# What does "lattice of abstraction" mean?

I've been thinking about specificity recently and decided to re-read [SotW: Be Specific](https://www.lesswrong.com/posts/NgtYDP3ZtLJaM248W/sotw-be-specific). In that post Eliezer writes the following:

> S. I. Hayakawa called this the ladder of abstraction.  I'm not sure if understanding the following section will really help with the skill of Being Specific, or help anyone construct exercises for the skill of being specific.  But a better theoretical understanding does sometimes prove useful.  So I will now digress to explain that abstraction isn't really a ladder, but a *lattice.*

I think I understand some of what he's saying. I think about it in terms of drawing boundaries around points in [Thingspace](https://www.lesswrong.com/lw/nl/the_cluster_structure_of_thingspace). So the concept of "Sunny Days" is drawing a boundary around the points:

*   `{ Sunny, Cool, Weekday }`
*   `{ Sunny, Cool, Weekend }`
*   `{ Sunny, Hot, Weekday }`
*   `{ Sunny, Hot, Weekend }`

And the concept of "Sunny Cool Days" is drawing a narrower boundary around the points:

*   `{ Sunny, Cool, Weekday }`
*   `{ Sunny, Cool, Weekend }`

And so we can say that "Sunny Cool Days" is more specific than "Sunny Days" because it draws a narrower boundary.

But I still have no clue what a lattice is. [Wikipedia's description](https://en.wikipedia.org/wiki/Lattice_(order)) was very intimidating:

> A **lattice** is an abstract structure studied in the [mathematical](https://en.wikipedia.org/wiki/Mathematical) subdisciplines of [order theory](https://en.wikipedia.org/wiki/Order_theory) and [abstract algebra](https://en.wikipedia.org/wiki/Abstract_algebra). It consists of a [partially ordered set](https://en.wikipedia.org/wiki/Partially_ordered_set) in which every pair of elements has a unique [supremum](https://en.wikipedia.org/wiki/Supremum) (also called a least upper bound or [join](https://en.wikipedia.org/wiki/Join_(mathematics))) and a unique [infimum](https://en.wikipedia.org/wiki/Infimum) (also called a greatest lower bound or [meet](https://en.wikipedia.org/wiki/Meet_(mathematics))). An example is given by the [power set](https://en.wikipedia.org/wiki/Power_set) of a set, partially ordered by [inclusion](https://en.wikipedia.org/wiki/Subset), for which the supremum is the [union](https://en.wikipedia.org/wiki/Union_(set_theory)) and the infimum is the [intersection](https://en.wikipedia.org/wiki/Intersection_(set_theory)). Another example is given by the [natural numbers](https://en.wikipedia.org/wiki/Natural_number), partially ordered by [divisibility](https://en.wikipedia.org/wiki/Divisibility), for which the supremum is the [least common multiple](https://en.wikipedia.org/wiki/Least_common_multiple) and the infimum is the [greatest common divisor](https://en.wikipedia.org/wiki/Greatest_common_divisor).

Maybe a lattice is simply what I described: a boundary around points in Thingspace. But I get a sense that "lattice" involves *order* in some way, and I am not seeing how order fits in to the question of how specific a concept is. I also think it's plausible that there are other aspects of lattices that are relevant to the discussion of specificity that I am missing.