ is canonically connective.
2. ∀f ∈ F the collection Cf = {l ∈ L; σ(f, l) = 1} is a canonical connection.
3. σ canonically segments all f ∈ F.
In preparation for the following sections, the atomisticity property will be avoided as much as possible in the proof.
Proof. 1 ⇔ 2: This part is analogous to the corresponding steps in the proof of Proposition 21, with Definitions 17 and 18 replaced by Definitions 24 and 25, respectively.
Throughout the rest of the proof, let f be any element of F, and let ΠA(L, Cf ) be the collec- tion of all A-partitions whose classes belong to Cf , i.e. all canonical partitions for every class c of which σ(f, c) = 1.
2 ⇒ 3: By Definition 23, item 1 of Definition 27 is satisfied. As for item 2, the empty supremum of ΠA(L, Cf ) is the partition π˜ defined for all a ∈ A by π˜(a) = a, (for which the axioms hold trivially) since the elements of A are atoms. By definition π˜ ∈ ΠA(L, Cf ).
Next, let {πi}i∈I , for some index set I, be a non-empty family in ΠA(L, Cf ) with π = W 3 i∈I πi. Let πC : A → Cf be defined by
_ πC (a) = {c ∈ Cf ; a ≤ c ≤ π(a)} (26) .
for all a ∈ A. (This is well defined since Cf being a connection, it is closed under the supremum of subsets with non-zero infima, and the set {c ∈ Cf ; a ≤ c ≤ π(a)} satisfies this latter property since a ∈ Cf .) Indeed, πC is a partition; the first axiom of partitions is satisfied, by item 2 of Definition 24, and as for the second, we observe that if for some a, b ∈ A it holds that πC (a)∧πC (b) 6= 0, then π(a)∧π(b) 6= 0, which implies that π(a) = π(b) — by construction of πC and item 3 of Definition 24, we then have πC (a) = πC (b).
Now, πC ≤ π, and for all i ∈ I and a ∈ A, πi(a) ≤ πC (a) since πi(a) ∈ Cf and a ≤ πi(a) ≤
π(a). Hence πC is an upper bound of {πi}i∈I , and π being the least such, we have π ≤ πC and altogether π = πC . Since clearly πC ∈ ΠA(L, Cf ), we deduce that ΠA(L, Cf ) is closed under non-empty suprema. Thus σ segments all f ∈ F canonically.
3 ⇒ 2: By definition, σ(f, l) = 1 for all l ∈ {0} ∪ A. Take any {ci}i∈I ⊆ Cf satisfying V i∈I ci 6= 0. This implies that
^ ∃l ∈ L \ {0}; l ≤ ci, (27) i∈I and l being the supremum of a subset of the sup-generating family A, we have that
^ ∃a0 ∈ A; a0 ≤ ci. (28) i∈I
3 This definition is equivalent to saying that πC assigns to each a ∈ A the connected component of π(a) greater than or equal to it. 18 Each ci induces a partition πi defined for all a ∈ A by ( πi(a) = ci if a ≤ ci, and (29) πi(a) = a otherwise.
The axioms of a partition are satisfied since all a ∈ A are atoms. It is clear that πi ∈
ΠA(L, Cf ) for all i ∈ I, and since σ canonically segments all f ∈ F, it follows that π = W W i∈I πi ∈ ΠA(L, Cf ). Define x = i∈I ci. We need show that x ∈ Cf . For any i ∈ I, the W definition of πi implies that πi(a0) = ci and since x = i∈I ci, it holds that πi(a0) ≤ x for all i ∈ I. Now define the partition π0 by ( π0(a) = x if a ≤ x, and (30) π0(a) = a otherwise
0 for all a ∈ A. For all i ∈ I it holds that πi ≤ π since πi(a0) = ci ≤ x, and since for a ∈ A, a πi(a0) we have the two easy cases
( 0 a ≤ x ⇒ πi(a) = a ≤ x = π (a), and 0 , (31) a x ⇒ πi(a) = a ≤ a = π (a). 0 Since π is the supremum of {πi}i∈I , it holds that π ≤ π , and thus
0 π(a0) ≤ π (a0) = x. (32)
To prove the inverse relation, we use the fact that, since πi ≤ π, we have ci = πi(a0) ≤ W π(a0) for all i ∈ I, from which follows by the definition of the supremum that x = i∈I ci ≤ π(a0). Thus by mutual inclusions we have shown that x = π(a0), whereby x ∈ Cf since
ΠA(L, Cf ) is closed under suprema. This completes the proof.
2.3.3 Step II: (Generally) Non-Atomistic Lattices
We now relax the condition of atomisticity. Note that the only parts in the proof of Theorem 28 that require atomisticity of L are the part in the proof of the implication 2 ⇒ 3 in which it is shown that the supremum of the empty family of partitions is in ΠA(L, Cf ), as well as the proof of the inverse implication, since atomisticity was implicitly needed in order for the functions πi, i ∈ I to be partitions. As for the first issue, the following proposition removes the need of atomisticity.
Proposition 29. Let L be a complete lattice sup-generated by a subset S ⊆ L. Let ΠS(L) be the com- plete lattice of all S-partitions on L, and denote by π its smallest element. Then for any S-connection C on L, π(s) ∈ C for all s ∈ S.
Proof. Given an S-connection C, consider the map πC : S → L defined by
_ πC (s) = {c ∈ C; s ≤ c ≤ π(s)} (33)
for all s ∈ S. Then πC is an S-partition, π(s) ∈ C for all s ∈ S, and in the lattice ΠS(L) we have πC ≤ π, the proofs of which statements are the same as the above given proofs of the 19 same statements for the map πC appearing in (26). Since however π is the smallest S-partition on L, it follows from πC ≤ π that πC = π, which completes the proof.
Remark. An immediate consequence of Proposition 29 is that for any set T and any family F of functions from S to T , every connective criterion on F is satisfied on (f, c) for each f ∈ F and each class c of the smallest S-partition on L. The following result now follows immediately from the proof of Theorem 28, the opening note of this section, and Proposition 29.
Theorem 30. Let L be a complete lattice sup-generated by a subset S ⊆ L. Let F be a family of functions f : S → T where T is an arbitrary set, and let σ be a criterion on F. The following statements are equivalent:
1. σ is S-connective.
2. ∀f ∈ F the collection Cf = {l ∈ L; σ(f, l) = 1} is an S-connection.
If any, and hence both, of these statements holds, then σ S-segments all f ∈ F.
The following example demonstrates that, in the general setting, the requirement of atom- isticity for the proof of the implication 3 ⇒ 2 of Theorem 28 is in fact indispensable.
Example 6. Let L be the set {0, 1, s0, s1, s2, s3, t} and define the partial order ≤ on L by
0 ≤ s0 ≤ s1 ≤ s3 ≤ 1, s0 ≤ s2 ≤ t ≤ 1, and s1 ≤ t, (34)
and by transitivity of order. This can be depicted as
0 / s0 / s1 / s3 / 1 (35) AA AA ~? AA AA ~~ AA A ~~ AA AA ~ A ~~ s2 / t where for l ∈ L and m ∈ L, l ≤ m if and only if there exists a directed path from l to m. It is easily checked that L is a lattice, hence a complete lattice by finiteness, and that it is sup- generated by the set S = {si, i = 0, 1, 2, 3}. Let T be any set, and F any family of functions from S to T . Define the criterion σ on F by ( σ(f, l) = 0 if l = t, and (36) σ(f, l) = 1 otherwise,
for all f ∈ F. Since t = s1 ∨ s2, and s1 ∧ s2 6= 0, this definition implies that σ is not connective. However, from the structure of L the only possible S-partition on L is the one whose unique class is 1, and since σ(f, 1) = 1, it follows trivially that σ S-segments f. This gives a counterexample to the conjecture that the implication 3 ⇒ 2 of Theorem 28 generalises to arbitrary non-atomistic complete lattices. We end this section by giving examples of non-atomistic lattices and partitions on them.
20 Example 7. The lattice R (the extended real line; see Example 1), is sup-generated by the set Q of rational numbers. One arrives however at the less interesting conclusion that the only Q-partition possible on R is the one whose unique class is ∞.
E Example 8. Consider the set R of extended real-valued functions on an arbitrary set E. By E [7], R is a complete lattice under the order ≤ given by
E ∀f, g ∈ R ; f ≤ g ⇔ (∀x ∈ E; f(x) ≤ g(x)) (37) where the order on the right hand side is the usual order on R. This lattice is sup-generated by the set P of all finite pulses, where a finite pulse is any function p = py,t : E → R for y ∈ E and t ∈ R such that ( py,t(x) = t if x = y, and (38) py,t(x) = −∞ otherwise.
From this follows that the relation - is transitive on P , hence an equivalence relation whose equivalence classes are the chains Cy = {py,t ∈ P ; t ∈ R} for each y ∈ E. The supremum of W each chain is the corresponding infinite pulse, i.e ∀y ∈ E, Cy is defined by
( W ( Cy)(x) = ∞ if x = y, and W (39) ( Cy)(x) = −∞ otherwise W This implies that for any chain y ∈ E and any p ∈ P , we have that Cy - p if and only E if p ∈ Cy. From this one can show that the smallest P -partition on R is the partition that to W each finite pulse p = py,t assigns the infinite pulse Cy. Note that the sup-generating set of finite pulses chosen above is not minimal; the set of rational-valued pulses is sufficient due to the denseness of Q in R. This is however immaterial as far as this example is concerned. It would be plausible to have a set-up where the equivalence between connectivity and segmentation, as of Serra’s theorem, holds even in the most general case. This is one moti- vation for generalising the concept of partial partitions, introduced in [4] and presented in the following section, to arbitrary complete lattices; as we shall see in Section 4, this provides such a set-up.
