Lax Extensions of Conical $\mathsf{I}$-Semifilter Monads

Abstract

For a quantale $\mathsf{I},$ the unit interval endowed with a continuous triangular norm, we introduce the canonical, op-canonical and Kleisli extensions of the conical $\mathsf{I}$-semifilter monad to $\mathsf{I}\text{-}\mathsf{Rel}.$ It is proved that the op-canonical extension coincides with the Kleisli extension.

1. Introduction and Preliminaries

Monoidal topology [1] provides a unification of settings to describe some important mathematical structures as $(\mathbb{T},\mathsf{Q},\widehat{\mathbb{T}})$-algebras (lax algebras for short) in which $\mathsf{Q}$ is a quantale and $\mathbb{T}$ is a monad on $\mathsf{Set}$ with a lax extension $\widehat{\mathbb{T}}$ to the category $\mathsf{Q}\text{-}\mathsf{Rel}$ of sets and $\mathsf{Q}$-relations.

Examples include:

• Metric spaces can be described as $(\mathbb{I},{\mathsf{P}}_{+},\overline{\mathbb{I}})$-algebras [2].
• Topological spaces can be characterized as $(\beta ,\mathsf{2},\overline{\beta })$-algebras [3,4].
• Approach spaces [5] can be viewed as $(\beta ,{\mathsf{P}}_{+},\overline{\beta })$-algebras [6].

Here, 2 denotes the two-element quantale, ${\mathsf{P}}_{+}=({[0,\infty ]}^{\mathrm{op}},+,0)$ is the Lawvere quantale, $\mathbb{I}$ is the identity monad with the identity extension, and $\beta$ is the ultrafilter monad with the Barr extensions $\overline{\beta }$ to $\mathsf{2}\text{-}\mathsf{Rel}$ ($\mathsf{Rel}$ for short) and $\mathsf{I}\text{-}\mathsf{Rel}$, respectively.

To study many-valued topologies within the monoidal topology framework, it is of importance to determine the counterpart of the filter monad in the many-valued context and investigate its lax extensions. Extensive studies have been conducted to develop many-valued filter monads and their lax extensions, including the $\mathfrak{B}$-valued filter monad [7], the ⊤-filter monad with its Kleisli extension to $\mathsf{Rel}$ [8], and the saturated prefilter monad with its Kleisli extension to $\mathsf{Rel}$ [9]. The lax algebras for the latter two are both CNS spaces, which are a kind of many-value topological spaces introduced in [10].

Lax extensions offer rich topological structures. For example, as demonstrated in [11], there are two lax extensions of the filter monad $\mathbb{F}$ to $\mathsf{Q}$-$\mathsf{Rel}:$ the canonical one $\widehat{\mathbb{F}}$ and the op-canonical one $\check{\mathbb{F}}$. When $\mathsf{Q}=2,$ the lax algebras with respect to the canonical extension are closure spaces, while those associated with the op-canonical extension are topological spaces. When $\mathsf{Q}={\mathsf{P}}_{+},$ the lax algebras with respect to the canonical extension are closeness spaces, while those for the op-canonical extension are approach spaces.

The approach adopted in this paper is motivated by an observation that the filter monad is the discrete restriction of two composite monads on $\mathsf{Ord}:$ up-set-ideal monad $\mathsf{IdeUp}$ and the down-set-filter monad $\mathsf{FilDn}.$ Furthermore, the canonical (op-canonical) lax extension of the filter monad can be induced from the lax extension of $\mathsf{IdeUp}$ ($\mathsf{FilDn}$) to $\mathsf{Dist}$.

In Section 2, we introduce the composite monads ${\mathsf{CP}}^{†}$ and ${\mathsf{C}}^{†}\mathsf{P}$ and show that the discrete restriction of them are the conical $\mathsf{I}$-semifilter monad [12], where $\mathsf{C}$ is the monad of $\mathsf{I}$-distributors generated by a forward Cauchy net that plays the role of the ordered-ideal monad $\mathsf{Ide}.$ The canonical and op-canonical extensions of the conical $\mathsf{I}$-semifilter monad to $\mathsf{I}\text{-}\mathsf{Rel}$ are also presented in this section. Section 3 focuses on the Kleisli extension of the conical $\mathsf{I}$-semifilter monad to $\mathsf{I}\text{-}\mathsf{Rel}$. The lax algebras for the Kleisli extension to $\mathsf{I}\text{-}\mathsf{Rel}$ are same to those for the Kleisli extension to $\mathsf{Rel}.$

In the remainder of this section, we introduce the many-valued context in which we work, including the quantale $\mathsf{I},$ $\mathsf{I}$-relations and $\mathsf{I}$-categories.

1.1. Monads

A monad on a category $\mathcal{A}$ is a triple $\mathbb{T}=(T,m,e),$ where $T:\mathcal{A}\to \mathcal{A}$ is an endfunctor and $m:T^2\to T, e:{\mathrm{id}}_{\mathcal{A}}\to T$ are natural transformations such that

$$m·eT=m·Te={\mathrm{id}}_{\mathcal{A}}\mathrm{and}m·mT=m·Tm.$$

Sometimes, we simply write T for $(T,m,e)$ if no confusion arises.

Given two monads $\mathbb{T}=(T,m,e)$ and $\mathbb{S}=(S,n,d),$ a morphism $\sigma :\mathbb{T}\to \mathbb{S}$ of monads is a natural transformation $\sigma :T\to S$ such that

$$d=\sigma ·e\mathrm{and}\sigma ·m=n·(\sigma \ast \sigma ),$$

where ∗ is the horizontal composition of natural transformations.

We let $(T,m,e)$ be a monad on $\mathcal{A}.$ A submonad of $(T,m,e)$ is a monad $(S,n,d)$ with a monad morphism $i:(S,n,d)\to (T,m,e)$ such that every component $i_X$ is monic. In this case, $i:S\to T$ is called the inclusion transformation. To keep notations simple, we write $(S,m,e)$ for submonad $(S,n,d).$

Given monad $\mathbb{T}=(T,m,e)$ on $\mathcal{A},$ an Eilenberg–Moore algebra for $\mathbb{T}$ ($\mathbb{T}$-algebra for short) is a pair $(X,a)$ consisting of an $\mathcal{A}$-object X and an $\mathcal{A}$-morphism $a:TX\to X$ subject to the following:

$$a·e_X=1_X\mathrm{and}a·m_X=a·Ta.$$

$(TX,m_X)$ is obviously a $\mathbb{T}$-algebra, which is called the free $\mathbb{T}$-algebra on $X.$

A $\mathbb{T}$-homomorphism $f:(X,a)\to (X^{\prime },a^{\prime })$ of $\mathbb{T}$-algebras is an $\mathcal{A}$-morphism $f:X\to X^{\prime }$ such that $a^{\prime }·Tf=f·a.$ $\mathbb{T}$-algebras and $\mathbb{T}$-homomorphisms assemble into a category ${\mathcal{A}}^{\mathbb{T}}$ which is called the Eilenberg–Moore category of $\mathbb{T}.$

Given a monad morphism $\sigma :\mathbb{S}\to \mathbb{T},$ there exists a functor $K_{\sigma }:{\mathsf{Set}}^{\mathbb{T}}\to {\mathsf{Set}}^{\mathbb{S}}$ induced by $\sigma ,$ which is identical on morphisms, sends the $\mathbb{T}$-algebra $(X,a)$ to the $\mathbb{S}$-algebra $(X,a·{\sigma }_X),$ and makes the diagram commute, where $G^{\mathbb{T}},G^{\mathbb{S}}$ are forgetful functors.

For more information on monads, we refer to [13,14]. Monads are useful for encoding general algebraic structures. The monograph by Plotkin [15] offers a comprehensive exploration of the algebraic aspects of database theory. Therefore, further research on the application of monads in the theory of databases is warranted.

Power-Enriched Monads

The powerset monad $\mathbb{P}$ is given by the covariant powerset functor $P:\mathsf{Set}\to \mathsf{Set}$ and two natural transformations:

$$\begin{array}{cc}\hfill {\{-\}}_X:& X\to PX,x\mapsto \left\{x\right\},\hfill \\ \hfill {\bigcup }_X:& P^{2}X\to PX,\mathfrak{A}\mapsto \bigcup \mathfrak{A}.\hfill \end{array}$$

The Eilenberg–Moore category of the powerset monad is isomorphic to the category $\mathsf{Sup}$ of complete lattices and sup-maps.

We consider monad $\mathbb{T}$ on $\mathsf{Set}$ equipped with monad morphism $\sigma :\mathbb{P}\to \mathbb{T}.$ By the functor $K_{\sigma }:{\mathsf{Set}}^{\mathbb{T}}\to {\mathsf{Set}}^{\mathbb{P}}$, every $\mathbb{T}$-algebra $(X,a)$ carries an order making X a complete lattice, and every morphism of $\mathbb{T}$-algebras is a sup-map. In particular, endowed with the order induced by the free $\mathbb{T}$-algebra structure on $X,$ every set $TX$ becomes a complete lattice.

