Tarski’s least fixed point theorem:
A predicative type theoretic formulation

First and foremost I would like to thank Giovanni Curi. His correspondence was essential for getting my bearings in this area of mathematics. I would also like thank Tom de Jong. He has already established much of predicative and constructive order theory (as well as domain theory) in a univalent setting. Beyond this he has provided me with countless hours of discussion and insight. Without Tom this paper would not exist. Finally, I would like to thank Martín Escardó, Ulrik Buchholtz, Jon Sterling, Jem Lord and many others from the Univalent Agda discord for small and large contributions to the present work.

Fixed point theorems, although of great interest and significance, are notorious for their non-constructive nature (e.g. Brouwer’s Fixed Point Theorem [Shu18]). Certain fixed point theorems do admit intuitionistic proofs (see [Coq95]). Of particular interest is the work [Esc03] which contains a constructive proof a Pataraia’s fixed point theorem: an analog of Tarski’s fixed point theorem for directed complete posets (DCPOs). However, some still view proofs like the ones above as not entirely constructive, due to their impredicative nature. What exactly is impredicativity? To assist explanation we focus our attention to one part of Tarski’s fixed point theorem: every monotone map $f$ on a complete lattice $L$ has a least fixed point. Traditional approaches define the least fixed point to be the infimum over the set of ‘deflationary’ points. Specifically, the least fixed point is

Note that the element we are trying to define is itself deflationary (as it is a fixed point) and thus an element of the very set we take the infimum over. This is an example of an impredicative construction. Alternatively, a predicative construction can be loosely characterized as one which builds an object up from things below. We will continue a critical discussion of predicativity in Section˜2.

Giovanni Curi provides a predicative proof of a variation to Tarski’s least fixed point theorem [Cur15] using constructive set theory (CZF). In Section 2 of his paper Curi recounts a paradigm shifting result in predicative order theory: any complete lattice with small carrier is necessarily trivial (for a type theoretic proof, see [JE21]). For this reason we are forced to work with large carriers if we want to study non-trivial complete lattices in a predicative setting. It is worth explicating that in the framework of CZF the small/large dichotomy is encoded by sets and proper classes. Curi then restricts attention to complete lattices that have a set of generators (along with other strong assumptions) to salvage a least fixed point theorem. The imposed restrictions will allow one to build the least fixed point inductively. Of course, in the presence of impredicative axioms the additional assumptions are satisfied vacuously. Curi concludes by observing his works dependence on the set theoretic framework, even stating that a \saysystem independent derivation is desirable.

Currently, we do not have the facilities to provide a derivation that is completely independent of any particular system. One may speculate that the notion of a predicative topos will provide canonical models of predicative mathematics — much in the same way that elementary topoi provide models of intuitionisitic mathematics. Progress has been made in this direction, but there is no agreed upon notion yet in place (see [Ber12]). Alternatively, a translation of [Cur15] into type theory may be a step towards such a system independent derivation; because type theory is a structural foundation.

There are additional benefits of working in a type theoretic setting. First, inductive constructions are first class citizens; so we don’t have to manually construct the object of interest as in [Cur15]. This allows us to cut away the complications that sideline the main arguments in [Cur15]. For this reason, a type theoretic derivation is arguably cleaner and more intuitive than the set theoretic counter part. Second, the proof can be routinely verified with a proof assistant. In fact, every stated definition and proposition in the present work has been implemented in Agda [Esc+23, Ray23]. An off shoot of this second benefit is that one could, in theory, port the proof into Cubical Agda and compute fixed points. Third, a proof in MLTT* + HITs is valid in any Grothendieck $\infty$-topos [Shu19] (MLTT* = MLTT + FunExt + PropExt, but not including full univalence [Uni13, Rij22]). Finally, it is worth noting that the type theoretic proof is universe polymorphic (see Section˜3) and thus, technically, more general than a proof in a set theoretic framework. We intend to work informally with type theory much like how mathematicians work informally with sets. The ideal audience of this paper is a (univalent) type theorist, but the informal style paired with no appeals to higher types will lend accessibility to any interested party.

