Slightly expand the draft.

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

content/blog/01_spa_agda_lattices.md

@@ -283,6 +283,7 @@ to in the main body of the text). The same is true for the set union and
 intersection operations, \((\cup)\) and \((\cap)\).
 {{< /sidenote >}}". For example, \((+\ \sqcap\ ?)\ =\ +\), because a variable
 that's both "any sign" and "positive" must be positive.
+{#lub-glub-or-and}
 
 There's just one hiccup: what's the greatest lower bound of `+` and `-`?
 it needs to be a value that's less than both of them, but so far, we don't have

content/blog/02_spa_agda_combining_lattices.md

@@ -188,7 +188,9 @@ Next, we need to define the \((\sqcup)\) and \((\sqcap)\) operators that match
 our definition of "less than or equal". Let's start with \((\sqcup)\). For two
 maps \(m_1\) and \(m_2\), the join of those two maps, \(m_1 \sqcup m_2\) should
 be greater than or equal to both; in other words, both sub-maps should be less
-than or equal to the join. Our newly-introduced condition for "less than or equal"
+than or equal to the join.
+
+Our newly-introduced condition for "less than or equal"
 requires that each key in the smaller map be present in the larger one; as
 a result, \(m_1 \sqcup m_2\) should contain all the keys in \(m_1\) __and__
 all the keys in \(m_2\). So, we could just take the union of the two maps:
@@ -198,7 +200,9 @@ maps could be distinct, and they might even be incomparable. This is where the
 second part of the condition kicks in: the value in the combination of the
 maps needs to be bigger than the value in either sub-map. We already know how
 to get a value that's bigger than two other values: we use a join on the
-values! Thus, define \(m_1 \sqcup m_2\) as a map that has all the keys
+values!
+
+Thus, define \(m_1 \sqcup m_2\) as a map that has all the keys
 from \(m_1\) and \(m_2\), where the value at a particular key is given
 as follows:
 
@@ -211,9 +215,25 @@ m_2[k] & k \notin m_1, k \in m_2
 \end{cases}
 {{< /latex >}}
 
+If you're familiar with set theory, this operation is like
+{{< sidenote "right" "map-union-note" "an extension of the union operator \((\cup)\)" >}}
+There are, of course, other ways to extend the "union" operation to maps.
+Haskell, for instance, defines it in a "left-biased" way (preferring the
+elements from the left operand of the operation when duplicates are encountered).<br>
+<br>
+However, with a "join" operation \((\sqcup)\) that's defined on the values
+stored in the map gives us an extra tool to work with. As a result, I would
+argue that our extension, given such an operator, is the most natural.
+{{< /sidenote >}} to maps. In fact, this begins to motivate
+the choice to use \((\sqcup)\) to denote this operation. A further bit of
+motivation is this:
+[we've already seen]({{< relref "01_spa_agda_lattices#lub-glub-or-and" >}})
+that the \((\sqcup)\) and \((\sqcap)\) operators correspond to "or"
+and "and". The elements in the union of two sets are precisely
+those that are in one set __or__ the other. Thus, using union here fits our
+notion of how the \((\sqcup)\) operator behaves.
+{#union-as-or}
+
 {{< todo >}}
 I started using 'join' but haven't introduced it before.
 {{< /todo >}}
-
-something something lub glub
-{#union-as-or}