If, for any sets $X,Y,$ the map

$${(-)}^{\mathbb{T}}:\mathsf{Set}(X,TY)\to \mathsf{Set}(TX,TY),f\mapsto m_Y·Tf$$

is monotone, where the hom-sets $\mathsf{Set}(-,TY)$ are ordered pointwise, then we refer to $(\mathbb{T},\sigma )$ as a power-enriched monad. Morphism $\sigma :(\mathbb{T},{\sigma }_1)\to (\mathbb{S},{\sigma }_1^{\prime })$ of power-enriched monads is monad morphism $\sigma :\mathbb{T}\to \mathbb{S}$ such that ${\sigma }_1^{\prime }=\sigma ·{\sigma }_1.$

1.2. $\mathsf{I}$-Settings

1.2.1. Continuous Triangular Norms

A triangular norm [16] (t-norm for short) is a binary operation & on the unit interval I subject to the following:

• & is associated;
• & is commutative;
• $a\&(-)$ is monotone for any $a\in I;$
• $a\&1=a$ for any $a\in I.$

A t-norm & is called continuous if map $\&:I^2\to I$ is continuous with respect to the standard topologies. We denote by $\mathsf{I}=(I,\&,1)$ the unit interval I endowed with a continuous t-norm $\&.$

Example 1.

There are three basic continuous t-norms.

(1)

The Gödel t-norm $a\&b=a\wedge b;$

(2)

The product t-norm $a\&b=a\times b;$

(3)

The Łukasiewicz t-norm $a\&b=max\{0,a+b-1\}.$

For each $a\in I,$ since $a\&(-):I\to I$ preserves arbitrary joints, then there exists a map $a\to (-):I\to I$ which is right adjoint to $a\&(-)$ and is determined by

$$a\&b\le c\iff b\le a\to c.$$

A continuous t-norm is said to satisfy condition (S); if it satisfies that, for each $a\in (0,1],$ map $a\to (-)$ is continuous on the interval $[0,a).$

The following proposition includes some basic properties of continuous t-norms.

Proposition 1.

For any $a,b,c\in I$ and ${\left\{a_i\right\}}_i\subset I,$

(1)

$a\&(a\to b)\le b;$

(2)

$1\to a=a;$

(3)

$a\to b=1\iff a\le b;$

(4)

$(a\&b)\to c=a\to (b\to c);$

(5)

$a\to \left({\bigwedge }_i{a}_i\right)={\bigwedge }_i(a\to a_i);$

(6)

$\left({\bigvee }_i{a}_i\right)\to a={\bigwedge }_i(a_i\to a).$

The reasons why we work with the particular quantale $\mathsf{I}$ include:

• Some important many-valued topological structures are considered as topologies valued in $\mathsf{I}=(I,\&,1)$ with & being certain t-norms. For example, fuzzy topologies can be seen as topologies valued in $(I,\wedge ,1),$ and since $(I,\times ,1)$ is isomorphic to the Lawvere quantale ${\mathsf{P}}_{+},$ approach spaces can be considered as topological spaces valued in $(I,\times ,1).$
• Many results about topologies valued in $\mathsf{Q}$ rely on the structure of $\mathsf{Q};$ due to the celebrated ordinal sum decomposition theorem [16,17], the structure of $\mathsf{I}$ is clear.

1.2.2. $\mathsf{I}$-Relations

An $\mathsf{I}$-relation r: X ⇸ Y is a map r: X × Y → I. The composition of r: X ⇸ Y, s: Y ⇸ Z is an $\mathsf{I}$-relation (s · r): X ⇸ Z given by

$$(s·r)(x,z)={\displaystyle \underset{y\in Y}{\bigvee }}r(x,y)\&s(y,z).$$

Sets and $\mathsf{I}$-relations assemble into a category

$$\mathsf{I}\text{-}\mathsf{Rel}.$$

Since the composition of $\mathsf{I}$-relations preserves arbitrary joins in each variable, for each r: X ⇸ Y and set Z, there are two maps (−) ⟜ r: $\mathsf{I}\text{-}\mathsf{Rel}$(X,Z) → $\mathsf{I}\text{-}\mathsf{Rel}$(Y,Z) and r $⊸$ (−): $\mathsf{I}\text{-}\mathsf{Rel}$(Z,Y) → $\mathsf{I}\text{-}\mathsf{Rel}$(Z,X) determined by

$$\begin{array}{cc}\hfill r·t\le s& \iff t\le s⟜r;\hfill \\ \hfill t^{\prime }·r\le s& \iff t^{\prime }\le r⊸s\hfill \end{array}$$

for any t ∈ $\mathsf{I}\text{-}\mathsf{Rel}$(Y,Z) and t′ ∈ $\mathsf{I}\text{-}\mathsf{Rel}$(Z,X).

For each r: X ⇸ Y, there is an $\mathsf{I}$-relation r^op: Y ⇸ X given by r^op(y,x) = r(x,y). For each map f: X → Y, graph f_∘: X ⇸ Y of f is given by

$$f_{\circ }(x,y)=\left\{\begin{array}{cc}1,\hfill & f\left(x\right)=y;\hfill \\ 0,\hfill & f\left(x\right)\ne y.\hfill \end{array}\right.$$

And the cograph f^∘ of f is given by f^∘ = (f_∘)^op. There are two functors:

$${(-)}_{\circ }:\mathsf{Set}\to \mathsf{I}\text{-}\mathsf{Rel}\mathrm{and}{(-)}^{\circ }:\mathsf{Set}\to \mathsf{I}\text{-}{\mathsf{Rel}}^{\mathrm{op}}.$$

1.2.3. Lax Extensions to $\mathsf{I}\text{-}\mathsf{Rel}$

We let $(T,m,e)$ be a monad on $\mathsf{Set}.$ A lax extension [18] of $(T,m,e)$ to $\mathsf{I}\text{-}\mathsf{Rel}$ is a triple $\widehat{\mathbb{T}}=(\widehat{T},m,e),$ where $\widehat{T}$ is given by a family of maps

$${\widehat{T}}_{X,Y}:\mathsf{I}\text{-}\mathsf{Rel}(X,Y)\to \mathsf{I}\text{-}\mathsf{Rel}(TX,TY)$$

subject to the following conditions:

(1)

Every ${\widehat{T}}_{X,Y}$ is monotone;

(2)

$\widehat{T}r·\widehat{T}s\le \widehat{T}(r·s);$

(3)

${\left(Tf\right)}_{\circ }\le \widehat{T}\left(f_{\circ }\right)$ and ${\left(Tf\right)}^{\circ }\le \widehat{T}\left(f^{\circ }\right);$

(4)

$s·e_X^{\circ }\le e_Y^{\circ }·\widehat{T}s;$

(5)

$\widehat{T}\widehat{T}s·m_X^{\circ }\le m_Y^{\circ }·\widehat{T}s$

for any sets $X,Y,Z,$ $\mathsf{I}$-relations s: X ⇸ Y, r: Y ⇸ Z and every map f: X → Y.

Morphism $\sigma :(\widehat{S},n,d)\to (\widehat{T},m,e)$ of lax extensions is a monad morphism $\sigma :(S,n,d)\to (T,m,e)$ such that $\widehat{S}r\le {\left({\sigma }_Y\right)}^{\circ }·\widehat{T}r·{\left({\sigma }_X\right)}_{\circ }$ for any $\mathsf{I}$-relation r: X ⇸ Y.

We let $\sigma :\mathbb{S}\to \mathbb{T}$ be a monad morphism and $\widehat{\mathbb{T}}$ a lax extension of $\mathbb{T}$ to $\mathsf{I}\text{-}\mathsf{Rel}.$ There is a lax extension of $\mathbb{S}$ given by

$$\widehat{S}r={\left({\sigma }_Y\right)}^{\circ }·\widehat{T}r·{\left({\sigma }_X\right)}_{\circ }$$

for any $\mathsf{I}$-relation r: X ⇸ Y. This lax extension $\widehat{\mathbb{S}}$ is called the initial extension of $\mathbb{S}$ induced by σ.

1.2.4. $\mathsf{I}$-Categories

An $\mathsf{I}$-category [2,19] is a pair $(X,r)$ consisting of a set X and a transitive and reflexive $\mathsf{I}$-relation $r,$ that is,

$$r(x,y)\&r(y,z)\le r(x,z)\mathrm{and}r(x,x)=1$$

for all $x,y,z\in X.$ For convenience, we simply use X to denote an $\mathsf{I}$-category $(X,r)$ and use $X(-,-)$ to denote $r(-,-).$

For every $\mathsf{I}$-category $X,$ the $\mathsf{I}$-relation $X^{\mathrm{op}}(x,y)=X(y,x)$ also gives an $\mathsf{I}$-category, which is called the dual of $X.$

Example 2.

