Dualizing sup-preserving endomaps of a complete lattice

The category SLatt of complete lattices and sup-preserving functions is a well-known $\ast$-autonomous category [2, 10, 9, 6]. For complete lattices $X,Y$, we denote by $[X,Y]_{\vee}$ the homset in this category. The two-element Boolean algebra $\mathbf{2}$ is a dualizing element and $[X,\mathbf{2}]_{\vee}$ is, as a lattice, isomorphic to the dual lattice $X^{op}$. More generally, the functor $(\,\cdot\,)^{\ast}=[\,\cdot\,,\mathbf{2}]_{\vee}$ is naturally isomorphic to the functor $(\,\cdot\,)^{op}$ where, for $f:Y\xrightarrow{\;\;\;\;\;}X$, $f^{op}:X^{op}\xrightarrow{\;\;\;\;\;}Y^{op}$ is the right adjoint of $f$, noted here by $\rho(f)$ (the left adjoint of an inf-preserving function $g:Y\xrightarrow{\;\;\;\;\;}X$ shall be denoted by $\ell(g):X\xrightarrow{\;\;\;\;\;}Y$).

Let us describe the internal structure of the homset $[X,Y]_{\vee}$ as a complete lattice. For $x\in X$ and $y\in Y$, we define the following elements of $[X,Y]_{\vee}$:

The tensor notation arises from the canonical isomorphism $[X,Y]_{\vee}\simeq(Y^{op}\otimes X)^{op}$ of $\ast$-autonomous categories. That is, $[X,Y]_{\vee}$ is dual to the tensor product $Y^{op}\otimes X$ and the functions $y\overline{\otimes}x$ correspond to elementary tensors of $Y^{op}\otimes X$. For $f\in[X,Y]_{\vee}$, $g\in[Y,Z]_{\vee}$, and $h\in[X,Z]_{\vee}$, let us recall that there exists uniquely determined maps $g\backslash h\in[X,Y]_{\vee}$ and $h/f\in[Y,Z]_{\vee}$ satisfying

The binary operations $\backslash$ and $/$ are known under several names: they yield left and right Kan extensions and (often when $X=Y=Z$) they are named residuals or division operations [7] or right and left implication [15, 6]. With the division operations at hand, let us list some elementary relations between the functions previously defined:

Let $[X,Y]_{\land}$ denote the poset of inf-preserving functions from $X$ to $Y$, with the pointwise ordering. Observe that, as a set, $[X,Y]_{\land}$ equals $[X^{op},Y^{op}]_{\vee}$. Yet, as a poset or a lattice, the equality $[X,Y]_{\land}=[X^{op},Y^{op}]_{\vee}^{op}$ is the correct one. As a matter of fact, we have $f\leq_{[X,Y]_{\land}}g$ iff $f(x)\leq_{Y}g(x)$, all $x\in X$, iff $g(x)\leq_{Y^{op}}f(x)$, all $x\in X$, iff $g\leq_{[X^{op},Y^{op}]_{\vee}}f$. Using standard isomorphisms of $\ast$-autonomous categories, we have

That is, the set of inf-preserving functions from $X$ to $Y$ can be taken as a concrete realization of the tensor product $Y\otimes X^{op}$. This should not come as a surprise, since it is well-known that the set of Galois connections from $X$ to $Y$—that is, pairs of functions $(f:X\xrightarrow{\;\;\;\;\;}Y,g:Y\xrightarrow{\;\;\;\;\;}X)$ such that $y\leq f(x)$ iff $x\leq g(y)$—realizes the tensor product $Y\otimes X$ in SLatt, see e.g. [19, 13] or [6, §2.1.2]. Such a pair of functions is uniquely determined by its first element, which is an inf-preserving functions from $X^{op}$ to $Y$.

Notice now that

from which we derive the following principle:

The map yielding the isomorphism in equation (1) is $\rho$, the operation of taking the right adjoint. The bijection stated in Fact 1.3 is therefore obtained by precomposing with $\rho$.