In Section˜2 we commence a more careful discussion of predicativity. In Section˜3 we fix the type theoretic framework and some conventions we employ to avoid technicalities. If you are unfamiliar with type theory then you may reasonably skip Section˜3 and still understand the bulk of the paper. Section˜4 develops a bit of order theory. In particular, we define posets, sup lattices and the notion of a basis. Section˜5 describes, what Curi calls, abstract inductive definitions in the language of type theory. Abstract inductive definitions serve to carve out a least subset of the basis that has some nice closure properties. As we will explore in Section˜7 there is a correspondence between deflationary points and a certain class of subsets that possess similar closure properties. The least closed subset will be closely related to the least fixed point via this correspondence. In Curi’s work this least closed subset is constructed manually via set theoretic axioms, while in type theory it manifests itself as a special quotient inductive type (QIT). In Sections˜6, 8 and 9 we impose further restrictions on abstract inductive definitions (viz. local and bounded) and sup lattices (viz. small presented). These restrictions serve to salvage a type theoretic formulation of the least fixed point theorem which is stated in Section˜10. We conclude the paper by investigating a condition on monotone maps that guarantees they have a least fixed point.

Unfortunately, the history behind predicativism is murky and wrought with philosophical motivations. In fact, there are many conflicting notions of predicativity; each with their own supporters and justifications. Classical notions of predicativity where explored by Poincaré, Russell and Weyl [Cro18, Cro22]. More recently Kreisel, Feferman and Sch$\ddot{\text{u}}$tte provided a proof theoretic analysis of predicativity [Cro17, Cro22]. When we say ‘predicative’ here we mean what is often referred to as constructive or generalised predicativity [Cro22, Cro18]. A generalised predicative system goes beyond Fefferman’s predicativity. In particular, generalised predicative systems have infinitary inductive types.

The systems of Constructive Zermelo Fraenkel set theory (CZF) and Martin L$\ddot{\text{o}}$f type theory (MLTT) are examples of such predicative systems. The Powerset axiom is an example of an impredicative principle and as such is not included in CZF. Similarly, the principle of propositional resizing is viewed as impredicative and thus not included in MLTT or any extensions we consider. In the absence of these strong principles we instead rely on inductive constructions. These constructions are argued to be philosophically sound with respect to such predicative systems [Cro22]. When working in extensions of MLTT we may apply similar philosophical arguments to justify the predicative admissibility of higher inductive types (HITs) as well. One may wish to avoid philosophical justifications entirely. One such justification would be a reduction of type theory extended by HITs to a system that is accepted as predicative. Such reductions have yet to be completely spelled out, but do seem possible. For example, if [SA21] could be extended to the schemas in [CHM18] in a constructive and predicative meta theory then we may achieve such a reduction (this observation is due to Ulrik Buchholtz).

Now that we have clarified the meaning of predicativity we want to return to the discussion of the standard formulation Tarski’s least fixed point theorem: every monotone map $f$ on a complete lattice $L$ has a least fixed point. The standard construction (see Section˜1) does not work in systems like CZF or MLTT*. Consider a non-trivial complete lattice $L$ and a monotone map $f:L\to L$, and observe that, since the carrier $L$ is necessarily large, the collection $\{x\in L\ |\ f(x)\leq x\}$ is not provably small and as such the join does not exist. In [Cur15] a predicative formulation of Tarski’s least fixed point theorem is proved. Although, as we shall see, this formulation contains serious concessions. It is very important to clarify: we have only argued that the standard construction is not predicatively viable, but it may be that some other construction is. It seems unlikely that there is a predicative proof of the standard formulation of Tarski’s least fixed point theorem, but ideally we would show that such a proof is impossible. We are actively exploring the prospect of showing that the standard formulation of Tarski’s least fixed point theorem is too strong of a result to have in a predicative theory — a predicative taboo so to speak. One could do this by showing that the standard formulation implies some impredicative axiom or by providing a model where the standard formulation fails. Showing that the standard formulation of Tarski’s least fixed point theorem is a predicative taboo would fully justify the content of this paper and that of [Cur15]. Regardless, we still feel that the failure of the standard construction is sufficient motivation to investigate a predicative version of Tarski’s least fixed point theorem.

One of the main goals of this paper is to maintain accessibility. For this reason, we will work informally with type theory much like a traditional mathematician works with set theory. Thus, in many occasions the technical type theoretic details of a proof will be underspecified. The purpose of this section is to assist in filling in these technical details to the interested parties and as such can be reasonably skipped by non-type theorists.

In traditional treatments of type theory one would find closure properties of universes as well as type formation, introduction and elimination rules stated in the language of natural deduction. We assume familiarity with such formal treatments (for more see [Rij22, Uni13]). Our starting point is Martin Löf Type Theory (MLTT): which has an empty type ($\boldsymbol{0}$), a unit type ($\boldsymbol{1}$), the natural numbers ($\mathbb{N}$), binary coproducts ($+$), dependent pair types ($\sum$), dependent functions types ($\prod$), intensional identity types ($=$) and general inductive types (W). We assume the existence of a base universe $\mathcal{U}_{0}$ as well as the operations successor $(\_)^{+}$ and join $\_\sqcup\_$ which satisfy the expected definitional equalities (see [Jon23, Section 2.1]). We attain an infinite tower of universes $\mathcal{U}_{0},\mathcal{U}_{1},\dots$ but we will often work with arbitrary universes such as $\mathcal{U,V,W,T},\dots$. We do not assume cumulativity of universes (e.g. for any $A:\mathcal{U}_{i}$ we have $A:\mathcal{U}_{i+1}$), but instead we employ a lifting operation: $\text{Lift}_{\mathcal{U,V}}:\mathcal{U}\to\mathcal{U\sqcup V}$ (see [Jon23, Section 2.1]). Universes will be presented à la Russell for convenience but should be understood to have an encoding à la Tarski. In the current state of development univalence is not required, but we do assume propositional and function extensionality as well as propositional truncation. There are some nuances in our base type theory that we should explicate. To maintain predicativity, we do not assume any resizing principles except explicitly in Corollary˜7.4.1. The propositional resizing principle, $\text{Prop-Resizing}_{\mathcal{U},\mathcal{V}}$, states that any proposition in the universe $\mathcal{U}$ is equivalent to a proposition in the universe $\mathcal{V}$ (see [JE21]). Additionally, we assume the existence of a special sort of quotient inductive type (QIT). Such an assumption may at first seem excessive, but investigation of this QIT reveals it is extremely tame comparatively. We will revisit this discussion in Section 4. The notion of equivalence serves as the standard notion of sameness in type theory (for a precise definition see Section 9 of [Rij22]). We say a function $f:A\to B$ is an equivalence when it has a left and right inverse and denote the type of such equivalences as $A\simeq B$. Finally, given universes $\mathcal{U}$ and $\mathcal{V}$, we say that a type $Y:\mathcal{U}$ is $\mathcal{V}$-small if there is a type $X:\mathcal{V}$ such that $X\simeq Y$ and a type $Y:\mathcal{U}$ is locally $\mathcal{V}$-small if $x=y$ is $\mathcal{V}$-small for all $x,y:Y$. In our type theoretic formulation the small/large dichotomy is encoded via those types that are $\mathcal{V}$-small and those that are not (in Section˜3 we fix some universes and the significance of the universe $\mathcal{V}$ will become clear).

We now discuss some conventions that will allow us to avoid technical type theoretic proofs. For example, the notion of transport is often motivated by the principle of indiscernibility of identicals. This principle, often taken for granted by mathematicians, says identified elements can be substituted for each other in expressions. Another type theoretic concept, often taken for granted, is the application of $f:X\to Y$ to an identification $x=x^{\prime}$ which yields an identification $f(x)=f(x^{\prime})$. As we are doing set-level mathematics we will employ such notions tacitly. Propositional truncation is a somewhat technical part of any formal treatment of type theory (see Section 14 of [Rij22]). In particular, given a type $A$ we have a proposition $||A||$ called the propositional truncation of $A$. Given a proposition $Q$ we can define a map $||A||\to Q$ by giving a map $A\to Q$. Propositional truncations can be formulated via universal properties or recursion principles. The observant reader will notice that when we must define a map out of a truncated type, the codomain is almost always a proposition. For that reason we will not explicitly mention that we are applying propositional truncation recursion and we will leave notions that are defined in terms of truncation (like surjections) underspecified. In the same vein we do not specify when we are applying propositional or function extensionality.