(1) The singleton $\{\ast \}$ set endowed with ${\left(\mathrm{id}\right)}_{\circ }$ is obviously an $\mathsf{I}$-category.

(2)

The set $I^X$ can be made an $\mathsf{I}$-category via

$${\mathrm{sub}}_X(\mu ,\nu )=\underset{x\in X}{\bigwedge }\mu \left(x\right)\to \nu \left(x\right).$$

An $\mathsf{I}$-functor $f:X\to Y$ is a map $f:X\to Y$ between $\mathsf{I}$-categories such that

$$X(x,y)\le Y\left(f\right(x),f(y\left)\right)$$

for all $x,y\in X.$ If the converse of the above inequality also holds, we refer to this $\mathsf{I}$-functor as fully faithful. $\mathsf{I}$-functors $f:X\to Y,g:Y\to X$ are called an adjunction $f⊣g$ if

$$Y\left(f\right(x),y)=X(x,g(y\left)\right)$$

for any $x\in X,y\in Y.$ In this case, we say f is left adjoint to $g.$

Example 3.

Given an $\mathsf{I}$-relation r: X ⇸ Y, there is an adjunction r_∨ ⊣ r_∧, in which r_∧,r_∨ are given by

$$\begin{array}{cc}\hfill & r_{\wedge }:I^X\to I^Y,\mu \mapsto \underset{x\in X}{\bigwedge }r(x,-)\to \mu \left(x\right);\hfill \\ \hfill & r_{\vee }:I^Y\to I^X,\nu \mapsto \underset{y\in Y}{\bigvee }r(-,y)\&\nu \left(y\right).\hfill \end{array}$$

$\mathsf{I}$-categories and $\mathsf{I}$-functors assemble into a category

$$\mathsf{I}\text{-}\mathsf{Cat}.$$

The forgetful functor $o:\mathsf{I}\text{-}\mathsf{Cat}\to \mathsf{Set}$ admits a left adjoint:

$$d:\mathsf{Set}\to \mathsf{I}\text{-}\mathsf{Cat},X\mapsto (X,1_X^{\circ }).$$

A locally small category is ordered if every hom-set carries an order such that the composition maps are monotone. A functor $F:\mathcal{A}\to \mathcal{B}$ between ordered categories is called a 2-functor if every $F_{A,B}:\mathcal{A}(A,B)\to \mathcal{B}(FA,FB)$ is monotone. A monad on an ordered category is called a 2-monad if the endfunctor is a 2-functor.

The underlying order of an $\mathsf{I}$-category X is given by

$$x{\le }_{X}y\iff X(x,y)=1.$$

An $\mathsf{I}$-category X is called separated if its underlying order is a partial order. $\mathsf{I}\text{-}\mathsf{Cat}$ is an ordered category with $\mathsf{I}\text{-}\mathsf{Cat}(X,Y)$ carrying the pointwise order.

Given an $\mathsf{I}$-category X and $p\in I,x\in X,$ the tensor of $(p,x)$ is an element $p\otimes x$ of X such that $X(p\otimes x,-)=p\to X(x,-);$ the cotensor of $(p,x)$ is an element $p\rightarrowtail x$ of X such that $X(-,p\rightarrowtail x)=p\to X(-,x).$

An $\mathsf{I}$-category X is called tensored (cotensored) if it fulfills that the tensor $p\otimes x$ (cotensor $p\rightarrowtail x$) exists for all $p\in I,x\in X.$

Proposition 2

([20]). The following statements are equivalent:

(1)

X is tensored, $(X,{\le }_X)$ is complete, and

$$X(\underset{i}{\bigvee }{x}_i,y)=\underset{i}{\bigwedge }X(x_i,y)$$

for all ${\left\{x_i\right\}}_i\subset X,y\in X;$

(2)

X is cotensored, $(X,{\le }_X)$ is complete, and

$$X(x,\underset{i}{\bigwedge }{y}_i)=\underset{i}{\bigwedge }X(x,y_i)$$

for all ${\left\{y_i\right\}}_i\subset X,x\in X.$

An $\mathsf{I}$-category is called complete if it satisfies the equivalent conditions stated above. For a complete $\mathsf{I}$-category, we have $p\otimes (-)⊣p\rightarrowtail (-).$

Example 4.

The $\mathsf{I}$-category $(I^X,{\mathrm{sub}}_X)$ is complete and separated. For any $p\in I,\mu \in I^X,$ the cotensor of $(p,\mu )$ is given by $p\to \mu .$

The following proposition is useful in ensuring the existence of adjunctions.

Proposition 3 ([20]).

We let $f:X\to Y,g:Y\to X$ be $\mathsf{I}$-functors between $\mathsf{I}$-categories. Then, $f⊣g$ is an adjunction if and only if $f⊣g:(Y,{\le }_Y)\to (X,{\le }_X)$ is an adjunction.

2. The Lax Extensions from the Laxly Extended Monads on $\mathsf{I}\text{-}\mathsf{Cat}$

2.1. $\mathsf{I}$-Distributors

Given two $\mathsf{I}$-categories, X and $Y,$ an $\mathsf{I}$-distributor [2] r: X ⇴ Y is an $\mathsf{I}$-relation such that

$$r·X\le r\mathrm{and}Y·r\le r.$$

If an $\mathsf{I}$-distributor r: X ⇴ Y is dummy in one variable, that is X = {∗} or Y = {∗}, then we simply write r(x) for r(x,∗) or r(∗,x). $\mathsf{I}$-categories and $\mathsf{I}$-distributors give rise to an ordered category

$$\mathsf{I}\text{-}\mathsf{Dist}.$$

The forgetful functor o: $\mathsf{I}\text{-}\mathsf{Dist}$ → $\mathsf{I}\text{-}\mathsf{Rel}$ admits a left adjoint:

$$\mathsf{d}:\mathsf{I}\text{-}\mathsf{Rel}\to \mathsf{I}\text{-}\mathsf{Dist},\mathsf{d}X=(X,1_X^{\circ }),r\mapsto r.$$

There are two 2-functors ${(-)}_{*}:\mathsf{I}\text{-}\mathsf{Cat}\to \mathsf{I}\text{-}{\mathsf{Dist}}^{\mathrm{co}}$ and ${(-)}^{*}:\mathsf{I}\text{-}\mathsf{Cat}\to \mathsf{I}\text{-}{\mathsf{Dist}}^{\mathrm{op}}$ defined on objects and morphisms by

$$\begin{array}{c}{\left(X\right)}_{*}=X,(f:X\to Y)\mapsto (f_{*}=(Y·f_{\circ }):X{\displaystyle ⇴}Y);\hfill \\ {\left(X\right)}^{*}=X,(f:X\to Y)\mapsto (f^{*}=(f^{\circ }·Y):Y{\displaystyle ⇴}X).\hfill \end{array}$$

We denote the set of $\mathsf{I}$-distributors from an $\mathsf{I}$-category X to $\{\ast \}$ by $\mathsf{P}X.$ Then, the set $\mathsf{P}X$ can be made an $\mathsf{I}$-category via

$$\mathsf{P}X(\mu ,\nu )=\nu ⟜\mu ={\mathrm{sub}}_X(\mu ,\nu ).$$

Furthermore, $\mathsf{P}$ can be made a 2-functor from $\mathsf{I}\text{-}{\mathsf{Dist}}^{\mathrm{op}}$ to $\mathsf{I}\text{-}\mathsf{Cat}$ via

$$(r:X{\displaystyle ⇴}Y)\mapsto \left(\mathsf{P}\right(r):\mu \mapsto \mu ·r).$$

It is routine to check that (−)^* is left adjoint to $\mathsf{P}$. The induced 2-monad ($\mathsf{P},\mathsf{s},\mathsf{y}$) on $\mathsf{I}\text{-}\mathsf{Cat}$ is called the presheaf monad.

Similarly, taking the $\mathsf{I}$-distributors of type $\{\ast \} {\displaystyle ⇴}$ X also gives rise to a 2-functor ${\mathsf{P}}^{†}:\mathsf{I}\text{-}{\mathsf{Dist}}^{\mathrm{op}}\to \mathsf{I}\text{-}\mathsf{Cat}$:

$$X\mapsto {\mathsf{P}}^{†}X,(r:X{\displaystyle ⇴}Y)\mapsto \left({\mathsf{P}}^{†}\right(r):\mu \mapsto r·\mu ),$$

in which

$${\mathsf{P}}^{†}X(\mu ,\nu )=\nu ⊸\mu ={\mathrm{sub}}_X^{\mathrm{op}}(\mu ,\nu )$$

for any μ, ν ϵ ${\mathsf{P}}^{†}X$. The functor $(—{)}_{*}$ is left adjoint to ${\mathsf{P}}^{†}$. The induced 2-monad (${\mathsf{P}}^{†},{\mathsf{s}}^{†},{\mathsf{y}}^{†}$) on $\mathsf{I}\text{-}\mathsf{Cat}$ is called the copresheaf monad.