3 Partial Connections & Partial Partitions
As has been seen, the theory of connective segmentation relies on the concepts of connections and partitions, which are in a sense linked by the theorem of Serra which we have gener- alised above. At each step, two lattices have been dealt with; a given complete lattice L sup-generated by a subset S ⊆ L, and the lattice of S-partitions on L. We now extend this study in another direction, by looking at operators that act on parti- tions. More specifically, we study a certain type of openings4 on partitions, and their relation to connections. This follows the work of Ronse, who has established this theory for the case of the subset lattice [5]. The framework will also be adjusted, since this part of the theory
4An introduction to openings on complete lattices was given in Section 2.2.1. 21 no longer relies on the axioms of connections and partitions as stated in Definitions 6 and 8 above, but rather on partial connections and partial partitions defined in [4]. There, the mo- tivation of this adjustment is given; its main argument is that in computations, considering singelton sets to be a priori connected becomes impractical, and this consideration is therefore removed. In general, this offers a great deal of flexibility to the theory, as we will see.
The outline of the study-to-come will be as follows. First, we quote the definitions of partial connections and partial partitions, followed by the definition of a certain class of in- tensive operators on partial partitions. Then we quote an important result by Ronse that links a subclass of this to the set of partial connections. We observe that the approach of Ronse is coherent with that of Serra (dealt with in the previous section), and we relate the two by a commutative diagram. All results in this section are formulated for the lattice 2E for a set E; more general lattices will be considered in Section 4.
3.1 Main Definitions
We begin by the definition a partial connection.
Definition 31 ([4]). A subset C of 2E for a set E is a partial connection on E if it satisfies
1. ∅ ∈ C, and T S 2. if {Ci}i∈I ⊆ C for some index set I, and i∈I Ci 6= ∅, then i∈I Ci ∈ C.
Elements of C are said to be C-connected, or connected if the partial connection is clear from the context. Note that in the case of partial connections, singleton sets are not a priori required to be connected. Connected components of partial connections are defined in the same way as for connections; the difference is that given a point x ∈ E, there may exist no component to which x belongs, i.e. the components of a partial connection do not constitute a partition of E. These properties might appear strange, especially if the reader has a background in topology. In practice, they allows partial connections to be viewed as connections on part of the set E. This is in line with the notion of partial partitions. Before defining these, we state a reformulation of the second axiom for partitions (Definition 8, item 2).
Proposition 32 ([4]). Given a set E, let π : E → 2E be a map that satisfies
∀x ∈ E; x ∈ π(x). (40) Then π is a partition of E if and only if it satisfies
∀x, y ∈ E; x ∈ π(y) ⇒ π(x) = π(y). (41)
The definition of partial partitions is then as follows.
Definition 33 ([4]). A partial partition of E is a map π : E → 2E that satisfies
1. ∀x ∈ E; π(x) = ∅ or x ∈ π(x), and 22 2. ∀x, y ∈ E; x ∈ π(y) ⇒ π(x) = π(y).
As in the case of partitions, when confusion is improbable, we will denote the image of the partial partition π, i.e. the collection of all its classes, by π as well. Thus the notation C ∈ π means that C is a class of π. Definition 33 implies that the classes of a partial partition do not necessarily cover E. The set of partial partitions on E is denoted Π∗(E); it is a complete lattice under the refinement order previously defined for partitions [4]. The greatest element in this lattice is the partition 1; 1(x) = E for all x ∈ E, while the smallest partial partition is 0; 0(x) = ∅ for all x ∈ E. It has been shown in [4] that the infimum of a non-empty family of partial partitions is given by the classwise infimum (i.e. intersection), and the supremum is the smallest partial partition greater than or equal to all members of the family.
For partial partitions, condition 33.2 is by [4] in general stronger than the statement that any two classes are either disjoint or equal, and using it avoids undesired properties that would arise when using this latter statement. A detailed study of partial connections and partial partitions is conducted in [4]. We instead move on to a class of operators on Π∗(E), defined in two steps as follows:
Definition 34 ([5]). Let E be a set.
1. A map ψ : 2E → Π∗(E) is set splitting if ψ(A)(x) ⊆ A for all x ∈ E.
2. Given a set splitting map ψ, the class splitting operator induced by ψ is the operator ψ∗ : ∗ ∗ ∗ ∗ W Π (E) → Π (E) such that for each partial partition π ∈ Π (E), ψ (π) = C∈π ψ(C).
From [5] one learns that ψ is set splitting if and only if for all A ⊆ E and x ∈ E \ A, ψ(A)(x) = ∅, and from item 1 of the definition, it follows that for a partial partition π, each class of ψ∗(π) is the image under ψ of a class of π. In [5] class splitting operators are called block splitting.
3.2 Openings on Partial Partitions and Ronse’s Corollary
Recall that an opening on a complete lattice is an intensive, idempotent and isotone operator on it. The set of openings on a complete lattice L is a subset of the complete lattice of operators on L (Example 3). It exhibits additional structure by the following lemma.
Lemma 35 ([7]). Let L be a complete lattice, and (L, L) the complete lattice of operators on L. The subset of (L, L) consisting of all openings on L is a dual Moore family in (L, L). It is thus a complete lattice under the order induced by the order on (L, L). In this lattice, the supremum resp. the infimum of a non-empty family P of openings is the supre- mum of P in (L, L) resp. the greatest opening smaller than or equal to the infimum of P in (L, L); the empty supremum is the constant map l 7→ 0 for all l ∈ L, and the empty infimum is the identity map l 7→ l for all l ∈ L.
Note that this does not mean that the lattice of openings is a sublattice of the lattice of operators, since the infimum (including the empty infimum) is defined differently. Moreover, 23 there is indeed a greatest opening less than or equal to a given element in (L, L), since the openings form a dual Moore family; it is for the same reason that the supremum in (L, L) of a family of openings is an opening. The following result by Ronse is central in the characterisation of partial connections in terms of class splitting openings. It will be referred to as Ronse’s corollary, as it appears as a corollary in the original article, from which we moreover know that the set of partial connections on E, denoted by Γ∗(E), is a complete lattice under the inclusion order.
Proposition 36. [5] The set Ω(E) of class splitting openings on Π∗(E) for a set E is a complete sub- lattice of the lattice of openings on Π∗(E). It is isomorphic to the lattice Γ∗(E) of partial connections on 2E. The isomorphism is given by the map
∗ ∗ λ :Γ (E) → Ω(E); C 7→ C• (42) ∗ ∗ W where for all π ∈ Π (E), C• (π)(x) = {C ∈ C; x ∈ C ⊆ π(x)} for all x ∈ E.
Put simply, λ maps each partial connection to the (class splitting) opening that splits the classes of a partial partition into their C-connected components. The proof and the justifica- tions of all statements in the proposition can be found in [5]. Our perhaps strange choice of notation is made to be coherent with the notation in Section 4 below.
3.3 Relating Ronse’s Approach to Serra’s
In [4], Serra’s theorem (Theorem 10) is augmented5 as well as modified to fit in the set-up of partial connections and partial partitions. In [5], the theorem is once again recalled, and it is stated that it is related to the approach of class splitting openings. The aim of this section is to precise this statement by giving an actual link in terms of a commutative diagram. In order that this link relate directly to the contents of this work, we rely on Serra’s theorem as stated above, and not on its modification in [4] to partial partitions and partial connections.
Theorem 37. Given sets E and T and a family F of functions f : E → T , let Σ(F,E) be the set6 of all connective criteria on F, Π(E) the complete lattice of partitions of E, Π∗(E) the complete lattice of partial partitions of E, Γ∗(E) the complete lattice of partial connections on E, and Ω(E) the complete lattice of class splitting openings on Π∗(E). Let moreover
• for each f ∈ F, φf : Σ(F,E) → Π(E) be the map that assigns to each connectice criterion σ its segmentation of f over E, and
• λ :Γ∗(E) → Ω(E) be the lattice isomorphism that assigns to each partial connection C the class ∗ splitting opening C• .
Then for each f ∈ F there exist maps µf and κ such that following diagram commutes
5This augmentation is discussed and generalised in Appendix B. 6This set is in fact a complete lattice, by [11]; however, neither Theorem 37 nor the coming treatment will depend on the lattice structure of this set.
24 φf Σ(F,E) / Π(E) (43) O µf κ Γ∗(E) / Ω(E) λ
In other words we have φf (σ) = κλµf (σ) for all f ∈ F and σ ∈ Σ(F,E). Specifically,
• µf is the map that assigns to each connective criterion σ the connection {C ⊆ E; σ(f, C) = 1}, and
• κ is the map that assigns to each class splitting opening γ∗ the partial partition γ∗(1), (where 1 is the greatest partition of E).
Remark. The map φf is due to Serra’s theorem (Theorem 10) and is well defined due to the uniqueness of the supremum on a complete lattice. The isomorphism λ, on the other hand, exists due to Ronse’s corollary (Proposition 36). This justifies the statement that this result relates the two approaches.
Proof. Take a function f ∈ F and a criterion σ ∈ Σ(F,E). Then µf (σ) = {C ⊆ E; σ(f, C) = 1}. ∗ Next, composition of µf with λ gives, that λµf (σ) = (µf (σ))•. Composing this with κ gives
∗ κλµf (σ) = (µf (σ))•(1), (44) where Proposition 36 gives that for all x ∈ E,
∗ _ _ (µf (σ))•(1)(x) = {C ∈ µf (σ); x ∈ C ⊆ 1(x)} = {C ∈ µf (σ); x ∈ C}, (45) ∗ since 1(x) = E for all x ∈ E. Now, (µf (σ))• is a partial partition, and moreover a partition, since µf (σ) is a connection, and hence every x ∈ E belongs to an element of µf (σ). By maximality of connected components, it is the largest partition all of whose classes belong to
µf (σ), since the right hand side of (45) is the µf (σ)-component to which x belongs. Hence it is the segmentation of f over E by σ, which is precisely φf (σ), and we arrive at
κλµf (σ) = φf (σ), (46) whereby the proof is complete.
4 Generalising Ronse’s Corollary
The results in the preceding section holds in the case of the lattice 2E for a set E. We have seen in Section 2 that the approach of connective segmentation, corresponding to the map φf in the commutative diagram of Theorem 37, generalises to the case of an arbitrary atomistic lattice, and, partially, even further. It is therefore of interest to investigate whether the isomorphism λ in the commutative diagram generalises as well; i.e. whether and how Ronse’s corollary can be generalised to arbitrary complete lattices. This section is concerned with this matter. To begin with, all definitions and results in [5] that underlie the corollary must be adapted to the general setting. 25 4.1 Basic Concepts
Following the lines of our treatment of partitions and connections on complete lattices sup- generated by a given subset, Definitions 31 and 33 generalise readily. We begin by the former. Note that, in contrast to the generalisation of the concept of a connection (Definition 24), this definition is independent of any sup-generating set. Indeed, the dependence of Definition 24 on a sup-generating set lies in its second item, which does not appear here.
Definition 38. Let L be a complete lattice. A subset C of L is a partial connection if it satisfies
1. 0 ∈ C, and V W 2. if {ci}i∈I ⊆ C for some index set I, and i∈I ci 6= 0, then i∈I ci ∈ C.
Before defining partial partitions, we generalise Proposition 32.
Proposition 39. Let L be a complete lattice sup-generated by a subset S ⊆ L, and let π : S → L be a map that satisfies
∀s ∈ S; s ≤ π(s). (47)
Then π is an S-partition if and only if it satisfies
∀s, t ∈ S; s ≤ π(t) ⇒ π(s) = π(t). (48)
Proof. Assume that π is an S-partition and assume that s ≤ π(t) for some s, t ∈ S. Since s ≤ π(s), we have π(s) ∧ π(t) 6= 0, and π being an S-partition this implies that π(s) = π(t). Assume now that (48) holds. By (47), π satisfies the first axiom of an S-partition. Assume that π(s) ∧ π(t) 6= 0 for some s, t ∈ S. Then ∃r ∈ S; r ≤ π(s) and r ≤ π(t). Then by (48), π(r) = π(s) and π(r) = π(t); thus π(s) = π(t). The claim follows by arbitrariness of the choice of s and t.
Now we define partial partitions, relying on Proposition 39 in order to remain consistent with the theory in [5].
Definition 40. Let L be a complete lattice sup-generated by a subset S ⊆ L.A partial S- partition on L is a map π : S → L that satisfies
1. ∀s ∈ S; π(s) = 0 or s ≤ π(s), and
2. ∀s, t ∈ S such that s ≤ π(t) it holds that π(s) = π(t).
In analogy with the above, we will use the same notation for a partial S-partition and its image (i.e. the set of its classes). The prefix S will moreover be dropped when the set S is clear from context.
The sets of partial connections and partial S-partitions on L, henceforth denoted by Γ∗(L) ∗ and ΠS(L), respectively, are indeed complete lattices. This is known [4] for the case where L = 2E for a set E. For the general case, the claim is proven in Appendix A.
26 Example 9. If L is a complete lattice sup-generated by S ⊆ L, then for each m ∈ L, the function
1m : S → L defined by 1m(s) = m if s ≤ m, and 1m(s) = 0 otherwise, is a partial S-partition with m as unique non-zero class. This partial partition will often be used in what follows. The class splitting operators become as follows.
Definition 41. Let L be a complete lattice sup-generated by a subset S ⊆ L.
∗ 1. A map ψ : L → ΠS(L) is element splitting if ψ(l) ≤ 1l for all l ∈ L; equivalently, if ∀l ∈ L; ∀s ∈ S; ψ(l)(s) ≤ l.
2. Given an element splitting map ψ, the class splitting operator induced by ψ is the oper- ∗ ∗ ∗ ∗ ∗ ator ψ :ΠS(L) → ΠS(L) such that for each partial S-partition π ∈ ΠS(L), ψ (π) = W c∈π ψ(c).
∗ The supremum in item 2 of the definition is well defined since ΠS(L) is a complete lattice. An example of an element splitting map and the class splitting operator it induces is given after Lemma 42 below.
4.2 Underlying Results
The propositions and lemmata of this section will be needed to prove the main result. They are generalisations of selected results proven for the subset lattice in [4] and, mainly, in [5], and their somewhat technical proofs generalise the original proofs. To begin with, the following result will be useful.
Lemma 42. Let L be a complete lattice sup-generated by a subset S ⊆ L. For each element splitting ∗ ∗ map ψ : L → ΠS(L), the induced class splitting operator ψ is intensive; moreover it is isotone if and only if ψ is isotone.
∗ ∗ W Proof. By definition, ψ(l) ≤ 1l for all l ∈ L, and thus for all π ∈ ΠS(L), ψ (π) = c∈π ψ(c) ≤ W 0 c∈π 1c = π; the last equality holds since 1c ≤ π for each class c of π, and given any π 0 such that 1c ≤ π for all c ∈ π, it holds for any s ∈ S with s ≤ c for some c ∈ π that 0 π(s) = c = 1c(s) ≤ π (s); if s c for all c ∈ π, then π(s)=0, hence π is the least upper bound of the set {1c, c ∈ π}. This proves intensiveness. 0 0 ∗ Assume next that ψ is isotone. If π ≤ π for partial partitions π, π ∈ ΠS(L), it follows from the order relation on partial partitions that for each c ∈ π there exists c0 ∈ π0 such that c ≤ c0, and since π0 is a partial partition, c0 is unique for c 6= 0. Since ψ is isotone, ψ(c) ≤ ψ(c0), and thus
_ _ _ _ ψ∗(π) = ψ(c) ≤ ψ(c) ≤ ψ(c0) ≤ ψ(c0) = ψ∗(π0), (49) c∈π c∈π\{0} c∈π\{0} c0∈π0\{0} where the inequalities follow from uniqueness of c0 for c 6= 0, and the fact that ψ(0) is the zero partition. Hence ψ∗ is isotone. ∗ Assume finally that ψ is isotone. If l ≤ m for l, m ∈ L, then 1l ≤ 1m since the only ∗ ∗ ∗ non-zero class of 1l is smaller than or equal to that of 1m. As ψ is isotone, ψ (1l) ≤ ψ (1m). ψ∗(1 ) = W ψ(c) = ψ(l) ∨ ψ(0) = ψ(l) ψ∗(1 ) = ψ(m) Now l c∈1l ; similarly, m . Thus we have ψ(l) ≤ ψ(m), i.e. ψ is isotone. This completes the proof. 27 ∗ Example 10. The map 1• : L → ΠS(L), l 7→ 1l is obviously element splitting. Its induced class ∗ ∗ ∗ W W splitting operator is 1• :ΠS(L) → ΠS(L); π 7→ c∈π 1c. By the above proof, c∈π 1c = π for ∗ ∗ any partial S-partition π; thus 1• is in fact the identity map on ΠS(L). The next proposition introduces the notion of a restriction. It generalises the concrete notion of a restriction of a partition on a set E to a class of a larger partial partition, as given W in [5]. The reader will note that the equality c∈π 1c = π above is a special case of this proposition.
0 ∗ Proposition 43. Let L be a complete lattice sup-generated by a subset S ⊆ L, and let π, π ∈ ΠS(L) 0 0 with π ≤ π . For each c ∈ π , the partial partition [π]c = π ∧1c, called the restriction of π to c, satisfies the following properties: W 1. π = c∈π0 [π]c.
0 0 2. If π1, π2 ≤ π , then π1 ≤ π2 if and only if ∀c ∈ π , [π1]c ≤ [π2]c.