We will frequently work with subsets in this paper and as such we need to address how formal we wish to be. In a standard treatment, ignoring universes, a subset of a type $X$ is a term $S$ in the powerset of $X$, $\mathcal{P}(X):\equiv X\to\text{Prop}$. We write $x\in S$ for the underlying type of $S(x)$. Given two subsets $S,S^{\prime}:\mathcal{P}(X)$ we write $S\subseteq S^{\prime}$ for the type of dependent functions $\prod_{x:X}x\in S\to x\in S^{\prime}$. We wish for mathematicians who are unfamiliar with the type theoretic encoding to follow our arguments. For this reason we will avoid using lambda notation when talking about subsets. For example, we define the union, $\bigcup\alpha$, of a family of subsets $\alpha:I\to\mathcal{P}(X)$, as $x\mapsto\exists i:I,x\in\alpha(i)$. When $X$ is a set we can define the singleton $\{a\}$, for some $a:X$, as $x\mapsto a=x$. Finally, since we define suprema (see Section˜3) in terms of families rather than subsets we also need to fix some notation that will unify the two concepts. Given a subset $S:\mathcal{P}(X)$ we define the total space of $S$, $\mathbb{T}(S):\equiv\Sigma_{x:X}x\in S$, with inclusion into $X$ given by $\text{inc}_{X}:\equiv\text{pr}_{1}:\mathbb{T}(S)\to X$. It is worth recalling that using sigma notation in this way is analogous to set builder notation (e.g. $\{x\in X\ |\ x\in S\}$). This will inform many of our translations of Curi’s work into type theory. We will often conflate a subset and its total space to aid in readability, but one should bare in mind the technical difference between them. Finally, since we are working predicatively we will not ignore universe levels. For every universe $\mathcal{U}$, there is a type of propositions $\Omega_{\mathcal{U}}:\equiv\sum_{A:\mathcal{U}}\text{is-prop}(A)$. For any type $X$ we may consider the $\mathcal{U}$-powerset of $X$, $\mathcal{P}_{\mathcal{U}}(X):\equiv X\to\Omega_{\mathcal{U}}$. We call the elements of $\mathcal{P}_{\mathcal{U}}(X)$ the $\mathcal{U}$-valued subsets of $X$. The above discussion extends to $\mathcal{U}$-valued subsets as well.

We commence this section by briefly recounting the notion of a set generated complete lattice as formulated in [Cur15]. A complete lattice $L$ is set-generated if it has a subset $B\subseteq L$ such that, for any $x\in L$,

1. 1.

   $\downarrow^{B}x=\{b\in B\ |\ b\leq x\}$ is a set.

2. 2.

   $x=\bigvee\downarrow^{B}x$.

We now work towards translating this notion into type theory. As we have established, it is well known that in a predicative setting there are no non-trivial examples of small (sufficiently) complete posets (see [JE21]). Thus, when working predicatively it is necessary to define sup lattices, and by extension, posets, in a universe polymorphic manner.

The anti-symmetry assumption guarantees that $P$ is a set (see Lemma 3.2.3 of [Jon23] for a proof following from Hedberg’s Lemma). Given universes $\mathcal{U}$ and $\mathcal{W}$ we denote the type of Posets with carrier in $\mathcal{U}$ and order valued in $\mathcal{W}$ as $\text{Poset}_{\mathcal{U},\mathcal{W}}$.

Of course it is easy enough to extend this to functions between different posets, but we focus our attention to endomaps as our goal is to study fixed points. We now define the notion of sup lattice. We follow the reasonable convention of [Jon23] in phrasing completeness with respect to small families rather than subsets.

Of course, we extend the notion of monotone endomaps to sup lattices. Given universes $\mathcal{U}$, $\mathcal{W}$ and  $\mathcal{V}$ we denote the type of sup lattices with carrier in $\mathcal{U}$, order valued in $\mathcal{W}$ and joins of families indexed in $\mathcal{V}$ as $\text{Sup-Lattice}_{\mathcal{U},\mathcal{W},\mathcal{V}}$, but often times we will simply say such a lattice is a $\mathcal{V}\text{-sup-lattice}$ and omit the other universes. As stated in Section˜3, we use the universe $\mathcal{V}$ as a point of reference for our relative notion of smallness and our language will often reflect this. In many concrete cases $\mathcal{U}\equiv\mathcal{V}^{+}$ and $\mathcal{W}\equiv\mathcal{V}$ (see Example˜4.5). Notice that if $\mathcal{W}\equiv\mathcal{V}$ then anti-symmetry implies that $L$ is locally $\mathcal{V}$-small.

We now define, analogous to the notion of a generating set from [Cur15], what it means for a sup lattice to have a basis. First recall, as stated in Section˜3, given a type $X:\mathcal{U}$ and a subset $U:\mathcal{P}_{\mathcal{T}}(X)$ the total space of $U$ is $\mathbb{T}(U):\equiv\sum_{x:X}x\in U$.