The following lemmas present some basic properties of $\mathsf{I}$-distributors.

Lemma 1

(Yoneda Lemma). For any $\nu \in {\mathsf{P}}^{†}X,\mu \in \mathsf{P}X,$ we have

$${\left({\mathsf{y}}_X\right)}_{*}(-,\mu )=\mu and{\left({{\mathsf{y}}^{†}}_X\right)}^{*}(\nu ,-)=\nu .$$

Lemma 2.

We let $f:X\to Y,g:Z\to Y$ be $\mathsf{I}$-functors. For any $\mu \in \mathsf{P}Z,\nu \in {\mathsf{P}}^{†}X,\varphi \in {\mathsf{P}}^{†}\mathsf{P}X,$ and $\psi \in {\mathsf{PP}}^{†}Z,$ we have the following statements:

(1)

${\left(\mathsf{P}g\right)}^{*}·{\left(\mathsf{P}f\right)}_{*}(-,\mu )={\mathsf{y}}_{\mathsf{P}X}(\mu ·g^{*}·f_{*});$

(2)

${\left({\mathsf{P}}^{†}g\right)}^{*}·{\left({\mathsf{P}}^{†}f\right)}_{*}(\nu ,-)={\mathsf{y}}_{{\mathsf{P}}^{†}Z}^{†}(g^{*}·f_{*}·\nu );$

(3)

${\left(\mathsf{P}g\right)}^{*}·{\left(\mathsf{P}f\right)}_{*}·\varphi =\varphi (-·g^{*}·f_{*});$

(4)

$\psi ·{\left({\mathsf{P}}^{†}g\right)}^{*}·{\left({\mathsf{P}}^{†}f\right)}_{*}=\psi (g^{*}·f_{*}·-).$

2.2. Composite Monads on $\mathsf{I}\text{-}\mathsf{Cat}$

We let $\mathbb{T}=(T,m,e)$ and $\mathbb{S}=(S,n,d)$ be monads. A distributive law of $\mathbb{T}$ over $\mathbb{S}$ is a natural transformation $\sigma :TS\to ST$ subject to some conditions. A composite monad of $\mathbb{T}$ and $\mathbb{S}$ is a monad $(ST,\mathfrak{m},d\ast e)$ such that $Se:S\to ST,dT:T\to ST$ are monad morphisms and $\mathfrak{m}$ satisfies that $\mathfrak{m}·\left(SedT\right)={\mathrm{id}}_{ST}.$ A distributive law σ yields a composite monad

$$(ST,(n\ast m)·S\sigma T,d\ast e).$$

This correspondence is bijective. Details can be found in [21].

A saturated class of weights is a submonad $\mathsf{A}$ of the presheaf monad $\mathsf{P}.$ It is easy to check that it also offers a submonad ${\mathsf{A}}^{†}$ of ${\mathsf{P}}^{†}$ by ${\mathsf{A}}^{†}X={\left(\mathsf{A}{X}^{\mathrm{op}}\right)}^{\mathrm{op}}$ for any $X.$

A distributive law $\sigma :{\mathsf{P}}^{†}\mathsf{A}\to {\mathsf{A}}^{†}\mathsf{P}$ of ${\mathsf{P}}^{†}$ over $\mathsf{A}$ also offers a distributive law of $\mathsf{P}$ over ${\mathsf{A}}^{†}$ whose components are given by

$$\begin{array}{cc}\hfill {\sigma }_X^{\prime }:{\mathsf{PA}}^{†}X={\left({\mathsf{P}}^{†}\mathsf{A}{X}^{\mathrm{op}}\right)}^{\mathrm{op}}\stackrel{{\sigma }_{X^{\mathrm{op}}}}{\to }& {\left({\mathsf{AP}}^{†}{X}^{\mathrm{op}}\right)}^{\mathrm{op}}={\mathsf{A}}^{†}\mathsf{P}X.\end{array}$$

One example of distributive laws is that the copresheaf monad distributes over the presheaf monad.

Proposition 4

([22]). There is a distributive law of ${\mathsf{P}}^{†}$ over $\mathsf{P},$ which offers the double presheaf 2-monad ${\mathsf{PP}}^{†}$ on $\mathsf{I}\text{-}\mathsf{Cat}.$

We let X be an $\mathsf{I}$-category. A forward Cauchy net [23] on X is a net ${\left\{x_i\right\}}_{i\in D}$ such that

$$\underset{i\in D}{\bigvee }\underset{k\ge j\ge i}{\bigwedge }X(x_j,x_k)=1.$$

A forward Cauchy net generates an $\mathsf{I}$-distributor $\mu :X{\displaystyle ⇴}$ {∗}:

$$\mu ={\displaystyle \underset{i\in D}{\bigvee }}{\displaystyle \underset{j\ge i}{\bigwedge }}X(-,x_j).$$

Example 5.

A directed set D of $(X,{\le }_X)$ is a forward Cauchy net ${\left\{x_i\right\}}_{i\in D}$ on $X.$ The $\mathsf{I}$-distributor generated by D is

$$\underset{d\in D}{\bigvee }X(-,d).$$

We denote by $\mathsf{C}X$ the set of all $\mathsf{I}$-distributors $\mu :X{\displaystyle ⇴}$ {∗} generated by forward Cauchy nets. The proof of that $\mathsf{C}$ is a saturated class of weights can be found in [24]. The following lemma offers a characterization of $\mathsf{C}X$ when X is complete and separated.

Lemma 3

(Proposition 4.8 in [25]). We let X be a complete separated $\mathsf{I}$-category. For every $\varphi \in \mathsf{C}X,$ we have that $D=\{x\in X\mid \varphi (x)=1\}$ is a directed set on $(X,{\le }_X)$ and

$$\varphi =\underset{d\in D}{\bigvee }X(-,d).$$

The existence of a distributive law of ${\mathsf{P}}^{†}$ over $\mathsf{C}$ depends on the structure of quantale $\mathsf{I}.$

Proposition 5

(Theorem 6.4 in [25]). There is a distributive law of ${\mathsf{P}}^{†}$ over $\mathsf{C}$ if and only if the continuous t-norm satisfies the condition (S).

In the remainder of this paper, we always assume that the continuous t-norm & satisfies the condition (S).

2.3. The Lax Extensions of Composite Monads to $\mathsf{I}\text{-}\mathsf{Dist}$

We let $(T,m,e)$ be a 2-monad on $\mathsf{I}\text{-}\mathsf{Cat}$. A lax extension of $(T,m,e)$ to $\mathsf{I}\text{-}\mathsf{Dist}$ is a family of maps

$${\widehat{T}}_{X,Y}:\mathsf{I}\text{-}\mathsf{Dist}(X,Y)\to \mathsf{I}\text{-}\mathsf{Dist}(TX,TY)$$

subject to the following conditions:

(1)

Every ${\widehat{T}}_{X,Y}$ is monotone;

(2)

$\widehat{T}r·\widehat{T}s\le \widehat{T}(r·s);$

(3)

${\left(Tf\right)}_{*}\le \widehat{T}\left(f_{*}\right)$ and ${\left(Tf\right)}^{*}\le \widehat{T}{f}^{*};$

(4)

$s·e_X^{*}\le e_Y^{*}·\widehat{T}s;$

(5)

$\widehat{T}\widehat{T}s·m_X^{*}\le m_Y^{*}·\widehat{T}s$

for any $\mathsf{I}$-categories $X,Y,Z,$ distributors $s:X{\displaystyle ⇴}$ Y, r: Y ${\displaystyle ⇴}$ Z and every $\mathsf{I}$-functor f: X → Y.

Theorem 1

(Theorem 8.5 in [26]). We let $\mathbb{T}$ be a 2-monad on $\mathsf{I}\text{-}\mathsf{Cat}.$ Then,

$$\widehat{T}r={\left(T\overleftarrow{r}\right)}^{*}·{\left(T{\mathsf{y}}_X\right)}_{*}:TX{\displaystyle ⇸}TY$$

defines a lax extension of $\mathbb{T}$ to $\mathsf{I}\text{-}\mathsf{Dist}$, where $\overleftarrow{r}$: Y → $\mathsf{P}$X, y ↦ r(−,y).

