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Varieties of non-distributive lattice [was: 'Closed over relations'; another possibly-quiz [was: Uniqueness of R00]]

Quote from AntC on October 12, 2019, 7:28 am


... What I'm trying to do is build on well-established work, to get insights into the lattice structure, to fix R00 as corresponding to DUM.

Specifically, I believe Relational Lattices exhibit this structure:

  • They do not contain M3 sub-lattices, per Pic 10 here.
  • They do not contain hexagon sub-lattices per the example here.
  • They do contain N5 (or Nj) sub-lattices, per Pic 11 here. In which case:
    • The nodes/edge labelled 1-a denote relations with the same heading, where a has one less tuple than 1.
    • The nodes/edge labelled c-b-...-0 denote relations with the same heading (where the ... denotes arbitrarily many).
    • Each step c, b, ..., 0 has one fewer tuple moving downwards.
    • The heading of c-b-...-0 is the heading of 1-a extended with one extra attribute, such that c is the Cartesian Product of 1 with every value in that extra attribute's type, and such that 0 is the Cartesian Product of a with every value in that type. [**]
  • Specifically in an N5/Nj sub-lattice where the node labelled 1 is R01 aka DEE, the node labelled a is R00 aka DUM.
    • IOW the nodes labelled c-b-...-0 have a single attribute/are degree 1; node labelled c is the relation with that singleton attribute and every possible tuple  with that heading; and node labelled 0 is the empty relation with that heading.
    • [**] exception to the 0 being a Cartesian Product: if node labelled a denotes an empty relation, 0 is an empty relation with heading the extra attribute. (Because the Cartesian Product with an empty relation is an empty relation, so actually not an exception.)

I'm not expecting all of the above points are necessary to specifying the structure; nor useful in fixing R00.

There's no N5/Nj sublattice if all of the attribute types are cardinality 1 (or less). So to get any useful algebraisation of the above, we need a (clunky, ad-hoc) axiom to the effect there's at least one attribute type of cardinality 2 or greater.