∗ 0 3. If {πi}i∈I is a non-empty family in ΠS(L) for an index set I, satisfying πi ≤ π for all i ∈ I, 0 W W V V then ∀c ∈ π , [ i∈I πi]c = i∈I [πi]c and [ i∈I πi]c = i∈I [πi]c. ∗ Proof. As for item 1, π is clearly an upper bound for all [π]c. Assume that π˜ ∈ ΠS(L) is another such upper bound. If for any s ∈ S it holds that π(s) 6= 0, then there is some c ∈ π0 such that s ≤ π(s) ≤ c, and so π(s) = [π]c(s) ≤ π˜(s). The claim follows. Regarding item 2, it is clear that π1 ≤ π2 implies that ∀c ∈ π, [π1]c ≤ [π2]c. The inverse implication follows from item 1, W W since if ∀c ∈ π, [π1]c ≤ [π2]c, then π1 = c∈π0 [π1]c ≤ c∈π0 [π2]c = π2. 0 W As for item 3, fix c ∈ π . To begin with, for all j ∈ I, πj ≤ i∈I πi and by item 2 [πj]c ≤ W W [ i∈I πi]c, whence [ i∈I πi]c is an upper bound of {[πi]c}i∈I . To prove that it is the smallest such, assume that π is any upper bound of {[πi]c}i∈I , and define π˜ by π˜(s) = (π ∧ 1 )(s) if s ≤ c, c π˜(s) = c0 if s ≤ c0 ∈ π0 \{c}, and (50) π˜(s) = 0 otherwise. 0 Then π˜ is a partial S-partition that satisfies π˜ ≤ π , [˜π]c = π ∧ 1c and [πi]c0 ≤ [˜π]c0 for all 0 0 i ∈ I and all c ∈ π . Thus by item 2, πi ≤ π˜ for all i ∈ I, and so by the definition of the W supremum, i∈I πi ≤ π˜. Another use of item 2 gives " # _ πi ≤ [˜π]c = π ∧ 1c ≤ π, (51) i∈I c W from which follows that [ i∈I πi]c is the smallest upper bound of {πi}i∈I . V V Next, for all j ∈ I, i∈I πi ≤ πj and as above, item 2 gives that [ i∈I πi]c is a lower bound of {[πi]c}i∈I . For any lower bound π it holds for all i ∈ I that π ≤ [πi]c ≤ πi. By V definition of the infimum we thus have that π ≤ i∈I πi snd since π ≤ [πi]c ≤ 1c it holds that V V V π ≤ ( i∈I πi)∧1c = [ i∈I πi]c. Thus [ i∈I πi]c is the greatest lower bound, whereby the proof is complete.
Next is a result on the structure of the set of element splitting maps, and how it transforms when passing to class splitting operators. The reader is asked to recall from Lemma 42 that class splitting operators are intensive. 28 Lemma 44. Let L be a complete lattice sup-generated by a subset S ⊆ L, and denote by ΨS(L) the ∗ ∗ set of all element splitting maps ψ : L → ΠS(L), and by I(ΠS(L)) the set of all intensive operators ∗ on ΠS(L). ∗ Then ΨS(L) and I(ΠS(L)) are complete lattices, under the usual order on maps of complete lat- ∗ tices. Moreover, the map ∗ :ΨS(L) → I(ΠS(L)) that assigns to each element splitting map ψ the class splitting operator ψ∗ induced by it is an injective complete morphism.
Proof. Recall the fact that given a complete lattice M and an element m ∈ M, the set Mm = {n ∈ M; n ≤ m} is a complete lattice, in which the smallest element coincides with that of M, ∗ and the largest element is m. We will show that both ΨS(L) and I(ΠS(L)) are obtained in this way. ∗ ∗ First, in the lattice (L, ΠS(L)) of all maps from L to ΠS(L), consider the map 1• : l 7→ 1l for all l ∈ L. By definition, every element splitting map ψ satisfies ψ ≤ 1•. Conversely, if φ ≤ 1• ∗ 1• for a map φ then φ(l) ≤ 1l for all l ∈ L. We have the equality ΨS(L) = (L, ΠS(L)) , and ΨS(L) is a complete lattice. Secondly, a map φ on a lattice M is intensive if and only if φ ≤ id, ∗ ∗ id where id is the identity map on M. From this follows the equality I(ΠS(L)) = (ΠS(L)) , and ∗ I(ΠS(L)) is a complete lattice.
Next we show that ∗ is complete. First, take a non-empty family {ψi, i ∈ I} of element W ∗ splitting maps for an index set I, and set ψ = i∈I ψi. Since by Lemma 42 ψ and, for all ∗ ∗ ∗ i ∈ I, ψi are intensive, then given any π ∈ ΠS(L) and any c ∈ π, the partial partitions [ψ (π)]c ∗ and [ψi (π)]c are defined. Further, " # ! ∗ _ 0 _ 0 [ψ (π)]c = ψ(c ) = ψ(c ) ∧ 1c = ψ(c), (52) c0∈π c c0∈π W 0 where the right-most equality holds since given s ∈ S; if s c, then (( c0∈π ψ(c )) ∧ 0 1c)(s) ≤ 1c(s) = 0 = ψ(c)(s), hence both sides are zero, and if s ≤ c, then ψ(c )(s) ≤ 1c0 (s) = 0 0 W 0 for all c 6= c, and thus (( c0∈π ψ(c )) ∧ 1c)(s) = ψ(c) ∧ 1c(s) = ψ(c)(s). Moreover, since the W partial order on maps is defined in terms of their images, it holds that ψ(c) = i∈I (ψi(c)). ∗ Using the analogue of (52) to each ψi yields [ψi (π)]c = ψi(c), and hence
_ ∗ _ [ψi (π)]c = ψi(c) = ψ(c), (53) i∈I i∈I ∗ W ∗ and combining (52) and (53) gives [ψ (π)]c = i∈I [ψi (π)]c. Item 3 of Proposition 43 then ∗ W ∗ gives that [ψ (π)]c = [ i∈I ψi (π)]c. Since this holds for all c ∈ π, item 2 of Proposition 43 ∗ W ∗ ∗ gives that ψ (π) = i∈I ψi (π), and as this holds for all π ∈ ΠS(L), we obtain
∗ _ ∗ ψ = ψi , (54) i∈I W ∗ and since the left hand side is precisely ( i∈I ψi) , it follows that ∗ commutes with non- empty suprema. The case of non-empty infima is analogous. ∗ The empty supremum in ΨS(L) is the map 0 : L → ΠS(L); l 7→ 0, where the last zero is ∗ ∗ ∗ W the zero element in ΠS(L). This implies that for all π ∈ ΠS(L), 0 (π) = c∈π 0(c) = 0, and ∗ hence it is the smallest element in I(ΠS(L)). Finally, the empty infimum in ΨS(L) is the map 29 ∗ ∗ W W 1•, and for all π ∈ ΠS(L), 1•(π) = c∈π 1c, and by the proof of Lemma 42, c∈π 1c = π; hence ∗ ∗ ∗ 1• is the identity map on ΠS(L), which is indeed the greatest element of I(ΠS(L)).
∗ ∗ Last, we show injectivity of ∗. Assume that for ψ1, ψ2 ∈ ΨS(L) it holds that ψ1 = ψ2 . From ∗ the last part of the proof of Lemma 42, we have ψj (1l) = ψj(l) for all l ∈ L and j ∈ {1, 2}. ∗ ∗ Hence by assumption, we have ψ1(l) = ψ1 (1l) = ψ2 (1l) = ψ2(l) for all l ∈ L; hence, ψ1 = ψ2 and ∗ is injective. This completes the proof.
The next result is in fact the generalisation of the main part of Serra’s theorem to partial partitions and partial connections on arbitrary complete lattices. Recall that Serra’s theorem was generalised to the partial setting on subset lattices in [4] (where an augmentation was added, which we generalise in Appendix B) and in the non-partial setting to arbitrary com- plete lattices in Theorems 28 and 30 above. Thus this result is a generalisation in two direc- tions; its proof follows closely the lines of the proof of Theorem 28, but will be given here (in an abridged form) due to some technical differences. The reader will note that the partial setting makes the statement hold true in full strength even in the most general case, i.e. it circumvents the need of atomisticity. For a complete lattice L sup-generated by a subset S, we denote the set of all partial S- ∗ partitions on L whose classes all belong to a subset C ⊆ L by ΠS(L, C).
Proposition 45. Let L be a complete lattice sup-generated by a subset S ⊆ L, and let C ⊆ L be a subset. The following statements are equivalent:
1. C is a partial connection.
∗ 2. The set ΠS(L, C) is closed under arbitrary suprema.
Remark. Note that this proposition does not use the concept of a criterion when relating con- nectivity to closedness under the supremum (i.e. to being a dual Moore family). It is in this sense more general.
∗ Proof. 1 ⇒ 2: The empty supremum in ΠS(L, C) is the zero partial partition, which is in ∗ ΠS(L, C) by item 1 of Definition 38. Let {πi}i∈I , for some index set I, be a non-empty family ∗ W in ΠS(L, C) with π = i∈I πi. Let πC be the partition that to each s ∈ S assigns its connected component of its class of π, i.e.
_ πC (s) = {c ∈ C; s ≤ c ≤ π(s)} (55) . ∗ By construction and by Definition 38, πC ≤ π and πC ∈ ΠS(L, C). (Indeed, if {c ∈ C; s ≤ c ≤ π(s)} = ∅, then πC (s) = 0 ∈ C, and if {c ∈ C; s ≤ c ≤ π(s)}= 6 ∅, then this set has s as a lower bound, so its infimum is non-zero, and by Item 2 of Definition 38, πC (s) ∈ C.)
Moreover, for all i ∈ I, each class of πi is smaller than or equal to a class of πC since each class of πi belongs to C and is smaller than or equal to a class of π. Hence πC is an upper bound ∗ of {πi}i∈I , and π being the supremum, we have π ≤ πC , and so π = πC . Hence ΠS(L, C) is closed under suprema.
