Order-Preserving Self-Maps of Complete Lattices

We study isotone self-maps of complete lattices and their fixed point sets, which are complete lattices contained as suborders, but not necessarily as subsemilattices. We develop a representation of such maps by means of relations and show how to navigate their fixed point lattices using a modification of the standard Next closure algorithm. Our approach is inspired by early work of Shmuely [8] and Crapo [1]. We improve and substantially extend our earlier publication [4].


Introduction
The interest in concept lattices [6] has stimulated the creation of algorithms for generating lattices, and the availability of fast algorithms may conversely have contributed to the popularity of concept lattices.Moreover, concept lattices have easy representations either by a binary relation or by a set of implications, both of which can conveniently be used as input for the algorithms.
Although all complete lattices are isomorphic to concept lattices, they sometimes come in a form for which the above-mentioned algorithms are not easy to apply.There are, for example, many families of sets which form complete lattices when ordered by the subset relation ⊆ , but are neither closure nor kernel systems.We provide a relational representation for such lattices and adapt one of the standard algorithms accordingly.
Throughout the paper, (L,≤) will be some abstract complete lattice.The supremum and infimum of a subset S ⊆ L will be denoted by ⋁ S and ⋀ S .The reader may assume, without much loss of generality, that (L,≤) is a powerset lattice ( (M), ⊆) .We use the abstract setting because we find it more transparent.

Representing relations
A mapping : → between two ordered sets and is called order-preserving 1 if x ≤ always implies ≤ .Let be an ordered set and let be a complete lattice.An easy way to give examples of order-preserving maps : is to choose an arbitrary relation between and and to define for all v ∈ P obviously is order-preserving.It is also evident that conversely every order-preserving map can be obtained via such a representing relation, since is a trivial solution.
For a more compact representation define to be a -proper preimage for if and denote the set of all such -proper preimages by P .We say that is represented on its ⋁ -proper preimages if holds for all .This is not always the case, a generalized finiteness condition is needed.The descending chain condition (dcc) requires that every non-empty subset has a minimal element.

Proposition 1 Let (P,≤) be an ordered set and let (L,≤) be a non-trivial complete lattice.
The following conditions are equivalent: 1. (P,≤) satisfies the dcc.
Proof If there was an element violating equation (1) in an ordered set (P,≤) satisfying the dcc, then there also is a minimal such violating element, say, v. Every w < v fulfills the equation A violating element cannot be a ⋁ -proper preimage, and thus showing that v does not violate the equation, a contradiction.For the converse assume that (P,≤) contains a non-empty subset without a minimal element.The order filter F generated by this subset has no minimal elements either.Mapping F to the largest and P ∖ F to the smallest element of (L,≤) yields a mapping that is order-preserving, but has no ⋁ -proper preimages at all, since each ⋁ -proper preimage would be minimal in F. □ An interesting question is how to find the ⋁ -proper preimages for a given order-preserving mapping φ.This is easy when a representing relation of reasonable size is given, as the following proposition shows.Proposition 2 also allows to check if two relations represent the same order-preserving mapping (assuming dcc): they must have the same ⋁ -proper preimages and produce the same images of these.Example 1 Let (P,≤) and (L,≤) both be equal to the powerset lattice of the three-element set {a,b,c}.We give three examples E 1 , E 2 , and E 3 in terms of representing relations , which we write in infix notation.
• Example E 1 is represented by the relation given as The resulting order-preserving map φ ⊳ is as follows: {a}, {b}, and {c} are indeed the ⋁ -proper preimages.

Proposition 3 All proper preimages of a
As a consequence we get that p ≠ ⋁ {q | q < p} , which shows that p must be ⋁ -irreducible.□

Order-preserving self-maps
Definition 1 Let (L,≤) be a complete lattice.An order-preserving mapping φ : for all x ∈ L, increasing 3  if φ(x) ≤ φ(φ(x)) for all x ∈ L, and for all x ∈ L.
Theorem 1 Figure 1 shows the logical hierarchy of the properties given in Definition 1.
In particular, if φ : L → L is order-preserving, then the following statements hold (as well as their duals): φ is idempotent iff φ is both increasing and decreasing.4. If φ is idempotent and extensive, then φ is dually tensive.

