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March 23, 2016 / porton

A new math abstraction, categories of sides

I introduce a new math abstraction, categories of sides, in order to generalize two theorems into one.

Category of sides \Upsilon is an ordered category whose objects are (small) bounded lattices and whose morphisms are maps between lattices such that every Hom-set is a bounded lattice and (for all relevant variables):

  1. (a \sqcup b) X = a X \sqcup b X
  2. (a \sqcap b) X \sqsubseteq a X \sqcap b X
  3. (\lambda x \in \mathfrak{A}: x \sqcap c) \in \Upsilon (\mathfrak{A}; \mathfrak{A}) for every c \in \mathfrak{A}
  4. a \bot = \bot
  5. \top X = \top unless X = \bot

I call morphisms of such categories sides.

The category of pointfree funcoids between boolean lattices is a category of sides. Also it seems (not checked yet) that the category of Galois connections between boolean lattices is a category of sides.

This way, it seems that I’ve found a common generalization of two theorems:

Theorem For category of pointfree funcoids, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

Theorem For category of Galois connections, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

The last theorem is a slight reformulation of theorem 3.8 in “Zahava Shmuely. The tensor product of distributive lattices. algebra universalis, 9(1):281–296.” (I borrowed the proof idea from that Zahava’s article.)

Common generalization:

Theorem For every category of sides, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

It is also conceivable to define pointfree reloids as filers on a (fixed) category of sides.

Note that the definition of “categories of sides” is preliminary, I may probably add more axioms in the future, if found convenient.

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