We exploit now the work done for $[X,Y]_{\vee}$ to recap the structure of $[X,Y]_{\land}$ as a tensor product. Consider the maps

By dualizing Lemma 1.1, we observe that the relation $y\underline{\otimes}x=\gamma_{y}\land\alpha_{x}$ holds, the maps $y\underline{\otimes}x$ realize the elementary tensors of the (abstract) tensor product $Y\otimes X^{op}$, $[X,Y]_{\land}$ is join-generated by these maps, and every $g\in[X,Y]_{\land}$ can be canonically written as $g=\bigvee_{x\in X}g(x)\underline{\otimes}x$.

Recall that a bimorphism $\psi:Y\times X^{op}\xrightarrow{\;\;\;\;\;}Z$ is a function that is sup-preserving in each variable, separately. This in particular means that infs in $X$ are transformed into sups in $Z$. The universal property of $[X,Y]_{\land}$ as a tensor product can be therefore stated as follows:

Observe now that, for $g:X^{\prime}\xrightarrow{\;\;\;\;\;}X$ and $f:Y\xrightarrow{\;\;\;\;\;}Y^{\prime}$, we have

implying that the following diagram commutes:

That is, Raney’s transform ${(\,\cdot\,)}{}^{\vee}$ is natural in both its variables. Let us remark on the way the following:

In order to simplify reading, we use $\psi$ both for a bimorphism $\psi:Y\times X^{op}\xrightarrow{\;\;\;\;\;}Z$ and for its extension to the tensor product $\tilde{\psi}:Y\otimes X^{op}\xrightarrow{\;\;\;\;\;}Z$.

For $f:X\xrightarrow{\;\;\;\;\;}Y$ in the category SLatt, let $j=\rho(f)\circ f$ and $o=f\circ\rho(f)$. Denote by $X_{j}$ (resp. $Y_{o}$) the set of fixed points of $j$ (resp., of $o$). Then, we have a standard (epi,iso,mono)-factorization

Thus, $f$ is mono if and only $j=id_{X}$ and $f$ is epic if and only if $o=id_{Y}$. Notice that $Y_{o}$ is the image of $X$ under $f$, while $X_{j}$ is the image of $Y$ under $\rho(f)$. We apply this factorization to Raney’s transforms.

By its definition, $[X,Y]_{\vee}^{\mathtt{t}}$ is the sub-join-semilattice of $[X,Y]_{\vee}$ generated by the $c_{y}\circ a_{x}$. Moreover, it is easily seen that $w\overline{\otimes}z\in[X,Y]_{\vee}^{\mathtt{t}}$, for each $w\in Y$ and $z\in X$, and that $c_{y}\circ a_{x}\leq w\overline{\otimes}z$ if and only of $y\leq w$ or $z\leq x$. From these relations, $[X,Y]_{\vee}^{\mathtt{t}}$ yields a concrete representation of Wille’s tensor product $Y\widehat{\otimes}X^{op}$, see [20], which, for finite lattices, coincides with the Box tensor product of [8].

Next, we list some immediate consequences of naturality of Raney’s transforms:

Let us recall that Raney’s Theorem [14] characterizes completely distributive lattices as those complete lattices satisfying the identity

This identity is exactly the identity $id={id}{}^{\land}{}{}^{\vee}$ or, as we have seen in Proposition 2.4, the identity $f={f}{}^{\land}{}{}^{\vee}$ holding for each $f\in[L,L]_{\vee}$. Since complete distributivity is autodual (at least in a classical context), we derive that Raney’s transform $(\,\cdot\,)^{\vee}:[L,L]_{\land}\xrightarrow{\;\;\;\;\;}[L,L]_{\vee}$ is surjective if and only if it is injective.

We conclude this section with a glance at the quantale $([L,L]_{\vee}^{\mathtt{t}},\circ)$ of tight maps, where $L$ is an arbitrary complete lattice $L$. We pause before for a technical lemma needed end the section and later on as well. Recalling the equations in (3), let us define

and observe the following:

Recall that a dualizing element in a quantale $(Q,\circ)$ is an element $0\in Q$ such that $0/(x\backslash 0)=(0/x)\backslash 0=x$, for each $x\in Q$. As consequences of Proposition 2.6, we obtain: