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 . The choice of integer 3 over the simpler 2 is due to the fact that 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 . For concreteness, lets enumerate all the vectors:

Again, we’ll operate with vector subspaces of rather than sets of vectors, thus enumerating basis vectors would provide less verbose specification, which in our case is

Our vector space has 4 one-dimensional subspaces, that is

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 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 that is , and the full subspace including all the 3 field elements: . Again, our preferred description would be in terms of operator basis, which is correspondingly the empty basis set , and the .

The lattice of subspaces of linear operators acting on the “full set of attributes”, that is 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 :

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…

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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:

However, in abstract form we’ll prefer conventional mathematical notation representing the lattice elements as ordered pairs

Now we are ready to define lattice operations: meet and join . As usual, we define them component-wise starting with the headers and .

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

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

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

Let’s work out a couple of cases from the running example .

1. The [scalar] operator space of the

expanded to the

is formally the solution of the equation

which is the subspace spanning

2. The [scalar] operator space of the

expanded to the full vector space is the operator subspace spanning

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?