We will typically conflate the two orders $\_\leq\_$ and $\_\leq^{B}\_$ and corresponding total spaces, as well as the joins of either. This is justified by Corollary˜4.8.1 – which essentially says that equivalences preserve joins. Notice, we have a subset $\_\leq^{B}a:\mathcal{P}_{\mathcal{V}}(B)$ such that $\operatorname{\downarrow^{B}}a\equiv\mathbb{T}(\_\leq^{B}a)$. We will often conflate the total space $\operatorname{\downarrow^{B}}a$ and this subset, and similarly for $\operatorname{\downarrow^{B}_{\mathcal{V}}}a$.

We will record the fact that taking the supremum preserves containment of subsets.

In line with earlier observations; it is well-known that large lattices containing all small suprema do not necessarily contain all small infima. One motivation for working with a basis is that the resulting sup lattices are better behaved. In particular we have the following result.

The next theorem and corollary are essential for our development. It says that reindexing families along a surjection (or an equivalence) does not change the supremum.

The notion of an abstract inductive definition considered in [Cur15] generalizes the inductive definitions in [Acz10]. Given a complete lattice $L$ with generating subset $B$ consider a subset $\Phi\subseteq B\times L$. In some sense $\Phi$ can be thought of as inductively defining a subset of $B$. To make this precise in set theoretic language we have to define appropriate closure conditions and then proceed to prove that there is in fact a subset satisfying these conditions. Curi does exactly that in [Cur15]. We briefly recount the desired closure conditions but for obvious reasons do not attempt to construct the desired subset. Consider a subclass $Y\subseteq B$,

1. 1.

   $Y$ is $\text{c}_{L}$-closed if for every subset $U\subseteq Y$ we have that $\operatorname{\downarrow^{B}}\bigvee U\subseteq Y$.

2. 2.

   $Y$ is $\Phi$-closed if for every $(b,a)\in\Phi$ then $\operatorname{\downarrow^{B}}a\subseteq Y\implies b\in Y$.

We denote the least closed class, under the $\text{c}_{L}$ and $\Phi$ closure conditions, as $\mathcal{I}(\Phi)$. In [Cur15], Curi proves that $\mathcal{I}(\Phi)$ always exists in CZF.

We will now translate this notion into type theory. Given a $\mathcal{V}\text{-sup-lattice}$ with a $\mathcal{V}\text{-basis}$ we can encode the notion of abstract inductive definitions à la CZF. Admittedly, the terminology is a bit unfortunate here as inductive definitions/constructions are first class in type theory and thus take on a more general meaning. To ameliorate this we will call them inductive generators or simply generators. We will also depart from Curi’s notation slightly, by using $\phi$ and $\mathcal{I}_{\phi}$, in the interest of differentiating between set theoretic and type theoretic constructions.

This section contains some unavoidable use of type theoretic concepts (see [Uni13, Chapter 6]). Using a generator $\phi$ as a parameter we construct a special higher inductive type (HIT) family. HITs are an active area of study so we will quickly comment on exactly what we must assume (for consistency of HITs see [LS20]). We do not need the full notion of HITs for our purposes. The type we are postulating is a quotient inductive type (QIT) family, where everything is quotiented. This amounts to having a propositional truncation constructor. Of course, details about how to construct QITs in general are beyond the scope of this note. For now we will simply state the definitions of our proposed type and hope that its tameness is apparent. First we provide some shorthand for the two closure properties:

What follows is our first use of Higher Inductive Types.

The truncation constructor restricts the possible codomains to be propositionally valued families indexed by $B$ and possibly $\mathcal{I}_{\phi}(b)$. When defining functions out of $\mathcal{I}_{\phi}$ we may use the induction principle or pattern match on the point constructors c-cl and $\phi$-cl. We need to show one crucial property of $\mathcal{I}_{\phi}$ before moving on: $\mathcal{I}_{\phi}$ is initial among all subsets of $B$ closed under containment and $\phi$.

In [Cur15] it is desirable to indicate when an abstract inductive definition yields a set, at least locally. We say an abstract inductive inductive definition $\Phi$ is local if for every $a\in L$ the class

is a set. When an abstract inductive definition $\Phi$ is local we may define a monotone operator as follows $\Gamma_{\Phi}(a)\equiv\bigvee\{b\in B\ |\ \exists a^{\prime}\in L,(b,a^{\prime})\in\Phi\land a^{\prime}\leq a\}$.