We let $\mathsf{A}$ be a saturated class of weights and assume that there is a distributive law $\sigma :{\mathsf{P}}^{†}\mathsf{A}\to {\mathsf{AP}}^{†}.$ Then, by Theorem 1, there are lax extensions of the monad ${\mathsf{AP}}^{†}$ and ${\mathsf{A}}^{†}\mathsf{P}$ given by

$$\begin{array}{cc}\hfill \overline{{\mathsf{AP}}^{†}}r& ={\left({\mathsf{AP}}^{†}\overleftarrow{r}\right)}^{*}·{\left({\mathsf{AP}}^{†}{\mathsf{y}}_X\right)}_{*};\hfill \\ \hfill \overline{{\mathsf{A}}^{†}\mathsf{P}}r& ={\left({\mathsf{A}}^{†}\mathsf{P}\overleftarrow{r}\right)}^{*}·{\left({\mathsf{A}}^{†}{\mathsf{Py}}_X\right)}_{*}.\hfill \end{array}$$

In [27], Lai and Tholen introduced a functor Γ which maps monads $(T,m,e)$ on $\mathsf{I}\text{-}\mathsf{Cat}$ with a lax extension $\widehat{T}$ to $\mathsf{I}\text{-}\mathsf{Dist}$ to monads on $\mathsf{Set}$ with a lax extension to $\mathsf{I}\text{-}\mathsf{Rel}:$

$$\begin{array}{cc}\hfill & \mathrm{\Gamma }(T,m,e)=(oTd,omd·oTϵTd,oed),\hfill \\ \hfill & \mathrm{\Gamma }\left(\widehat{T}\right)r=\mathsf{o}\widehat{T}\mathsf{d}\left(r\right),\hfill \end{array}$$

in which ϵ is the counit of the adjunction $d⊣o.$

It is routine to check that $\mathrm{\Gamma }\left({\mathsf{A}}^{†}\mathsf{P}\right)=\mathrm{\Gamma }\left({\mathsf{AP}}^{†}\right).$ We denote this monad by $({\mathsf{U}}_{\mathsf{A}},\mathsf{n},\mathsf{d}).$

For the lax extensions, using Lemma 2, we can compute as follows: for any $\mathsf{I}$-relation $r:X⇸$ Y, $\varphi$, ∈ ${\mathsf{AP}}^{†}X$, ${\varphi }^{\prime }\in {\mathsf{AP}}^{†}Y$, $\psi \in {\mathsf{A}}^{†}\mathsf{P}X$, and ${\psi }^{\prime }\in {\mathsf{A}}^{†}\mathsf{P}X$,

$$\begin{array}{cc}\hfill \overline{{\mathsf{AP}}^{†}}r(\varphi ,{\varphi }^{\prime })& =\left({\left({\mathsf{AP}}^{†}\overleftarrow{r}\right)}^{*}·{\left({\mathsf{AP}}^{†}{\mathsf{y}}_X\right)}_{*}\right)(\varphi ,{\varphi }^{\prime })\hfill \\ \hfill & ={\mathsf{PP}}^{†}X(\varphi ,{\varphi }^{\prime }·{\left({\mathsf{P}}^{†}\overleftarrow{r}\right)}^{*}·{\left({\mathsf{P}}^{†}{\mathsf{y}}_X\right)}_{*})\hfill \\ \hfill & ={\mathsf{PP}}^{†}X\left(\varphi ,{\varphi }^{\prime }({\overleftarrow{r}}^{*}·{\mathsf{y}}_{X_{*}}·-)\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \overline{{\mathsf{A}}^{†}\mathsf{P}}r(\psi ,{\psi }^{\prime })& =\left({\left({\mathsf{A}}^{†}{\mathsf{Py}}_X\right)}^{*}·{\left({\mathsf{A}}^{†}\mathsf{P}\overleftarrow{r}\right)}_{*}\right)(\psi ,{\psi }^{\prime })\hfill \\ \hfill & ={\mathsf{P}}^{†}\mathsf{P}Y({\left(\mathsf{P}\overleftarrow{r}\right)}^{*}·{\left({\mathsf{Py}}_X\right)}_{*}·\psi ,{\psi }^{\prime })\hfill \\ \hfill & ={\mathsf{P}}^{†}\mathsf{P}Y(\psi (-·{\overleftarrow{r}}^{*}·{\mathsf{y}}_{X_{*}}),{\psi }^{\prime }).\hfill \end{array}$$

Thus, we obtain the following result.

Proposition 6.

We let ${\mathsf{AP}}^{†}$ be a composite monad. There are two lax extensions of the monad$({\mathsf{U}}_{\mathsf{A}},\mathsf{n},\mathsf{d})$:

$$\begin{array}{cc}\widehat{{\mathsf{U}}_{\mathsf{A}}}r(\varphi ,\psi )=\underset{\mu \in I^X}{\bigwedge }\varphi \left(\mu \right)\to \psi \left({\left(r^{\mathrm{op}}\right)}_{\vee }\left(\mu \right)\right),\hfill & \hfill \left(\mathrm{canonical}\right)\\ {\displaystyle \check{{\mathsf{U}}_{\mathsf{A}}}}r(\varphi ,\psi )=\underset{\nu \in I^Y}{\bigwedge }\psi \left(\nu \right)\to \varphi \left(r_{\vee }\left(\nu \right)\right),\hfill & \hfill \left(\mathrm{op}\text{-}\mathrm{canonical}\right)\end{array}$$

where $r:X⇸Y$ is an $\mathsf{I}$-relation, $\varphi \in {\mathsf{U}}_{\mathsf{A}}X,\psi \in {\mathsf{U}}_{\mathsf{A}}Y.$

2.4. The Conical $\mathsf{I}$-Semifilter Monad

A conical $\mathsf{I}$-semifilter [12] on set X is a function $\varphi :I^X\to I$ subject to the following:

(F1)

$\varphi \left(1_X\right)=1;$

(F2)

$\varphi (\mu \wedge \nu )=\varphi \left(\mu \right)\wedge \varphi \left(\nu \right);$

(F3)

${\mathrm{sub}}_X(\mu ,\nu )\le \varphi \left(\mu \right)\to \varphi \left(\nu \right);$

(F4)

$\varphi ={\bigvee }_{\varphi \left(\xi \right)=1}{\mathrm{sub}}_X(\xi ,-).$

Proposition 7.

The elements of ${\mathsf{CP}}^{†}dX$ are exactly the conical $\mathsf{I}$-semifilters.

Proof. 

Given a conical $\mathsf{I}$-semifilter ϕ on $X,$ it follows from (F2) that $\{\mu \mid \varphi (\mu )=1\}$ is a directed set of ${\mathsf{P}}^{†}dX;$ hence, by (F4), we have $\varphi \in {\mathsf{CP}}^{†}dX.$

We let $\varphi \in {\mathsf{CP}}^{†}dX.$ Since ${\mathsf{P}}^{†}dX$ is separated and complete, by Lemma 3, it holds that

$$\varphi =\underset{\varphi \left(\nu \right)=1}{\bigvee }{\mathsf{P}}^{†}dX(-,\nu )=\underset{\varphi \left(\nu \right)=1}{\bigvee }{\mathrm{sub}}_X(\nu ,-).$$

Hence, (F1), (F3) and (F4) are obvious. For (F2),

$$\varphi ({\mu }_1\wedge {\mu }_2)=\underset{\varphi \left(\nu \right)=1}{\bigvee }\left({\mathsf{P}}^{†}dX({\mu }_1,\nu )\wedge {\mathsf{P}}^{†}dX({\mu }_2,\nu )\right)=\varphi \left({\mu }_1\right)\wedge \varphi \left({\mu }_2\right),$$

the last equality holds because $\{\nu \mid \varphi (\nu )=1\}$ is directed. □

For every set $X,$$o{(\mathsf{y}\ast {\mathsf{y}}^{†})}_{dX}$ maps $x\in X$ to ${\mathsf{P}}^{†}dX(-,{\mathsf{y}}_{dX}\left(x\right))=(-)\left(x\right);$${(o(\mathsf{s}\ast {\mathsf{s}}^{†})d·o\mathsf{C}\sigma {\mathsf{P}}^{†}d·o{\mathsf{CP}}^{†}ϵ{\mathsf{CP}}^{†}d)}_X$ maps $\mathrm{\Phi }\in {{\mathsf{U}}_{\mathsf{C}}}^{2}X$ to the conical $\mathsf{I}$-semifilter

$$\varphi :{\mathsf{P}}^{†}dX\to I,\mu \mapsto \mathrm{\Phi }\left({\mu }^{♯}\right),$$

where ${\mu }^{♯}$ belongs to ${\mathsf{P}}^{†}do{\mathsf{CP}}^{†}dX$ and maps every $\psi \in do{\mathsf{CP}}^{†}dX$ to $\psi \left(\mu \right).$ Therefore, the monad $({\mathsf{U}}_{\mathsf{C}},\mathsf{n},\mathsf{d})$ is exactly the conical $\mathsf{I}$-semifilter monad in [12]. We adopt the notation from [12] and denote $({\mathsf{U}}_{\mathsf{C}},\mathsf{n},\mathsf{d})$ by $(\mathsf{CSF},\mathsf{n},\mathsf{d}).$

Corollary 1.