30 V 2 ⇒ 1: By definition, for 0 ∈ L we have 0 ∈ C. Take any {ci}i∈I ⊆ C satisfying i∈I ci 6= 0. V This implies that ∃s0 ∈ S; s0 ≤ i∈I ci. Each ci induces a partial partition πi defined for all s ∈ S by ( πi(s) = ci if s ≤ ci, and (56) πi(s) = 0 otherwise. ∗ W ∗ It is clear that πi ∈ ΠS(L, C) for all i ∈ I, and thus π = i∈I πi ∈ ΠS(L, C). Define W x = i∈I ci. We need show that x ∈ C. For any i ∈ I, the definition of πi implies that W πi(s0) = ci and since x = i∈I ci, it holds that πi(s0) ≤ x for all i ∈ I. Consider the partial partition π0 defined in analogy to (56) by ( π0(s) = x if s ≤ x, and (57) π0(s) = 0 otherwise 0 for all s ∈ S. For all i ∈ I it holds that πi ≤ π since πi(s0) = ci ≤ x. Now, π being the supremum it holds that π ≤ π0, and thus
0 π(s0) ≤ π (s0) = x. (58)
The inverse relation holds as well, since as πi ≤ π, we have ci = πi(s0) ≤ π(s0) for all W i ∈ I, from which follows that x = i∈I ci ≤ π(s0). Thus x = π(s0), whereby x ∈ C since ∗ ΠS(L, C) is closed under suprema. This completes the proof.
Before continuing, let us contemplate for a moment the usefulness of partial partitions and partial conditions in this general setting. The proof of Proposition 45 is, as stated above, analogous to that of Theorem 28, which, as we have seen, does not generalise directly to the generally non-atomistic case. The proof of Proposition 45 however requires no atomisticity. This depends both on the fact that the smallest partial partition is that with unique class 0, and on the fact that not every element of the sup-generating set is smaller than or equal to its class of a partial partition. In other words, partitions are too exhaustive to serve the purpose, and this matters only when the lattice is not assumed to be atomistic.
The construction in the following example will be useful. Example 11. Let L be a complete lattice sup-generated by a subset S ⊆ L. Let l ∈ L. Any ∗ partial connection C ∈ Γ (L) defines a partial S-partition Cl by
_ Cl(s) = {c ∈ C; s ≤ c ≤ l} (59)
for all s ∈ S. Thus Cl(s) = 0 if s l (but the inverse implication is in general false, since ∗ C is a partial connection). The map C• : L → ΠS(L), l 7→ Cl is element splitting, and induces ∗ ∗ ∗ the class splitting operator C• :ΠS(L) → ΠS(L). One sees, using Proposition 43.1–2 and the ∗ ∗ proof of Lemma 44, that Inv(C• ) = ΠS(L, C), which is a dual Moore family by Proposition 45. ∗ In other words, C• assigns to a partial S-partition π the partial S-partition whose classes are the connected components of the classes of π; its invariance set is the set of partitions whose classes are already connected. Finally, the following result will be needed. 31 Lemma 46. Let L be a complete lattice sup-generated by a subset S ⊆ L. Let {ψi}i∈I be a non-empty ∗ ∗ ∗ family of isotone element splitting maps ψi : L → ΠS(L). Then the greatest opening γ on ΠS(L) ∗ V ∗ ∗ that satisfies γ ≤ i∈I (ψi ) is a class splitting operator on ΠS(L).
Remark. Note that the notion of the greatest opening smaller than or equal to an arbitrary operator on a complete lattice M is well defined, since by Lemma 35 the set of openings on M is a dual Moore family in the lattice (M, M).
Proof. We begin by characterising the opening in terms of its domain of invariance (see the V remark preceding Definition 13). Set ψ = i∈I ψi. Then by Lemma 44, ψ is element splitting ∗ V ∗ and ψ = i∈I (ψi ). Since all ψi are isotone, so is ψ (by an elementary proof), and hence Lemma 42 gives that ψ∗ is isotone and intensive. Consider the set Inv(ψ∗). This set is closed ∗ ∗ under suprema, since ψ being intensive, we have, for the empty supremum 0 ∈ ΠS(L) that ∗ ∗ ∗ ψ (0) ≤ 0 ⇒ ψ (0) = 0, and if {πj}j∈J (J being an index set) is a non-empty family in Inv(ψ ), then by intensiveness ∗ _ _ ψ πj ≤ πj, (60) j∈J j∈J and since ψ∗ is isotone, it holds that for all j ∈ J, ∗ ∗ _ πj = ψ (πj) ≤ ψ πj , (61) j∈J ∗ W W thus ψ ( j∈J πj) is an upper bound of {πj}j∈J which is smaller than or equal to j∈J πj; ∗ W W W ∗ ∗ hence necessarily ψ ( j∈J πj) = j∈J πj, and j∈J πj ∈ Inv(ψ ). Hence Inv(ψ ) is a dual ∗ ∗ ∗ ∗ Moore family in ΠS(L), and there exists a unique opening γ on ΠS(L) such that Inv(γ ) = Inv(ψ∗), and γ∗ is defined by
_ γ∗(π) = {π0 ∈ Inv(ψ∗); π0 ≤ π}, (62)
∗ ∗ for all π ∈ ΠS(L). We now show that γ is the greatest opening smaller than or equal to ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ψ . Take any π ∈ ΠS(L); then γ (π) = ψ (γ (π)) since γ (π) ∈ Inv(ψ ). Now, since γ is intensive (being an opening), γ∗(π) ≤ π, and so ψ∗(γ∗(π)) ≤ ψ∗(π) since ψ∗ is isotone. Hence γ∗ ≤ ψ∗. If γ0∗ is another opening such that γ0∗ ≤ ψ∗, then since ψ∗ is intensive,
Inv(γ0∗) ⊆ Inv(ψ∗) = Inv(γ∗). (63)
0∗ ∗ 0∗ 0∗ ∗ 0∗ As γ is idempotent, this means that for all π ∈ ΠS(L), γ (π) ∈ Inv(γ ), thus γ (γ (π)) = γ0∗(π), which implies that γ0∗ ≤ γ∗ since γ∗(γ0∗(π)) ≤ γ∗(π) by intensiveness of γ0∗ and iso- tonity of γ∗. Hence γ∗ is the greatest opening smaller than or equal to ψ∗.
In the remainder of the proof, for any set-splitting map φ, set F(φ) = {l ∈ L; φ(l) = 1l}.
∗ ∗ ∗ Next we show that γ is class splitting. We begin by showing that Inv(γ ) = ΠS(L, F(ψ)), ∗ ∗ or equivalently Inv(ψ ) = ΠS(L, F(ψ)), which we do by mutual inclusions.
32 ∗ ∗ ∗ Let π ∈ Inv(ψ ), and c ∈ π. Then ψ (π) = π, and ψ (π) ∧ 1c = π ∧ 1c = 1c. Now, ∗ ∗ ψ(c) = ψ (π) ∧ 1c, by the proof of Lemma 44. Thus for all π ∈ Inv(ψ ), and all c ∈ π, ∗ ψ(c) = 1c, hence π ∈ ΠS(L, F(ψ)). ∗ If on the other hand π ∈ ΠS(L, F(ψ)), then for all c ∈ π, ψ(c) = 1c; hence
∗ _ _ ψ (π) = ψ(c) = 1c = π. (64) c∈π c∈π ∗ ∗ Thus ψ (π) = π for all π ∈ ΠS(L, F(ψ)). ∗ ∗ ∗ Altogether we have Inv(γ ) = Inv(ψ ) = ΠS(L, F(ψ)). Thus this latter is a dual Moore family, and satisfies item 2 of Proposition 45; hence F(ψ) is a partial connection. Moreover, as has been seen in Example 11,
∗ ∗ ΠS(L, F(ψ)) = Inv(F(ψ)•). (65)
∗ ∗ If one can show that F(ψ)• is an opening on ΠS(L), then by the one-one correspondence ∗ ∗ between dual Moore families and openings, it necessarily holds that γ = F(ψ)•; hence that ∗ ∗ ∗ γ is class splitting. In fact, we will show that C• is an opening on ΠS(L) for any partial ∗ ∗ 0 ∗ connection C. Indeed, C• is intensive by Lemma 42. Let π ∈ ΠS(L), and take any π ∈ ΠS(L) ∗ 0 0 ∗ 0 0 such that C• (π ) ≤ π. By construction, each class c of C• (π ) belongs to C and c ≤ c for a class ∗ 0 ∗ c of π. By definition of C• , this implies that c is smaller than or equal to a class of C• (π). Thus ∗ 0 ∗ ∗ C• (π ) ≤ C• (π), and C• is an opening, whereby the statement follows.
4.3 The Corollary Generalised
Now we are ready to generalise Ronse’s corollary (Proposition 36).
Theorem 47. Let L be a complete lattice sup-generated by a subset S ⊆ L. Let ΩS(L) be the set of all ∗ class splitting openings on ΠS(L). Then ΩS(L) is a complete sublattice of the lattice of openings on ∗ ∗ ΠS(L). As a complete lattice, it is isomorphic to the lattice Γ (L) of partial connections on L.
∗ Proof. We begin by proving the first part. The smallest opening on ΠS(L) is obviously the map 0 : π 7→ 0 mapping each partial S-partition to the zero partial partition. This is class splitting, as induced by the element splitting map that assigns to each element l ∈ L the zero partial ∗ partition. The greatest opening on ΠS(L) is the identity map; as has been seen in Example ∗ 10 this is the class splitting operator 1•. It follows that the greatest and smallest elements of ΩS(L) coincide with those of the lattice of openings, and that the infimum and the supremum of an empty family of class splitting openings are both class splitting openings. ∗ Consider next a non-empty family of class splitting openings {ψi }i∈I for an index set I. By ∗ injectivity of ∗ (Lemma 44), each ψi is induced by a unique element splitting map ψi, which ∗ is moreover isotone by Lemma 42 since ψi is an opening, hence isotone itself. By Lemma 35, W ∗ W the supremum i∈I (ψi ) is an opening, and Lemma 44 gives that i∈I ψi is element splitting and that
!∗ _ ∗ _ (ψi ) = ψi , (66) i∈I i∈I
33 W ∗ ∗ whereby i∈I (ψi ) is a class splitting opening, which is the smallest upper bound of {ψi }i∈I ∗ in the lattice of openings. The infimum in the lattice of openings on ΠS(L) of a non-empty set is by Lemma 35 the greatest opening smaller than or equal to the infimum of this set in ∗ the lattice of operators on ΠS(L). Since all ψi, i ∈ I are isotone, Lemma 46 applies, and the ∗ infimum of {ψi }i∈I in the lattice of openings is a class splitting opening. This proves the first part of the theorem.
