## Fibration Lattices

In section 2.3 “Relational Lattices, (Op-)Fibrations and the Grothendieck Construction” of the paper by Litak & Mikulas & Hidders, the authors cast category theory perspective to hint that one can recruit categories other than sets to construct relational-like lattices. Finding examples of such lattices, which we’ll promptly call fibration lattice, has been left as an exercise to the reader. In this notes we’ll work out an example of fibration lattice from the category of Linear Vector Spaces.

Relational lattice elements are structured as pairs with the first component being the set of attributes, and the second one being the set of tuples. First, we’ll substitute the set of attributes by linear subspace of some linear space. Second, we replace the set of tuples by the subspace of linear operators acting on those subspaces.

Conventionally, the theory of Linear Vector Spaces — Linear Algebra — is constructed over the field of real or complex numbers. These sets are uncountable, let alone infinite. Our goal is intuitive illustration of fibration lattice via Hasse diagram, therefore we’ll work with finite fields, such as $\mathbb{F} _3 = \mathbb{Z}/ 3\mathbb{Z}$. The choice of integer 3 over the simpler 2 is due to the fact that $\mathbb{F} _2$ is essentially Boolean Algebra in disguise, while we want fibration lattice example outside of the familiar Relational Lattices.

With graphical illustration via Hasse diagram in mind let’s fix the dimension of vector space as the minimal , but not trivial number. Therefore, the main mathematical object of this essay is two dimensional vector space $\mathbb{F} _3 ^2$. For concreteness, lets enumerate all the vectors: $\{ \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}0\\1\end{pmatrix}, \begin{pmatrix}0\\2\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix}, \begin{pmatrix}1\\1\end{pmatrix}, \begin{pmatrix}1\\2\end{pmatrix}, \begin{pmatrix}2\\0\end{pmatrix}, \begin{pmatrix}2\\1\end{pmatrix}, \begin{pmatrix}2\\2\end{pmatrix} \}$

Again, we’ll operate with vector subspaces of $\mathbb{F} _3 ^2$ rather than sets of vectors, thus enumerating basis vectors would provide less verbose specification, which in our case is $\{ \begin{pmatrix}0\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix} \}$

Our vector space has 4 one-dimensional subspaces, that is $\{ \begin{pmatrix}0\\1\end{pmatrix} \} ,\{ \begin{pmatrix}1\\0\end{pmatrix} \}, \{ \begin{pmatrix}1\\1\end{pmatrix} \} , \{ \begin{pmatrix}1\\2\end{pmatrix} \}$

Finally, there is also one zero-dimensional subspace consisting of the lone zero vector, alternatively described via the empty vector basis.

After having completed the description of “relation attributes”, let’s move on onto “relation tuples”. Those are the linear spaces of linear operators acting on the linear spaces that we have just introduced. As we have represented abstract vectors as column vectors, we’ll represent linear operators as matrices.

We’ll start with one-dimensional subspaces of $\mathbb{F} _3 ^2$ because linear operators acting on them are plain scalars. Therefore, for each of the 4 subspaces, we have 2 subspaces of linear operators: the zero subspace consisting of just one scalar value from $\mathbb{F} _3$ that is $\{ 0 \}$ , and the full subspace including all the 3 field elements: $\{ 0,1,2 \}$ . Again, our preferred description would be in terms of operator basis, which is correspondingly the empty basis set $\{ \}$ , and the $\{ 1 \}$.

The lattice of subspaces of linear operators acting on the “full set of attributes”, that is $\mathbb{F} _3 ^2$ is a little bit more sophisticated. Again, to keep the discription less verbose with the sets of basis vectors, which in our case are 2×2 matrices : $\{ \}$ $\{ \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix} \}$ $\{ \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix} \}$ $\{ \begin{pmatrix}0 & 0\\1 & 0\end{pmatrix} \}$ $\{ \begin{pmatrix}0 & 0\\0 & 1\end{pmatrix} \}$ $\{ \begin{pmatrix}1 & 1\\0 & 0\end{pmatrix} \}$ $\{ \begin{pmatrix}1 & 2\\0 & 0\end{pmatrix} \}$ $\{ \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix} \}$ $\{ \begin{pmatrix}1 & 1\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix} \}$ $\dots$ $\{ \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0\\1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0\\0 & 1\end{pmatrix} \}$

To formally define elements of fibration lattice of vector spaces we’ll mimic relational database notation, when relation header is placed over the set of tuples:

[DEPTNO  DNAME       LOC]
10      ACCOUNTING  LONDON
20      SALES       PARIS
30      RESEARCH    ROME
40      OPERATIONS  LONDON

Adapted to vector/operator spaces we’ll have lattice elements represented like this: $\{ \begin{pmatrix}0\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix} \} \over \{ \begin{pmatrix}1 & 2\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1\\0 & 0\end{pmatrix} \}$

However, in abstract form we’ll prefer conventional mathematical notation representing the lattice elements as ordered pairs $\langle x_h,x_t \rangle$

Now we are ready to define lattice operations: meet $\langle x_h,x_t \rangle \wedge \langle y_h,y_t \rangle$ and join $\langle x_h,x_t \rangle \vee \langle y_h,y_t \rangle$. As usual, we define them component-wise starting with the headers $x_h$ and $y_h$ .

The join of two vector subspaces $x_h$ and $y_h$ is their set intersection. With judicious choice of basis it is also the intersection of the sets of basis vectors. The meet of $x_h$ and $y_h$ is the minimal subspace containing their set-theoretic union, or the space spanning set-theoretic union of the basis vectors. Alternatively, $x_h \wedge y_h$ is the sum of $x_h + y_h$ of subspaces with the sum elements being paiwise sums of the elements from $x_h$ and $y_h$ .

Since join $x_h \vee y_h$ is subspace of the operands $x_h$ and $y_h$ , the result of $x_t \vee y_t$ is easier to describe. Take all the operators from the first operand $x_t$ and find their action on the subspace $x_h \vee y_h$. Likewise, take all the operators from the second operand $y_t$ and find their action on the subspace $x_h \vee y_h$. Operator sum of both operator subspaces is the resulting operator set $x_t \vee y_t$ acting on $x_h \vee y_h$ .

In the case of lattice meet $x_t \wedge y_t$, both operands operate on subspaces $x_h$ and $y_h$ of $x_h \wedge y_h$ so we have to expand their action onto $x_h \wedge y_h$, first. We’ll define it as the the maximum operator space which action on $x_h$ is $x_t$. Correspondingly, we also need the expansion of $y_t$ onto $x_h \wedge y_h$. Then we just take the intersection of both expansions.

Let’s work out a couple of cases from the running example $\mathbb{F} _3 ^2$ .

1. The [scalar] operator space of the $\{ \begin{pmatrix}0\\1\end{pmatrix} \} \over \{ \}$

expanded to the $\{ \begin{pmatrix}0\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix} \}$

is formally the solution of the equation $\begin{pmatrix}0\\0\end{pmatrix} = \begin{pmatrix}a_{11}& a_{12} \\ a_{21}& a_{22} \end{pmatrix} \begin{pmatrix}0\\1\end{pmatrix}$

which is the subspace spanning $\{ \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0\\1 & 0\end{pmatrix} \}$

2. The [scalar] operator space of the $\{ \begin{pmatrix}0\\1\end{pmatrix} \} \over \{ 1\}$

expanded to the full vector space $\{ \begin{pmatrix}0\\1\end{pmatrix}, \begin{pmatrix}1\\0\end{pmatrix} \}$ is the operator subspace spanning $\{ \begin{pmatrix}1 & 0\\0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0\\1 & 0\end{pmatrix}, \begin{pmatrix}0 & 0\\0 & 1\end{pmatrix} \}$

In the case of relational lattices the sublattice of headers was distributive. This is no longer the case, which prompts revision of the relational lattice axioms consistent with distributivity of the header sublattice when extrapolated to more general setting of fibration lattices.

The final remarks concern algebraic signature extensions and applications. Linear spaces equipped with scalar product gives rise to orthomodular lattice of lattice headers. This opens a venue to introduce fibration lattice analogs for unary operations of tuple and attribute complement. Furthermore, orthomodular lattices constitute a foundation of Quantum Logic, with quite distinct flavor as compared to Boolean Algebra. Boolean Algebra is Propositional Logic cast into algebraic perspective, while Quantum Logic is remarkably non-classic. Quantum Logic encompasses a wealth of literature, but the issue what might be the Quantum Predicate Calculus received relatively little attention. Our opinion is that if relational lattice can be considered as an algebraization of a fragment of Predicate Calculus, then perhaps fibration lattice of the linear spaces and operators can be considered as algebraic perspective to Quantum Predicate Calculus?