# Does this lattice construction have a name?

Suppose we are given complete lattices $$(L_i,\le_i)$$ indexed over some set $$I$$, such that $$L_i\ne 2$$ and $$L_i\cap L_j=\emptyset$$ for all $$i\ne j$$. There is a sort of "amalgamation" that can be constructed from these lattices. Consider $$L$$ as the union of the sets $$L_i\setminus\{0_L,1_L\}$$ alongisde two objects $$0,1\not\in\bigcup_{i\in I} L_i$$. Construct a partial order as follows: $$0\le x \le 1$$ for all $$x\in L$$, and $$x\le y$$ if $$x,y\in L_i\setminus\{0_L,1_L\}$$ for some $$i$$ and $$x\le_i y$$. This makes $$(L,\le)$$ a complete lattice which glues together all of the original lattices by identifying the least and greatest elements.

Does this construction have a name? I have searched for it all over the internet and books but cannot seem to find anything about it. Maybe it is a particular case of a different lattice construction?

I have seen this is called the Horizontal Sum of the bounded lattices $$(L_i, \land_i, \lor_i, 0_i, 1_i)$$.

As is often the case in order theory, the term "Horizontal Sum" means something different depending on how much structure you have. For mere posets $$(P_i, \leq_i)$$, the horizontal sum is defined as

$$x_i \leq y_j \iff i = j \text{ and } x_i \leq_i y_j$$

(so we place the posets next to each other horizontally). Davey and Priestley call this the "disjoint union" of two posets, denoted $$\overset{\cdot}{\cup}$$.

However, when our posets are bounded, that is, when there is a $$0_i, 1_i \in P_i$$ with $$0_i \leq x \leq 1_i$$ for each $$x \in P_i$$, then we can ask for all of our poset operations to respect these bounds. In this case we write $$(P_i, \leq_i, 0_i, 1_i)$$ to indicate that there is extra structure we're keeping track of.

In this case, a "Horizontal Sum" is the construction you've described: We take the horizontal sum as mere posets, but then identify all the top (resp. bottom) elements.

I'm struggling to find a textbook reference for this, but you can see this definition in Chajda and Länger's Horizontal Sums of Bounded Lattices or implicit in Giuntini, Mureşan, and Paoli's Ordinal and Horizontal Sums Constructing PBZ*-lattices.

I've checked my lattice theory references, but I'm not finding this particular definition anywhere (even though I know I've seen it). I'll look through my combinatorics references too to see if it's in one of those, and I'll update this answer if I'm able to find a textbook reference.

Edit: Aha! I thought it was in Davey and Priestley's Introduction to Lattices and Order. It just wasn't where I expected. You can find this definition relegated to exercise $$3.12$$, on page $$84$$ of my edition.

I hope this helps ^_^

• Thank you! I had not found those references. Digging a little deeper I found about the notion of "pushout" in category theory. It appears that, for two bounded lattices, their pushout along $\{0,1\}$ coincides with their horizontal sum, I think. So maybe this Horizontal Sum could be also called an "amalgamated sum". Commented Oct 10, 2021 at 6:18
• I just checked Dave and Priestley's book and was thrilled to see that the definition appears as an exercise in the chapter about Formal Concept Analysis, which is exactly the topic that led me to look for a name for this construction in the first place. Commented Oct 10, 2021 at 6:22
• Be careful! The notion of "pushout" depends on what category you're working in. This is the same pitfall I mentioned earlier about "horizontal sum" meaning a different thing for posets vs bounded posets. I avoided categorical terminology in my answer, but your construction is the coproduct in the category of bounded posets, or, as you've noticed, a certain pushout in the category of posets. Importantly, though, it is not a coproduct/pushout in any category of lattices! Commented Oct 10, 2021 at 6:33
• Re: Formal Concept Analysis, I'm excited to hear you're excited about Davey and Priestley's book! It's a really excellent little reference, and I'm sure you'll learn a lot from it! I know I did ^_^. Commented Oct 10, 2021 at 6:35