The considerations below were with an error, see the comment.
Product order of posets (for where is some index subset) is defined by the formula . (By the way, it is a product in the category of posets.)
By definition the lambda-function for a form depended on variable .
It is easy to show that for a product of distributive lattices with least elements we have lattice-theoretic difference whenever every is defined. Compare also (where denotes supremum of two elements), whenever every is defined.
I conjecture that this equality can be generalized to a relatively wide class of functions of a finite number of elements: .
I do not hold the claim of originality of this conjecture. Moreover, I ask you to notify me (email@example.com) if you know a work where a similar theory was described.
Now (to formulate the conjecture precisely) it is required to lay some formalistic.
I call an an order logic a set of propositional axioms (without quantifiers) with variables , , , … and two binary relations and including at least the following axioms:
- ; ; (equality axioms);
- ; ; (partial order axioms).
By definition a partial axiomatic function for a given order logic is a partial function of a finite number of arguments which is unambigously defined by some set of additional propositional formulas (the definition). I mean that we have some finite set of propositional formulas , , , … such that it can be proved that is unambiguously determined by , …, , , …, , , …, , …
For example, the function is defined by the formula:
That this definition of is unambiguous is a well known fact. Note that is general this function of two arguments is partial (as not every order is a semilattice).
Distributive lattices, Heyting algebras, and boolean algebras can be defined as examples of order logics.
Similarly we can define as partial axiomatic functions lattice-theoretic difference , boolean lattice complement and even co-brouwerian pseudodifference .
Conjecture: For every partial axiomatic function we have whenever every is defined (Is it equivalent to to be defined?).
Please notify me (firstname.lastname@example.org) if you know a solution of this conjecture.
Further generalization may be interesting. For example, our above consideration does not consider the formula (where is the supremum on our poset), because supremum is a function of infinite arity, while we considered only finite relations.
Can it also be generalized for categorical product (not only in category )?
In a short (4 pages) article I define pointfree binary relations, a generalization of binary relations which does not use “points” (elements).
In a certain special case (of endo-relations) pointfree binary relations are essentially the same as binary relations.
It seems promising to research filters on sets of pointfree relations, generalizing the notion of reloids (that is filters on cartesian products) from my my research book
We need a more abstract way to define reloids:
For example filters on a set are isomorphic to triples where is a filter on , as well as filters of boolean reloids (that is pairs of functions , such that (for all and ).
I propose a way to encompass all ways to describe reloids as follows:
Let call a filtrator of pointfree reloid a pair of a filtrator and an associative operation on its core. Then call abstract reloids pointfree reloids isomorphic (both a filtrators and as semigroups) to reloids.
I am yet unsure that this structure encompasses all essential properties of reloids (just like as primary filtrators encompass all properties of filters on posets).
There were several errors in the section “Open maps” of my online book.
I have rewritten this section and also moved the section below in the book text.
However, the new proof of the theorem stating that composition of open maps between funcoids is an open map now uses a proof referring to a particular point .
It is currently an open problem to rewrite this proof in pointfree style.