∗ Given a partial connection C ∈ Γ (L), consider the element splitting map C• : L → ∗ ∗ ∗ ∗ ΠS(L), l 7→ Cl of Example 11, and the induced class splitting operator C• :ΠS(L) → ΠS(L). In the proof of Lemma 46 we showed that this operator is an opening. We will now show that the map
∗ ∗ λ :Γ (L) → ΩS(L); C 7→ C• (67)
∗ is an isomorphism of complete lattices. It is surjective, since any ψ ∈ ΩS(L) is an opening, hence the greatest opening smaller than or equal to ψ∗ = V{ψ∗}; hence, by the proof of ∗ ∗ Lemma 46, F(ψ) is a partial connection, and ψ = (F(ψ))•. To show injectivity, assume that ∗ ∗ ∗ ∗ ∗ there exist partial connections C, D ∈ Γ (L) such that C• = D•. Then since C• ≤ D•, we have
∗ ∗ ∀c ∈ C, 1c = C• (1c) ≤ D•(1c) ≤ 1c (68) where the equality holds since c ∈ C, and the rightmost inequality is due to intensiveness ∗ of D•. This implies that at each inequality we have equality, and the rightmost equality thus ∗ ∗ ∗ obtained implies that 1c ∈ Inv(D•). Thus c ∈ D and C ⊆ D. From the inequality D• ≤ C• we deduce D ⊆ C, whence C = D. This proves injectivity.
∗ ∗ ∗ It remains to be shown that for C, D ∈ Γ (L), it holds that C ⊆ D ⇔ C• ≤ D•. The implica- tion “⇐” was shown as part of the proof of injectivity. To prove the inverse implication, note that if C ⊆ D, then for each l ∈ L and s ∈ S,
Cl(s) ∈ C ⊆ D. (69)
For a given s ∈ S, either Cl(s) = 0 or s ≤ Cl(s) ≤ l. Thus by definition of Dl, Cl(s) ≤ Dl(s) for all s ∈ S, hence ∀l ∈ L; Cl ≤ Dl, and so C• ≤ D•. Since ∗ is a complete morphism, we have ∗ ∗ C• ≤ D•, and the proof is complete.
5 Relating Serra’s & Ronse’s Approaches in General Lattices
In Section 3 we showed that in the case of subset lattices, the two approaches studied above are related by a commutative diagram. It is thus natural to ask if the generalisations made in Sections 2 and 4 respect this; in other words, whether the commutative diagram of Theorem 37 holds in the general setting. Before we begin, we recall the part of Serra’s theorem that does not generalise entirely; namely, for a non-atomistic lattice L sup-generated by a set S, given a family F of functions f : S → T for a set T , it may happen that a criterion that S-segments all f ∈ F is not 34 S-connective (see Example 6, Section 2.3.3). As far as Theorem 37 is concerned, this is im- material, since in the theorem one starts with S-connective criteria, and hence only relies on the implication that S-connective criteria give an S-segmentation. We will provide two gen- eralisations of Theorem 37. The first relies on Theorem 30, which is the generalisation of the (original version of) the theorem of Serra. The second uses Proposition 45, which generalises not Serra’s theorem, but its adaptation in [4] to partial partitions and partial connections. While the first approach is in this sense more faithful, the second is more general.
5.1 Maintaining the Non-Partial Setting
We begin by generalising Theorem 37 in the light of Theorem 30. This leads to the following
Theorem 48. Let L be a complete lattice, sup-generated by a subset S ⊆ L, T a set, and F a family of functions f : S → T . Define ΣS(F, L) to be the set of all S-connective criteria on F, ΠS(L) the ∗ ∗ complete lattice of S-partitions on L, ΠS(L) the complete lattice of partial S-partitions on L, Γ (L) the complete lattice of partial connections on L, and ΩS(L) the complete lattice of class splitting openings ∗ on ΠS(L). Let moreover
• for each f ∈ F, φf :ΣS(F, L) → ΠS(L) be the map that assigns to each S-connectice criterion σ its S-segmentation of f, and
∗ • λ :Γ (L) → ΩS(L) be the lattice isomorphism that assigns to each partial connection C the ∗ class splitting opening C• .
Then for each f ∈ F there exist maps µf and κ such that following diagram commutes
φf ΣS(F, L) / ΠS(L) (70) O µf κ
∗ Γ (L) / ΩS(L) λ
In other words we have φf (σ) = κλµf (σ) for all f ∈ F and σ ∈ ΣS(F, L). Specifically,
• µf is the map that assigns to each S-connective criterion σ the S-connection {c ∈ L; σ(f, c) = 1}, and
• κ is the map that assigns to each class splitting opening γ∗ the partial S-partition γ∗(1), (where 1 is the greatest S-partition on L).
Remark. In this generalised version, the map φf is well-defined due to Theorem 30. The iso- morphism λ is due to Theorem 47 and its proof.
Proof. Fix a function f ∈ F and take any criterion σ ∈ ΣS(F, L). Then µf (σ) = {c ∈ ∗ L; σ(f, c) = 1}, and composing µf with λ gives that λµf (σ) = (µf (σ))•. Composition with κ gives
∗ κλµf (σ) = (µf (σ))•(1), (71)
35 where for all s ∈ S,
∗ _ _ (µf (σ))•(1)(s) = {c ∈ µf (σ); s ≤ c ≤ 1(s)} = {c ∈ µf (σ); s ≤ c}, (72)
∗ i.e. (µf (σ))•(1)(s) is equal to the µf (σ)-connected component greater than or equal to s, if it exists, and otherwise it is equal to 0. Since µf (σ) is a partial connection, it follows from ∗ the axioms that (µf (σ))•(1) is a partial S-partition all of whose classes belong to µf (σ). It is moreover an S-partition, since µf (σ) is an S-connection, and hence every s ∈ S is smaller than or equal to an element of µf (σ), and thus to a µf (σ)-connected component. This is the largest S-partition all of whose classes belong to µf (σ), since there exist no c ∈ µf (σ) that is strictly greater than a µf (σ)-component. Hence it is the S-segmentation of f by σ, and we arrive at
κλµf (σ) = φf (σ), (73)
which completes the proof.
5.2 Using Partiality
We now consider the analogue of Serra’s theorem for partial connections and partial parti- tions, as introduced in [4]. By Proposition 45 above this generalises unconstrained to the case of arbitrary complete lattices, and it is on this result that the following study relies. We first need the notion of partially connective criteria, and partial segmentation. These have been introduced by Ronse (see e.g. [4]) for the case of the subset lattice. We give here the generali- sation to arbitrary complete lattices, in coherence with Definitions 25 and 27 above.
Definition 49. Let L be a complete lattice sup-generated by a subset S ⊆ L. A criterion σ on a family F of functions from S to a set T is partially connective if for each f ∈ F the set {c ∈ L; σ(f, c) = 1} is a partial connection.
Definition 50. Let L be a complete lattice sup-generated by a subset S ⊆ L. Given f ∈ F ∗ where F is a family of functions from S to a set T , and a criterion σ on F, let ΠS(L, σf ) be the family of all partial S-partitions π of L such that σ is validated on each class of π. The ∗ criterion σ is said to S-segment f partially if ΠS(L, σf ) is closed under the supremum of partial S-partitions. ∗ In that case, the supremum of ΠS(L, σf ) is the partial S-segmentation of f under σ.
Proposition 45 then immediately implies the following
Corollary 51. Let L be a complete lattice sup-generated by a subset S ⊆ L, F a family of functions f : S → T for a set T , and σ a criterion on F. The following statements are equivalent:
1. σ is partially connective.
2. σ S-segments all f ∈ F partially.
We are now ready to state the desired generalisation of Theorem 37.
36 Theorem 52. Let L be a complete lattice, sup-generated by a subset S ⊆ L, T a set, and F a family ∗ ∗ of functions f : S → T . Define Σ (F, L) to be the set of all partially connective criteria on F, ΠS(L) the complete lattice of partial S-partitions on L, Γ∗(L) the complete lattice of partial connections on L, ∗ and ΩS(L) the complete lattice of class splitting openings on ΠS(L). Let moreover
0 ∗ ∗ • for each f ∈ F, φf :Σ (F, L) → ΠS(L) be the map that assigns to each partially connectice criterion σ its partial S-segmentation of f, and
∗ • λ :Γ (L) → ΩS(L) be the lattice isomorphism that assigns to each partial connection C the ∗ class splitting opening C• .
0 Then for each f ∈ F there exist maps µf and κ such that following diagram commutes
φ0 ∗ f ∗ Σ (F, L) / ΠS(L) (74) O 0 µf κ
∗ Γ (L) / ΩS(L) λ 0 0 ∗ In other words we have φf (σ) = κλµf (σ) for all f ∈ F and σ ∈ Σ (F, L). Specifically,
0 • µf is the map that assigns to each partially connective criterion σ the partial connection {c ∈ L; σ(f, c) = 1}, and
• κ is the map that assigns to each class splitting opening γ∗ the partial S-partition γ∗(1), where 1 is the greatest S-partition on L.