There are two lax extensions of the conical $\mathsf{I}$-semifilter monad $(\mathsf{CSF},\mathsf{n},\mathsf{d})$:

$$\begin{array}{cc}\widehat{\mathsf{CSF}}r(\varphi ,\psi )=\underset{\mu \in I^X}{\bigwedge }\varphi \left(\mu \right)\to \psi \left({\left(r^{\mathrm{op}}\right)}_{\vee }\left(\mu \right)\right),\hfill & \hfill \left(\mathrm{canonical}\right)\\ \check{\mathsf{CSF}}r(\varphi ,\psi )=\underset{\nu \in I^Y}{\bigwedge }\psi \left(\nu \right)\to \varphi \left(r_{\vee }\left(\nu \right)\right),\hfill & \hfill \left(\mathrm{op}\text{-}\mathrm{canonical}\right)\end{array}$$

where $r:X⇸$ Y is an $\mathsf{I}$-relation, $\varphi \in \mathsf{C}\mathsf{S}\mathsf{F}X,\psi \in \mathsf{C}\mathsf{S}\mathsf{F}Y.$

Remark 1.

Here, we prove that the continuous t-norm satisfies the condition (S) is a sufficient condition for conical $\mathsf{I}$-semifilters to give rise to a monad. In fact, it is also a necessary condition; see [12].

3. The Kleisli Extensions of $({\mathsf{U}}_{\mathsf{A}},\mathsf{n},\mathsf{d})$

3.1. The $\mathsf{I}$-Powerset Monad

For each set $X,$ we let $P_{\mathsf{I}}X=I^X.$ Then, $P_{\mathsf{I}}$ can be made a functor from $\mathsf{I}\text{-}{\mathsf{Rel}}^{\mathrm{op}}$ to $\mathsf{Set}$ by letting

$$P_{\mathsf{I}}\left(r\right)\left(\mu \right)=r_{\vee }\left(\mu \right)=\underset{y\in Y}{\bigvee }\mu \left(y\right)\&r(-,y)$$

for each $\mathsf{I}$-relation $r:X⇸$ Y and μ ∈ I^Y. It is routine to check that (—)° is left adjoint to $P_{\mathsf{I}}$. The induced monad is called the $\mathsf{I}$-powerset monad and is denoted by ${\mathbb{P}}_{\mathsf{I}}=(P_{\mathsf{I}},\mathsf{m},\mathsf{e})$. We spell it out here: for any maps f: X → Y and $\mathit{\mu }\in P_{\mathsf{I}}$,

$$\begin{array}{cc}\hfill P_{\mathsf{I}}\left(f\right)\left(\mu \right)& :y\mapsto \underset{f\left(x\right)=y}{\bigvee }\mu \left(x\right),\hfill \\ \hfill {\mathsf{e}}_X& :x\mapsto 1_x,\hfill \\ \hfill {\mathsf{m}}_X& :\varphi \mapsto \underset{\mu \in P_{\mathsf{I}}X}{\bigvee }\varphi \left(\mu \right)\&\mu ,\hfill \end{array}$$

where 1_A is defined as 1_A(x)={ $\begin{array}{cc}1,\hfill & x\in A,\hfill \\ 0,\hfill & x\notin A,\hfill \end{array}$ and 1_x denotes 1_{x}.

It is easy to check that the $\mathsf{I}$-powerset monad is power-enriched by

$${\theta }_X:PX\to P_{\mathsf{I}}X,A\mapsto 1_A.$$

It also holds that ${\mathbb{P}}_{\mathsf{I}}=\mathrm{\Gamma }(\mathsf{P},\mathsf{s},\mathsf{y})=\mathrm{\Gamma }({\mathsf{P}}^{†},{\mathsf{s}}^{†},{\mathsf{y}}^{†}).$

3.2. $\mathsf{I}$-Power-Enriched Monads

An $\mathsf{I}$-power-enriched monad is a pair $(\mathbb{T},\sigma )$ composed of a monad $(T,m,e)$ on $\mathsf{Set}$ and a monad morphism $\sigma :{\mathbb{P}}_{\mathsf{I}}\to \mathbb{T}$ such that $(\mathbb{T},\sigma ·\theta )$ is a power-enriched monad. A morphism $\sigma :(\mathbb{T},{\sigma }_1)\to (\mathbb{S},{\sigma }_2)$ of $\mathsf{I}$-power-enriched monads is a monad morphism $\sigma :\mathbb{T}\to \mathbb{S}$ such that ${\sigma }_2=\sigma ·{\sigma }_1.$

We let ${\mathsf{AP}}^{†}$ be a composite monad. Since there is a monad morphism ${\mathsf{yP}}^{†}:{\mathsf{P}}^{†}\to {\mathsf{AP}}^{†},$ by applying the functor $\mathrm{\Gamma },$ we obtain the following Proposition.

Proposition 8.

The monad $({\mathsf{U}}_{\mathsf{A}},\mathsf{n},\mathsf{d})$ is $\mathsf{I}$-power-enriched by κ whose components are given by

$${\kappa }_X:P_{\mathsf{I}}X\to {\mathsf{U}}_{\mathsf{A}}X,\mu \mapsto {\mathrm{sub}}_X(\mu ,-).$$

An $\mathsf{I}$-action in $\mathsf{Sup}$ is a complete lattice X endowed with a map $-\otimes -:I\times X\to X$ subject to the following: for any $p,q\in I$ and $x\in X$

(1)

$p\otimes -$ and $-\otimes x$ are sup-maps;

(2)

$(p\&q)\otimes x=p\otimes (q\otimes x)$ and $1\otimes x=x.$

A morphism of $\mathsf{I}$-actions is a sup-map $f:X\to Y$ such that $p{\otimes }_{Y}f\left(x\right)=f\left(p{\otimes }_{X}x\right)$ for any $p\in I$ and $x\in X.$ $\mathsf{I}$-actions in $\mathsf{Sup}$ and their morphisms assemble into a category ${\mathsf{Sup}}^{\mathsf{I}}.$

It is shown in [28] that ${\mathsf{Sup}}^{\mathsf{I}}$ is isomorphic to the Eilenberg–Moore category of the $\mathsf{I}$-powerset monad and there exists a functor $\mathrm{\Lambda }:{\mathsf{Set}}^{{\mathbb{P}}_{\mathsf{I}}}\to \mathsf{I}\text{-}\mathsf{Cat}.$

Explicitly, we let $(X,a)$ be a ${\mathbb{P}}_{\mathsf{I}}$-algebra; by functor $K_{\theta }:{\mathsf{Set}}^{{\mathbb{P}}_{\mathsf{I}}}\to {\mathsf{Set}}^{\mathbb{P}},$ X can be made a complete lattice. The $\mathsf{I}$-action on X in $\mathsf{Sup}$ is given by

$$-\otimes -:I\times X\to X,(p,x)\mapsto a(p\&1_x).$$

Conversely, an $\mathsf{I}$-action $(X,-\otimes -)$ yields a ${\mathbb{P}}_{\mathsf{I}}$-algebra structure as follows:

$$a:P_{\mathsf{I}}X\to X,\mu \mapsto \underset{x}{\bigvee }\mu \left(x\right)\otimes x.$$

The functor Λ maps a ${\mathbb{P}}_{\mathsf{I}}$-algebra $(X,a)$ to

$$\mathrm{\Lambda }(X,a)(x,y)=a^{⊣}\left(y\right)\left(x\right),$$

where $a⊣a^{⊣}:(X,{\le }_X)\to (P_{\mathsf{I}}X,{\le }_{P_{\mathsf{I}}X})$ is an adjunction. Furthermore, we have the following proposition.

Proposition 9.

Every $\mathsf{I}$-category $\mathrm{\Lambda }(X,a)$ is complete.

Proof. 

