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

**2**

Quote from AntC on October 12, 2019, 9:10 amQuote 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 M
_{3}sub-lattices, per Pic 10 here.- They do not contain hexagon sub-lattices per the example here.
- They do contain N
_{5}(or N_{j}) sub-lattices, per Pic 11 here. In which case:

- The nodes/edge labelled
1-adenote relations with the same heading, whereahas one less tuple than1.- The nodes/edge labelled
c-b-...-0denote relations with the same heading (where the...denotes arbitrarily many).- Each step
c, b, ..., 0has one fewer tuple moving downwards.- The heading of
c-b-...-0is the heading of1-aextended with one extra attribute, such thatcis the Cartesian Product of1with every value in that extra attribute's type, and such that0is the Cartesian Product ofawith every value in that type. [**]- Specifically in an N
_{5}/N_{j}sub-lattice where the node labelled1is`R01`

aka`DEE`

, the node labelledais`R00`

aka`DUM`

.

- IOW the nodes labelled
c-b-...-0have a single attribute/are degree 1; node labelledcis the relation with that singleton attribute and every possible tuple with that heading; and node labelled0is the empty relation with that heading.- [**] exception to the
0being a Cartesian Product: if node labelledadenotes an empty relation,0is 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 N

_{5}/N_{j}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.

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 M
_{3}sub-lattices, per Pic 10 here. - They do not contain hexagon sub-lattices per the example here.
- They do contain N
_{5}(or N_{j}) 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. [**]

- The nodes/edge labelled
- Specifically in an N
_{5}/N_{j}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.)

- IOW the nodes labelled

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

.

There's no N_{5}/N_{j} 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.

**2**