Moreover, there are examples of order-preserving mappings falsifying other implications, as indicated in the diagram. 4
Fig. 1 The result of an attribute exploration [5] for order-preserving self-maps, see Theorem 1 2 A synonym is intensive 3 Following Shmuely [8].Tarski [9] uses "increasing" in the sense of "order-preserving" 4 Assertions 1., 3. and 4. hold even without assuming that φ is order preserving, as one of the reviewers has pointed out.
Order (2023) 40:455-468 In Example 1 above we have used infix notation for the relation , writing instead of .Note that always implies , but that the converse does not hold in general.It will be convenient to have a short notation for this case as well.We therefore define and keep in mind that implies .We say that entails a relation R ⊆ L × L , when and represent the same mapping.

Proof For arbitrary x ∈ L we have
The two images are the same if and this holds if for all (r,s) ∈ R But and r ≤ x imply s ≤ φ ⊳ (r) ≤ φ ⊳ (x), showing that the condition is sufficient.It is also necessary, because if s ≰  ⊳ (r) for some (r,s) ∈ R, then φ ⊳ and φ ⊳∪R are different since s ≤ φ ⊳∪R (r).□

Proposition 5
The order-preserving mapping φ ⊳ represented by is 1. extensive iff holds for all x ∈ L, 2. increasing iff holds for all x ∈ L, and 3. decreasing iff and together always imply .

Proof
The first two claims are immediate from the definitions of being extensive (x ≤ ).The third requires a few more words: The condition of being decreasing is φ ⊳ (φ ⊳ (x)) ≤ φ ⊳ (x), which obviously is equivalent to This is, up to notation, exactly the condition given in the proposition.□ We say that is an upward relation iff Proposition 6 Let (L,≤) be a complete lattice and φ : L → L be order-preserving.φ has an upward representing relation if and only if φ is tensive.

Closed and fixed points
Definition 2 If φ : L → L is a mapping and x ∈ L, then we say that x is a fixed point of φ iff φ(x) = x, and that x is a closed point of φ iff φ(x) ≤ x.

Proposition 7 Every fixed point is closed. If φ is order-preserving and increasing, and x is closed, then φ(x) is fixed.
Proof The first statement is obvious.Suppose that x is closed, i.e., that x ≥ φ(x).Then φ(x) ≥ φ(φ(x)) ≥ φ(x), when φ is order-preserving and increasing.We conclude that φ(x) = φ(φ(x)) and thus φ(x) is fixed.□ The proposition may suggest a pairing between fixed and closed elements.But note for example that when φ is the function which maps everything to the least element of (L,≤), then every element of (L,≤) is closed, but only the least element is fixed.
A function that is both idempotent and order-preserving is called a closure operator on (L,≤) if it is extensive, and is a kernel operator if contractive.The set of fixed points of a closure operator is called a closure system.It is well known that the closure systems are precisely the ⋀ -subsemilattices.Each complete meet-subsemilattice of a complete lattice is itself a complete lattice, because the join operation can be expressed in terms of the meet operation: the join of a subset S equals the meet of all upper bounds of S.However, this join operation usually is not identical with the join in the original complete lattice.The meet-subsemilattice therefore is a complete lattice, but not a complete sublattice in general.In a closure system of sets, for example, the join of two elements is usually not given by their set union, but by the closure of this union.Thus a closure system, ordered by set inclusion, is a complete lattice, but not necessarily a sublattice.The fixed points of a kernel operator are closed under arbitrary joins and thus form a ⋁ -subsemilattice, also called a kernel system.Again we get the second operation from the first, so that each kernel system also is a complete lattice.This shows that closure systems are not the only subsets yielding order-embedded complete lattices.In fact, the following is well known5 :

Lemma 1 A subset of a complete lattice (L,≤), with the induced order, is a complete lattice if and only if it is the image of an idempotent order-preserving function φ : L → L.
Proof Suppose that F = { (x) | x ∈ L} for some idempotent order-preserving function φ : L → L. We claim that for any subfamily S ⊆ F the element ( ⋀ S) is the infimum of S in F .Clearly ⋀ S ≤ s holds for every s ∈ S .Since φ is order-preserving, we get that For the converse suppose that F ⊆ L is a complete lattice and define a function φ : L → L by (x) ∶= sup F {f ∈ F | f ≤ x} (where sup F denotes the supremum in F ).This function is clearly idempotent and order-preserving, and its image is F .□ Lemma 1 adds a kind of converse to the celebrated Knaster-Tarski theorem [7,9], which states that the set of fixed points of any order-preserving function on a complete lattice is itself a complete lattice: Corollary 2 A subset F ⊆ L of a complete lattice (L,≤), with the induced order, is a complete lattice if and only if F is the set of fixed points of some order-preserving function.
The second part of the proof of Lemma 1 is more informative than required, since the function which was used not only is order-preserving and idempotent, but has an additional property:

Proposition 8
The function which was used in the proof of Lemma 1, is tensive.
Theorem 2 Let (L,≤) be a complete lattice and let F ⊆ L be an order embedded complete lattice.Then there is an upward relation over (L,≤) such that F is the set of fixed points of φ ⊳ .Conversely it holds for every upward relation over (L,≤) that the set of fixed points of φ ⊳ is an order embedded complete lattice.In both cases the fixed points of φ ⊳ are exactly the images of �  ⊳ .

The Next fixed poiNt algorithm
Many years ago the author suggested a simple algorithm [3] for finding all closed sets of a given closure operator ϕ on a (finite, linearly ordered) set G. One starts with the closure A := ϕ(∅) of the empty set and then repeats the procedure shown in Fig. 2, using the output of each application as the input of the next one, until it returns ⊥.The algorithm is extremely useful for browsing and navigating in closure systems.And since it is so simple, many variations and generalizations have been invented, see [5].
It is easy to generalize the algorithm to closure operators on complete lattices, not only powerset lattices.It therefore seems natural to ask if a modification of Next closure can be used for generating all images of any given idempotent, order-preserving, and tensive function.If such a mapping is given as a "black box" only, then unfortunately, the answer is "no".Our pessimism is prompted by the following example: Example 2 Let A ⊆ L be an antichain complete lattice (L,≤), let 0 L be the least and and 1 L be the greatest element of (L,≤), and let f be an element of A. The function is idempotent, order-preserving, and tensive.
In this example it is tedious to determine the fixed points by repeated invocation of φ.Since the number of antichains may be exponential 6 in the size of L, it seems difficult to find an algorithm which determines the fixed point f reasonably fast.Stronger assumptions are needed.What we shall assume is that an upward representation of reasonable size for φ is given. 7o a given upward relation we associate an extensive operator, which then will be iterated to compute a closure.This operator will be set valued.Its base set, named G in Fig. 2, will be (considered as a set of pairs).

𝜑(x) ∶=
Fig. 2 The Next closure algorithm, from [5] with u ≤ S(A) and v ≰ S(A) .We expect that the idea behind the well-known liNclosure algorithm can be carried over to this recursive construction.We conclude the section with an example.Example 3 Let (L,≤) = {0 < 1 < 2 < 3}×{0 < 1 < 2 < 3} be the direct product of two 4-element chains, and let be given as follows: For the linear order of the base set , as required by the Next closure algorithm, we use the order in the table.For a shorter notation, we leave out parentheses.Writing abcd instead of ((a,b),(c,d)), we obtain This relation represents the following order-preserving map φ ⊳ : This table is so small that we can easily read off the final result: φ ⊳ has eight fixed points.They are marked in boldface.

Discussion
Apart from closure and kernel systems, there are many "lattices of sets", i.e., families of sets which form complete lattices, when ordered by set inclusion.We have studied those and, more generally, subsets of arbitrary complete lattices which, endowed with the induced order, are complete lattices themselves.We have shown that each such complete lattice can be described by an "upward" relation, in a way which is very similar to the representation of closure systems by implications.The Next closure algorithm can be tweaked to work with this representation, so that we were able to give an algorithm for generating such lattices.We did not discuss complexity questions in detail, because we expect that there is room for improvements.Many mathematical questions remain open.For example, we did not investigate how to construct an upward representation from a given "non-upward" one.
An important question is if embedded complete lattices have a natural and useful interpretation.The work of Shmuely [8] gives interesting hints.Her u − v-connections generalize Galois connections and seem to be related to what we construct.One might hope that these can be derived from formal contexts with additional, meaningful structure.Our results may help to study more substantial examples.
Funding Open Access funding enabled and organized by Projekt DEAL.

(
⋀ S) ≤ (s) = s for all s ∈ S , which shows that ( ⋀ S) is a lower bound of S .But any lower bound b of S must satisfy b ≤ s for all s ∈ S and therefore b ≤ ⋀ S .If b ∈ F , then b = (b) ≤ ( ⋀ S) , as desired.