For every $p\in I,$ since $p\otimes -$ and a are sup-maps, we have the following adjunctions:

$$X{\displaystyle \underset{\underset{p\otimes -}{\leftarrow }}{\overset{\stackrel{p\rightarrowtail -}{\to }}{\top }}}X{\displaystyle \underset{\underset{a}{\leftarrow }}{\overset{\stackrel{a^{⊣}}{\to }}{\top }}}{P}_{\mathsf{I}}X$$

To show X is cotensored by ↣, we can follow these steps:

$$\begin{array}{cc}\hfill \mu \le p\to a^{⊣}\left(x\right)& \iff p\&\mu \le a^{⊣}\left(x\right)\hfill \\ \hfill & \iff a(p\&\mu )\le x\hfill \\ \hfill & \iff \underset{t}{\bigvee }(p\&\mu \left(t\right))\otimes t\le x\hfill \\ \hfill & \iff p\otimes \left(\underset{t}{\bigvee }\mu \left(t\right)\otimes t\right)\le x\hfill \\ \hfill & \iff p\otimes a\left(\mu \right)\le x\hfill \\ \hfill & \iff a\left(\mu \right)\le p\rightarrowtail x\hfill \\ \hfill & \iff \mu \le a^{⊣}(p\rightarrowtail x).\hfill \end{array}$$

□

Thus, the tensor of $\mathrm{\Lambda }(X,a)$ is given by its $\mathsf{I}$-action, the cotensor is given by the right adjoint of its $\mathsf{I}$-action. That is the reason why we use the same notations.

Example 6.

For a composite monad ${\mathsf{AP}}^{†},$ since $({\mathsf{U}}_{\mathsf{A}}X,{\mathsf{n}}_X·{\kappa }_{{\mathsf{U}}_{\mathsf{A}}X})=K_{\kappa }({\mathsf{U}}_{\mathsf{A}}X,{\mathsf{n}}_X)$ is a ${\mathbb{P}}_{\mathsf{I}}$-algebra, ${\mathsf{U}}_{\mathsf{A}}X$ can be made a complete $\mathsf{I}$-category via

$${\mathsf{U}}_{\mathsf{A}}(\varphi ,\psi )={({\mathsf{n}}_X·{\kappa }_{{\mathsf{U}}_{\mathsf{A}}X})}^{⊣}\left(\psi \right)\left(\varphi \right)={\mathrm{sub}}_{I^X}(\psi ,\varphi )=\underset{\mu \in I^X}{\bigwedge }\psi \left(\mu \right)\to \varphi \left(\mu \right).$$

The tensor of $(p,\varphi )$ in ${\mathsf{U}}_{\mathsf{A}}X$ is given by

$$({\mathsf{n}}_X·{\kappa }_{{\mathsf{U}}_{\mathsf{A}}X})(p\&1_{\varphi })=\underset{\psi \in {\mathsf{U}}_{\mathsf{A}}X}{\bigwedge }\left(p\&1_{\varphi }\left(\psi \right)\to \psi \right)=p\to \varphi .$$

3.3. Kleisli Extensions

Given an $\mathsf{I}$-power-enriched category $(\mathbb{T},\sigma ),$ for any $\mathsf{I}$-relations $r:X\to Y$, the composite ${\mathbb{P}}_{\mathsf{I}}$-homomorphism

$$\begin{array}{ccc}\hfill (TY,m_Y)\stackrel{T_{({\sigma }_X·r^{♭})}}{\to }& (T^{2}X,m_{TX})\stackrel{m_X}{\to }& (TX,m_X)\hfill \end{array}$$

offers an $\mathsf{I}$-functor $r^{\sigma }:TY\to TX,$ where $r^{♭}:Y\to P_{\mathsf{I}}X,y\mapsto r(-,y).$

According to Section 4.5 in [18], there is a lax extension $\widehat{\mathbb{T}}$ of $\mathbb{T}$ to $\mathsf{I}\text{-}\mathsf{Rel}$ named the Kleisli extension, which is given by

$$\widehat{T}r(\varphi ,\psi )=TX(\varphi ,r^{\sigma }\left(\psi \right))$$

for any $\varphi \in TX,\psi \in TY$ and every $\mathsf{I}$-relation $r:X⇸$ Y.

Proposition 10.

For a composite monad ${\mathsf{AP}}^{†}$ the Kleisli extension of $({\mathsf{U}}_{\mathsf{A}},\mathsf{n},\mathsf{e})$ is given by

$$\overline{{\mathsf{U}}_{\mathsf{A}}}r(\varphi ,\psi )={\mathsf{U}}_{\mathsf{A}}X(\varphi ,r^{\kappa }\left(\psi \right))={\displaystyle \underset{\mu \in I^X}{\bigwedge }}\psi \left(r_{\wedge }\left(\mu \right)\right)\to \varphi \left(\mu \right),$$

where $r:X⇸$ Y is an $\mathsf{I}$-relation, $\varphi \in {\mathsf{U}}_{\mathsf{A}}X,\psi \in {\mathsf{U}}_{\mathsf{A}}Y.$

Theorem 2.

For the monad ${\mathsf{U}}_{\mathsf{P}},$ the op-canonical extension to $\mathsf{I}\text{-}\mathsf{Rel}$ coincides with the Kleisli extension to $\mathsf{I}\text{-}\mathsf{Rel}.$

Proof. 

For any $\mathsf{I}$-relation $r:X⇸$ Y and $\varphi \in {\mathsf{U}}_{\mathsf{P}}X$, by Lemma 2, the $\mathsf{I}$-distributor

$$\ast {\displaystyle \stackrel{\varphi }{⇴}}\mathsf{P}dX{\displaystyle \stackrel{{\left({\mathsf{Py}}_{dX}\right)}_{*}}{⇴}}{\mathsf{P}}^{\mathsf{2}}dX{\displaystyle \stackrel{{\left(\mathsf{P}\overleftarrow{\mathsf{d}r}\right)}^{*}}{⇴}}\mathsf{P}dY$$

is given by ϕ(− · ${\left(\overleftarrow{r}\right)}^{*}$ · ${\left({\mathsf{y}}_{\mathsf{d}\mathsf{X}}\right)}_{*}$) = ϕ($r_{\vee }$(−)). Thus, mapping ϕ to ϕ($r_{\vee }$(−)) is an $\mathsf{I}$-functor f: ${\mathsf{U}}_{\mathsf{P}}\mathsf{X}\to {\mathsf{U}}_{\mathsf{P}}\mathsf{Y}$.

To show the op-canonical extension to $\mathsf{I}\text{-}\mathsf{Rel}$ coincides with the Kleisli extension to $\mathsf{I}\text{-}\mathsf{Rel},$ by Proposition 3, it suffices to show that $f⊣r^{\kappa }:({\mathsf{U}}_{\mathsf{P}}Y,{\le }_{{\mathsf{U}}_{\mathsf{P}}Y})\to ({\mathsf{U}}_{\mathsf{P}}X,{\le }_{{\mathsf{U}}_{\mathsf{P}}X})$ is an adjunction. For any $\chi \in {\mathsf{U}}_{\mathsf{P}}X,\psi \in {\mathsf{U}}_{\mathsf{P}}Y,$ since $r_{\vee }⊣r_{\wedge }$ we have

$$(r^{\kappa }·f)\left(\chi \right)=\chi ·r_{\vee }·r_{\wedge }{\ge }_{{\mathsf{U}}_{\mathsf{P}}X}\chi \mathrm{and}(f·r^{\kappa })\left(\psi \right)=\psi ·r_{\wedge }·r_{\vee }{\le }_{{\mathsf{U}}_{\mathsf{P}}Y}\psi .$$

This completes the proof. □

Since

$$\check{\mathsf{CSF}}r(\varphi ,\psi )=\check{{\mathsf{U}}_{\mathsf{P}}}r\left(i_X\left(\varphi \right),i_Y\left(\psi \right)\right)\mathrm{and}\overline{\mathsf{CSF}}r(\varphi ,\psi )=\overline{{\mathsf{U}}_{\mathsf{P}}}r\left(i_X\left(\varphi \right),i_Y\left(\psi \right)\right)$$

for any $\varphi \in \mathsf{CSF}X,\psi \in \mathsf{CSF}Y,r:X⇸$ Y, where $i:\mathsf{CSF}\to {\mathsf{U}}_{\mathsf{P}}$ is the inclusion transformation, we have the following corollary.

Corollary 2.

For the conical $\mathsf{I}$-semifilter monad, the op-canonical extension to $\mathsf{I}\text{-}\mathsf{Rel}$ coincides with the Kleisli extension to $\mathsf{I}\text{-}\mathsf{Rel}$.

Proposition 11.

We let $\lambda :(\mathbb{S},\sigma )\to (\mathbb{T},{\sigma }^{\prime })$ be a morphism of $\mathsf{I}$-power-enriched monads. Then, λ is a morphism of the Kleisli extensions to $\mathsf{I}\text{-}\mathsf{Rel}$. Furthermore, every component ${\lambda }_X:SX\to TX$ is fully faithful if and only if the initial extension of $\mathbb{S}$ induced by λ is the Kleisli extension of $\mathbb{S}$.

Proof. 