0 Remark. Here, the map φf is well defined due to Corollary 51. The proof is similar to that of Theorem 48, and we therefore give it in a short form.
Proof. Take a function f ∈ F and a criterion σ ∈ Σ∗(F, L). Then as in the proof of Theorem 0 ∗ 48, κλµf (σ) = (µf (σ))•(1), which is as given in the last proof. Again, since µf (σ) is a partial ∗ connection, we have that (µf (σ))•(1) is a partial S-partition all of whose classes belong to µf (σ), and which is the greatest partial S-partition with this property, by the definition of connected components of a partial connection. Hence it is the partial S-segmentation of f by σ, which completes the proof.
We end this section by emphasizing that the main difference between Theorems 48 and 52 is that the former keeps each approach in its original setting; thus, the map due to the generalisation of Serra’s theorem is defined on connective rather than partially connective criteria. Theorem 52, on the other hand, is entirely based on the partial set-up.
6 Concluding Remarks
In one sentence, this work has been preoccupied with studying two theories in connective segmentation, finding the relation between them, and generalising the theories and the rela- tion from the concrete setting of the lattice of the subsets of a given set, to the abstract setting of an arbitrary complete lattice sup-generated by a given subset. 37 Nevertheless, such a summary does not give an accurate idea of the disposition of the work. Indeed, the fundamental task was that of generalising the theorem of Serra, which was done slowly and steadily. The main results in this part of the work are Theorem 28 on atom- istic lattices, to which the theorem generalises entirely, and Theorem 30 on arbitrary complete lattices, where it holds only in part. The study of viscous lattices, leading to Proposition 21, may either be seen as a step between the original result and its generalisations (as was indeed done above), or as a special case that is more complicated than the case of subset lattices. This study was succeeded by the generalisation of the corollary of Ronse. This required generalising a larger number of lemmata and propositions in order to be accomplished. How- ever, as much of the needed apparatus was already developped in the process of generalising Serra’s theorem, most such lemmata and propositions were generalised without much trou- ble and with little surprise, by generalising their original proofs. The most important of these results is Proposition 45, from the proof of which one realises that the partial setting allows Serra’s theorem to generalise in full to the case of arbitrary complete lattices (see also Corol- lary 51). The main result of this section is clearly the generalisation of the corollary itself, i.e. Theorem 47. The fact that Proposition 45 and Theorem 47 hold in the most general case shows an advantage that partial connections and partial partitions have over their non-partial counterparts — an advantage that matters only in the generalised setting. An important theorem that is not a generalisation of any existing result is Theorem 37 that links the two approaches of Ronse and Serra, and provides a unity to this work. Equally important are Theorems 48 and 52, which generalise this link. The first Appendix below contains proofs of basic fundamental results used in the above, and the second entails a generalisation of an additional augmentation to the theorem.
It is the hope of the author that this work has contributed, in a generalising and clarifying manner, to the algebraic theory of connective segmentation.
38 A Appendix: On Partitions, Partial Partitions and Partial Con- nections
Here we state and prove some results that have been used at different instances in the above sections.
Proposition 53. 1. Let L be a complete lattice, sup-generated by a subset S ⊆ L. Let ΠS(L) be 0 the set of all S-partitions on L, ordered by refinement: for all π, π ∈ ΠS(L),
π ≤ π0 ⇔ ∀s ∈ S; π(s) ≤ π0(s). (75)
With respect to this order, ΠS(L) is a complete lattice.
∗ 2. Let L be a complete lattice, sup-generated by a subset S ⊆ L. Let ΠS(L) be the set of all partial ∗ S-partitions on L, ordered by refinement. With respect to this order, ΠS(L) is a complete lattice.
3. Let L be a complete lattice. Let Γ∗(L) be the set of all partial connections on L. Ordered by inclusion, Γ∗(L) is a complete lattice.
Proof. 1. The empty infimum, i.e. the maximal element, of ΠS(L) is the partition that as-
signs to each s ∈ S the maximal element 1 ∈ L. For a non-empty family {πi}i∈I ⊆
ΠS(L), where I is an index set, define the function π˜ : S → L for all s ∈ S by
^ π˜(s) = πi(s), (76) i∈I
which is well-defined since L is a complete lattice. Now π˜ satisfies the axioms of an
S-partition: firstly, since πi is a partition for all i ∈ I, it holds for all s ∈ S and all i ∈ I
that s ≤ πi(s), and, by the definition of the infimum, that s ≤ π˜(s) as well. Secondly, if π˜(s) ∧ π˜(t) 6= 0 for some s, t ∈ S, then for all i ∈ I
π˜(s) ≤ πi(s) ⇒ πi(s) ∧ π˜(t) 6= 0 (77)
and since similarly ∀i ∈ I;π ˜(t) ≤ πi(t) we get πi(s) ∧ πi(t) 6= 0 for all i ∈ I. But since
every πi is a partition, this implies that
∀i ∈ I; πi(s) = πi(t) (78)
and then of course
^ ^ π˜(s) = πi(s) = πi(t) =π ˜(t), (79) i∈I i∈I
since the sets {πi(s)}i∈I and {πi(t)}i∈I are equal and hence have equal infima. More-
over, it is clearly a lower bound of {πi}i∈I and, by construction and by the definition of
the infimum in L, it is the greatest lower bound. Altogether we have shown that ΠS(L)
39 is a closed under arbitrary infima.7
V From the above we infer that ΠS(L) belongs to ΠS(L) and is its smallest element, hence it is the supremum of the empty family. To show that the supremum exists for all
non-empty families of partitions as well, take a non-empty family {πi}i∈I ⊆ ΠS(L), and define the set
I = {π ∈ ΠS(L); ∀i ∈ I; πi ≤ π}. (80)
V By the above, I is well-defined. It is an upper bound of {πi}i∈I , since for each i ∈ I,
πi ≤ π for all π ∈ I, hence πi is a lower bound of I, hence smaller than or equal to its greatest lower bound. From the definition of I it follows that V I is the smallest upper
bound of {πi}i∈I . Thus, ΠS(L) is also closed under arbitrary suprema, and hence a complete lattice.
∗ 2. As in ΠS(L), the maximal element of ΠS(L) is the partition that assigns to each s ∈ S ∗ the maximal element 1 ∈ L. For a non-empty family {πi}i∈I ⊆ ΠS(L), where I is an index set, define as above the function π˜ : S → L for all s ∈ S by
^ π˜(s) = πi(s), (81) i∈I
The axioms of a partial S-partition are satisfied: first, for s ∈ S, either πi(s) = 0 for V some i ∈ I, and thus π˜(s) = 0, or s ≤ πi(s) for all i ∈ I, whence s ≤ π˜(s) = i∈I πi(s). Secondly, if s ≤ π˜(t) for some s, t ∈ S, then for all i ∈ I we have s ≤ πi(t), and since
every πi is a partial partition, it holds that
∀i ∈ I; πi(s) = πi(t) (82)
and then of course
^ ^ π˜(s) = πi(s) = πi(t) =π ˜(t). (83) i∈I i∈I
Moreover, π˜ is clearly a lower bound of {πi}i∈I and, by construction, it is the greatest ∗ lower bound. Hence ΠS(L) is closed under arbitrary infima.
V ∗ ∗ From the above, ΠS(L) belongs to ΠS(L) and is its smallest element (in fact, this is the partial partition with unique class 0), hence it is the supremum of the empty family. To show that the supremum exists for all non-empty families of partial partitions, take ∗ such a family {πi}i∈I ⊆ ΠS(L), and define the set
7In fact, there is a result in lattice theory that states that a partially ordered set containing a maximal element, and where every non-empty infimum is defined, is a complete lattice. We will not use this result, however, anywhere in the proof of this proposition, and rather show explicitely that the partially ordered sets at hand are closed under arbitrary suprema, obtaining, in doing so, a more constructive proof. 40 ∗ I = {π ∈ ΠS(L); ∀i ∈ I; πi ≤ π}. (84) V Then I is well-defined and is the smallest upper bound of {πi}i∈I (see the proof of ∗ item 1). Hence, ΠS(L) is closed under arbitrary suprema, and thus a complete lattice.
3. The smallest element of Γ∗(L) is the partial connection {0} ⊆ L. The largest element is L, which is a partial connection since it trivially satisfies the axioms. These are, respec- tively, the empty supremum and the empty infimum in Γ∗(L). ∗ For a non-empty family {Ci}i∈I ⊆ Γ (L), where I is an index set, consider the collection ˜ T ˜ C = i∈I Ci; indeed 0 ∈ C since 0 ∈ Ci for all i ∈ I. If now for an index set J, {cj}j∈J is a ˜ T V W family of elements in C = i∈I Ci such that j∈J cj 6= 0, then j∈J cj ∈ Ci for all i ∈ I, W ˜ ˜ ˜ and hence j∈J cj ∈ C, whereby C is a partial connection. Being the intersection, C is the greatest lower bound of {Ci}i∈I for the inclusion order. ∗ Consider next I = {C ∈ Γ (L); ∀i ∈ I, Ci ⊆ C}. This is a non-empty family of partial connections, since it contains the partial connection L. By the above, the intersection of its members is the partial connection V I. Being the intersection, V I satisfies that V ∀i ∈ I, Ci ⊆ I, hence it is an upper bound of {Ci}i∈I and by construction it is the least upper bound. Hence Γ∗(L) is a complete lattice. The proof is complete.
B Appendix: On an Augmentation to the Theory
B.1 The Non-Partial Setting
In Section 3.3, it is mentionned that the theorem of Serra was augmented in [4]. The aug- mentation consists in a fourth statement being proven equivalent to the three of Theorem 10; namely that the criterion in question segments the restriction of the given function to any sub- set of the set at hand, and not only the whole set. More precisely, the following was proven.
Proposition 54. [4] Let E be a set, and C ⊆ 2E a collection of subsets of E. For any B ⊆ E, denote by Π(B, C) the set of all partitions of B all of whose classes belong to C. Then Π(E, C) is closed under suprema if and only if for all B ⊆ E, Π(B, C) is closed under suprema.
Remark. The notion of a criterion is not explicitely mentioned in the proposition; of course, the collection C can be taken to be the collection of all subsets that satisfy a given criterion for a given function. This way, the proposition provides the desired augmentation to Serra’s theorem. This property in fact generalises to atomistic lattices, whereby the following augmentation to Theorem 28 is obtained. Recall from Example 4 (Section 1.7.1) that given a complete lattice L and an element m ∈ L, the set Lm = {l ∈ L; l ≤ m} is a complete lattice. It is clear that if L is sup-generated by a subset S ⊆ L, then Lm is sup-generated by the set Sm = {s ∈ S; s ≤ m}.
41 Notation Throughout this appendix, given a complete lattice L sup-generated by S ⊆ L, an m m element m ∈ L \ 0, and a subset C ⊆ L, we denote by ΠS(L ) the set of all S -partitions on m m m m L , and by ΠS(L , C) the set of all S -partitions on L whose classes are all in C. Likewise, ∗ m m m ∗ m we let ΠS(L ) denote the set of all partial S -partitions on L , and ΠS(L , C) the set of all partial Sm-partitions on Lm whose classes are all in C.
Throughout this treatment, we require m 6= 0 in order to avoid lattices that contain only one element. The generalisation is as follows.
Proposition 55. Let L be an atomistic complete lattice, A the set of all atoms of L, and C ⊆ L a subset. m Then ΠA(L, C) is closed under suprema if and only if for all m ∈ L \ 0, ΠA(L , C) is closed under suprema.
m 1 Proof. If for all m ∈ L\0, ΠA(L , C) is closed under suprema, then so is ΠA(L, C) = ΠA(L , C).
If ΠA(L, C) is closed under suprema, then it is closed under the empty supremum, i.e. the partition π˜ : A → L, a 7→ a belongs to ΠA(L, C). Hence for all a ∈ A, a ∈ C. Take any m ∈ L\0. m The empty supremum in ΠA(L ) is clearly the partition
π˜m : Am → Lm, a 7→ a (85)
m whose clases thus belong to C. Hence ΠA(L , C) is closed under the empty supremum. m Next, take a non-empty family {πi}i∈I of partitions in ΠA(L , C) (where I is an index W set), and set π = i∈I πi. These partitions induce A-partitions on L as follows. Define ι : m ΠA(L ) → ΠA(L) by ( ι(ρ)(a) = ρ(a) if a ≤ m, and (86) ι(ρ)(a) = a otherwise,
m for all ρ ∈ ΠA(L ) and a ∈ A. Note first that ι is injective. By the first part of the proof, and W by construction, ι(πi) ∈ ΠA(L, C) for all i ∈ I, and hence by assumption i∈I ι(πi) ∈ ΠA(L, C). If we can show that ! _ _ ι(πi) = ι πi = ι(π) (87) i∈I i∈I
then since the left hand side is in ΠA(L, C), so is the right hand side, from which it then m follows that π ∈ ΠA(L , C), since every class of π is a class of ι(π).
To begin with, ι(πi) ≤ ι(π) holds for all i ∈ I since π is an upper bound of {πi}i∈I , and m W since ι(π)(a) = ι(πi)(a) for all i ∈ I and a ∈ A \ A , and thus i∈I ι(πi) ≤ ι(π). This implies m W m W that for all a ∈ A , ( i∈I ι(πi))(a) ≤ m, and for all a ∈ A \ A , we have ( i∈I ι(πi))(a) = a. 0 m Hence there exists a partition π ∈ ΠA(L ) such that
_ 0 ι(πi) = ι(π ). (88) i∈I By injectivity of ι, π0 is unique, and by the above ι(π0) ≤ ι(π). Then necessarily π0(a) ≤ m 0 π(a) for all a ∈ A , and hence π ≤ π. Finally, by the definition of an upper bound, ι(πi) ≤ 42 0 0 0 ι(π ) for all i ∈ I, which in the same way necessitates πi ≤ π for all i ∈ I. Hence π is an upper bound of {πi}i∈I that is smaller than or equal to π, whence by the definition of the supremum it follows that π0 = π, and by (88)
_ ι(πi) = ι(π) (89) i∈I as desired, whereby the proof is complete.
In non-atomistic lattices, the corresponding result is in general false; indeed, taking the situation of Example 6 (Section 2.3.3), and setting m = t there provides a counterexample. However, the following strengthening of Theorem 30 holds.
Proposition 56. Let L be a complete lattice sup-generated by a subset S ⊆ L. Let F be a family of functions f : S → T where T is an arbitrary set, and let σ be a criterion on F. For each f ∈ F, let Cf denote the set of all elements l ∈ L such that σ(l, f) = 1. If σ is S-connective, then for each f ∈ F m and each m ∈ L \ 0, the set ΠS(L , Cf ) is closed under suprema.
Proof. Fix f ∈ F, and let {πi}i∈I , for some index set I, be a (possibly empty) family in m W m ΠS(L , Cf ) with π = i∈I πi. Define πC : S → Cf by
_ πC (s) = {c ∈ Cf ; s ≤ c ≤ π(s)} (90)
. m V for all s ∈ S . This is well-defined since Cf is an S-connection, and {c ∈ Cf ; s ≤ c ≤ m m π(s)} = s 6= 0. Also, for all s ∈ S, πC (s) ≤ π(s) ≤ m, and πC is an S -partition of L (the m axioms are checked as in the proof of Theorem 28) and moreover πC ∈ ΠS(L , Cf ). Now,
πC ≤ π, and we have two cases: m m If I = ∅, then π is the smallest S -partition on L , and thus it follows that πC = π and m m hence π ∈ ΠS(L , Cf ). Thus ΠS(L , Cf ) is closed under the empty supremum. m If I 6= ∅, then for all i ∈ I and s ∈ S , πi(s) ≤ πC (s) since πi(s) ∈ Cf and s ≤ πi(s) ≤ π(s).
Hence πC is an upper bound of {πi}i∈I , and π being the least such, we have π ≤ πC , and thus m m π = πC . Since πC ∈ ΠS(L , Cf ), we have shown that ΠS(L , Cf ) is closed under non-empty suprema as well. The statement follows.
B.2 The Partial Setting
Recall that in Proposition 45 of Section 4.2, the adaptation of Serra’s theorem to the language of partial partitions and partial connections, made in [4], was generalised entirely to arbitrary complete lattices. In this section we will generalise the augmentation to this adaptation, which was also made in [4], and reads:
Proposition 57. [4] Let E be a set, and C ⊆ 2E a collection of subsets of E. For any B ⊆ E, denote by Π∗(B, C) the set of all partial partitions on B all of whose classes belong to C. Then Π∗(E, C) is closed under suprema if and only if for all B ⊆ E, Π∗(B, C) is closed under suprema.
Indeed, we have the following
43 Proposition 58. Let L be a complete lattice sup-generated by a subset S ⊆ L, and let C ⊆ L be a ∗ ∗ m subset. Then ΠS(L, C) is closed under suprema if and only if for all m ∈ L \ 0, ΠS(L , C) is closed under suprema.
Note that as in Proposition 55, the “if”-statement is trivial. The following is a proof of the “only if”-statement.
∗ Proof. If ΠS(L, C) is closed under suprema, then it is closed under the empty supremum, i.e. ∗ the zero partial S-partition belongs to ΠS(L, C). Hence for 0 ∈ L we have 0 ∈ C. Take any ∗ m m m ∈ L \ 0. The empty supremum in ΠS(L ) is clearly the zero partial S -partition, whose unique class 0 ∈ L thus belongs to C. ∗ m Next, take a non-empty family {πi}i∈I ⊆ ΠS(L , C) (where I is an index set), and set W ∗ m ∗ π = i∈I πi. Define ι :ΠS(L ) → ΠS(L) by ( ι(ρ)(s) = ρ(s) if s ≤ m, and (91) ι(ρ)(s) = 0 otherwise,
∗ m ∗ for all ρ ∈ ΠS(L ) and s ∈ S. Note that ι is injective. We have ι(πi) ∈ ΠS(L, C) for all W ∗ i ∈ I, and hence by assumption i∈I ι(πi) ∈ ΠS(L, C). We will show that ! _ _ ι(πi) = ι πi = ι(π) (92) i∈I i∈I ∗ m from which then follows that π ∈ ΠS(L , C). W As in the proof of Proposition 55, one sees that i∈I ι(πi) ≤ ι(π). This implies that for all m W m W s ∈ S , ( i∈I ι(πi))(s) ≤ m, and for all s ∈ S \ S , we have ( i∈I ι(πi))(s) = 0. Hence there 0 m W 0 exists (by injectivity of ι a unique) π ∈ ΠS(L ) such that i∈I ι(πi) = ι(π ), and moreover 0 0 0 0 ι(π ) ≤ ι(π). Again we deduce that π ≤ π and that πi ≤ π for all i ∈ I. Hence π = π, giving that
_ ι(πi) = ι(π). (93) i∈I The proof is complete.
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