We now translate these notions into type theory. For the remainder of this note we work in the context of a $\mathcal{V}$-sup-lattice $L$ with a basis $\beta:B\to L$.

As is convention we will conflate the total space $S_{\phi,a}$ and its corresponding subset.

We can now state what it means for a generator to be local.

We now define, for any local generator, the following map. Here we are taking advantage of the existence of small joins.

Not surprisingly this map is monotonic. It is worth noting that the formalized proof must account for the implicit use of Proposition˜4.8 in the Definition of $\Gamma_{\phi}$ which adds a layer of difficulty. This is one of the reasons why Proposition˜4.8 is so essential.

Perhaps more surprisingly, every monotone map provides a canonical local generator such that the induced map equals the orignal map.

In [Cur15] it is shown that there is a correspondence between $\text{c}_{L}$ and $\phi$-closed subsets and elements $a\in L$ such that $\Gamma_{\Phi}(a)\leq a$. Further it is shown that if $\mathcal{I}(\Phi)$ is a set then $\Gamma_{\Phi}$ has a least fixed point. We now translate these results into type theory.

Assuming a generator $\phi$ is local we can show that the $\mathcal{V}$-small subsets that are both c-closed and $\phi$-closed correspond to deflationary points with respect to $\Gamma_{\phi}$. For the remainder of this section we work in a context with a local generator $\phi$.

One can observe that $\text{is-c-}\phi\text{-closed}(P)$ and $\text{is-deflationary}(a)$ are propositions as they are made up of propositions. We now prove the proposed correspondence, which reveals itself in type theory as an equivalence.

We can now show, under certain smallness assumption on the QIT family $\mathcal{I}_{\phi}$, that $\Gamma_{\phi}$ has a least fixed point. First we collect some important constructions that can be built under the assumptions of the following theorem.

If for all $b:B$ the type $b\in\mathcal{I}_{\phi}$ is $\mathcal{V}$-small, then we can construct the following:

1. 1.

   For each $b:B$, a type $\mathcal{I}_{\phi}^{\mathcal{V}}(b):\mathcal{V}$ that is a proposition and an equivalence $\mathcal{I}_{\phi}^{\mathcal{V}}(b)\simeq b\in\mathcal{I}_{\phi}$.

2. 2.

   A subset $\mathcal{I}_{\phi}^{\mathcal{V}}:\mathcal{P}_{\mathcal{V}}(B)$ that witnesses to the fact that $\mathbb{T}(\mathcal{I}_{\phi})$ is $\mathcal{V}$-small; that is, $\mathbb{T}(\mathcal{I}_{\phi}^{\mathcal{V}})\simeq\mathbb{T}(\mathcal{I}_{\phi})$.

3. 3.

   Intiality of $\mathcal{I}_{\phi}^{\mathcal{V}}$, which follows from that of $\mathcal{I}_{\phi}$ via the equivalence.

In light of the previous result we gain an impredicative version of the least fixed point theorem which follows from propositional resizing.

In [Cur15] a further restriction on abstract inductive definitions is imposed. Given an abstract inductive definition $\Phi$ we say it is bounded if

1. 1.

   $\{b\in B\ |\ (b,a)\in\Phi\}$ is a set for every $a\in L$

2. 2.

   There is a set $\alpha$ such that, whenever $(b,a)\in\Phi$ there is $x\in\alpha$ such that the set $\downarrow^{B}a$ is the image of $x$.

Notice if $\Phi$ is a set, rather than a class, then $\Phi$ is automatically bounded.

We will now explore the translation of these notions into type theory. For the remainder of this section we work in the context of a $\mathcal{V}$-generated sup lattice $L$ with basis $\beta:B\to L$.

We now show that if a generator is bounded then it is local (see Definition˜6.3).

Recall, Proposition˜6.6 states that every monotone map determines canonical a local generator: $(b,a)\in\phi:\equiv b\leq^{B}f(a)$. Unfortunately, this generator is not particularly well behaved. To illustrate this we give the following example.

A similar situations occurs for other simple functions like the identity function. In light of this peculiarity we may extend the boundedness restriction to monotone endomaps as follows.

In [Cur15] a further restriction of set-generated complete lattices is explored. We depart from Curi here and instead use an equivalent formulation of set presentation due to [Acz10]. A complete lattice $L$ is set-presented if there is a subset $R\subseteq B\times\mathcal{P}(B)$ such that for any $b\in B$ and $X\subseteq B$