We denote $\mathbb{T}=(T,m,e) \mathrm{and} \mathbb{S}=(S,n,d)$. By the commutative diagram ${\lambda }_X:(SX,n_X)\to (TX,m_X·{\lambda }_{TX})\mathrm{is} \mathrm{an} \mathbb{S}\text{-}\mathrm{homomorphism}; \mathrm{hence}, \mathrm{it} \mathrm{is} \mathrm{an} \mathsf{I}\text{-}\mathrm{functor}:$

$$\widehat{S}r(\alpha ,\beta )=SX(\alpha ,r^{\sigma }\left(\beta \right))\le TX\left({\lambda }_X\left(\alpha \right),{\lambda }_X\left(r^{\sigma }\left(\beta \right)\right)\right).$$

By the commutative diagram we have

$$TX\left({\lambda }_X\left(\alpha \right),{\lambda }_X\left(r^{\sigma }\left(\beta \right)\right)\right)=TX\left({\lambda }_X\left(\alpha \right),r^{{\sigma }^{\prime }}\left({\lambda }_Y\left(\beta \right)\right)\right)=\widehat{T}r({\lambda }_X\left(\alpha \right),{\lambda }_Y\left(\beta \right)).$$

This completes the proof. □

An element of I^X is called bounded if $\bigwedge \mu$ > 0. A conical $\mathsf{I}$-semifilter ϕ is called bounded if ϕ(μ) < 1 for any unbounded μ. Conical bounded $\mathsf{I}$-semiflters also give rise to a monad ($\mathsf{ConBSF},\mathsf{n},\mathsf{d}$), and there is a monad morphism η: $\mathsf{CSF}\to \mathsf{ConBSF}$

$${\eta }_X:\mathsf{CSF}X\to \mathsf{ConBSF}X,\varphi \mapsto {\displaystyle \underset{\begin{array}{c}\varphi \left(\mu \right)=1\\ \bigwedge \mu >0\end{array}}{\bigvee }}{\mathrm{sub}}_X(\mu ,-);$$

see [12] for details.

Example 7.

(1)

The Kleisli extension of the conical $\mathsf{I}$-semifilter monad to $\mathsf{I}\text{-}\mathsf{Rel}$ coincides with the initial extension induced by the inclusion transformation $i:\mathsf{CSF}\to {\mathsf{U}}_{\mathsf{P}}.$

(2)

The conical bounded $\mathsf{I}$-semifilter monad is $\mathsf{I}$-power-enriched by $\eta ·\kappa ,$ and $\eta :(\mathsf{CSF},\kappa )\to (\mathsf{ConBSF},\eta ·\kappa )$ is a morphism of $\mathsf{I}$-power-enriched monads. Since κ is not fully faithful, the Kleisli extension $\overline{\mathsf{CSF}}$ does not coincide with the initial extension induced by $\kappa .$

3.4. Lax Algebras

Given a lax extension $\widehat{\mathbb{T}}$ of $\mathbb{T}$ to $\mathsf{I}\text{-}\mathsf{Rel},$ a $(\mathbb{T},\mathsf{I},\widehat{\mathbb{T}})$-algebra (lax algebra for short) is a pair $(X,a:TX⇸$ X) so that

$${\left(1_X\right)}_{\circ }\le a·{\left(e_X\right)}_{\circ }\mathrm{and}a·\widehat{T}a\le a·{\left(m_X\right)}_{\circ }.$$

A morphism f: (X,a) → (Y,b) of lax algebras is a map f: X → Y subject to

$$f_{\circ }·a\le b·{\left(Tf\right)}_{\circ }.$$

Lax algebras and morphisms of lax algebras form a category denoted by

$$(\mathbb{T},\mathsf{I},\widehat{\mathbb{T}})\text{-}\mathsf{Cat}.$$

When the involved lax extension is clear, we simply write $(\mathbb{T},\mathsf{I})\text{-}\mathsf{Cat}$.

Lax extensions $\widehat{\mathbb{T}}$ of monad $\mathbb{T}$ to $\mathsf{Rel}$ and lax algebras of $(\mathbb{T},\mathsf{2},\widehat{\mathbb{T}})$ are defined in a manner similar to those of lax extensions to $\mathsf{I}\text{-}\mathsf{Rel}$ and lax algebras of $(\mathbb{T},\mathsf{I},\widehat{\mathbb{T}}).$ Given an $\mathsf{I}$-power-enriched monad $(\mathbb{T},\sigma ),$ it can be extended to $\mathsf{Rel}$ via

$$\alpha \left(\overline{T}r\right)\psi \iff \varphi {\le }_{TX}{r}^{\sigma }\left(\psi \right),$$

which is called the Kleisli extension of $\mathbb{T}$ to $\mathsf{Rel},$ where r is a $\mathsf{2}$-relation and $r^{\sigma }$ is defined by treating r as the $\mathsf{I}$-relation $r(x,y)=\left\{\begin{array}{cc}1,\hfill & xry,\hfill \\ 0\hfill & otherwise.\hfill \end{array}\right.$

The following proposition affirms that, at the level of lax algebras, there is no distinction between the Kleisli extension to $\mathsf{I}\text{-}\mathsf{Rel}$ and the Kleisli extension to $\mathsf{Rel}.$

Proposition 12

(Proposition 6.1 in [18]). We let $(X,\sigma )$ be an $\mathsf{I}$-power-enriched category. Then, there is an isomorphism

$$(\mathbb{T},\mathsf{I})\text{-}\mathsf{Cat}\cong (\mathbb{T},\mathsf{2})\text{-}\mathsf{Cat},$$

in which the lax extensions are the Kleisli extensions.

In [9], it is proven that

$$(\mathsf{CSF},\mathsf{2},\overline{\mathsf{CSF}})\text{-}\mathsf{Cat}\cong \mathsf{CNS},$$

where $\mathsf{CNS}$ is the category of CNS spaces. Therefore, we have the following corollary.

Corollary 3.

There is an isomorphism:

$$(\mathsf{CSF},\mathsf{I},\overline{\mathsf{CSF}})\text{-}\mathsf{Cat}\cong \mathsf{CNS}.$$

When & is the product t-norm, the conical bounded $\mathsf{I}$-semifilter monad is isomorphic to the functional ideal monad, and by [29], we have

$$(\mathsf{ConBSF},\mathsf{2},\overline{\mathsf{ConBSF}})\text{-}\mathsf{Cat}\cong \mathsf{App},$$

where $\mathsf{App}$ is the category of approach spaces and $\overline{\mathsf{ConBSF}}$ is the Kleisli extension to $\mathsf{Rel}.$

Since $\eta :(\mathsf{CSF},\kappa )\to (\mathsf{ConBSF},\eta ·\kappa )$ is a morphism of the $\mathsf{I}$-power-enriched category, by Theorem 11, it is a morphism of the Kleisli extensions. Hence, it induces an algebraic functor as follows:

Proposition 13.

If & is the product t-norm, there is a functor $A_{\kappa }:\mathsf{CNS}\to \mathsf{App}:$

$$(X,{(-)}^{\circ })\mapsto (X,\mathfrak{A})$$

that maps a CNS space X to the approach space $(X,\mathfrak{A}),$ where the bounded approach system ${\left\{\mathfrak{A}\left(x\right)\right\}}_{x\in X}$ is given by

$$\mathfrak{A}\left(x\right)=\{\mu \in {[0,\infty ]}^X\mid \underset{\begin{array}{c}{\omega }^{\circ }\left(x\right)=1\\ \bigwedge \omega >0\end{array}}{\bigvee }{\mathrm{sub}}_X(\omega ,e^{-\mu })=1\},$$

in which ${(-)}^{\circ }$ is the interior operator of the CNS space $X.$

4. Conclusions

In order to find the many-valued version of the filter monad, we begin with the composite monads ${\mathsf{CP}}^{†},{\mathsf{C}}^{†}\mathsf{P}$ on $\mathsf{I}\text{-}\mathsf{Cat}$ and then restrict them to $\mathsf{Set}$ to obtain the monad ${\mathsf{U}}_{\mathsf{C}}.$ This $\mathsf{Set}$-based monad ${\mathsf{U}}_{\mathsf{C}}$ is precisely the conical $\mathsf{I}$-semifilter monad. Three lax extensions of the conical $\mathsf{I}$-semifilter monad to $\mathsf{I}\text{-}\mathsf{Rel}$ are presented: the canonical, op-canonical and Kleisli extensions. We prove that the op-canonical extension coincides with the Kleisli extension. Lax algebras of this extension can be described using relations rather than $\mathsf{I}$-relations; hence, they are CNS spaces.

Problem 1.

When considering the canonical extension of the conical $\mathsf{I}$-semifilter monad, what are the lax algebras?

As for the future research direction, exploring the connections between monoidal topology and nonstandard analysis [30,31] is of interest.