Add a tiny section about the tropical semiring to polynomial post

content/blog/search_polynomials.md

@@ -202,7 +202,7 @@ way, polynomials like this give us _less_ information than our original ones
 we're just warming up - there are more interesting semirings for us to
 consider!
 
-#### Polynomials over Sets of Paths, \\(\\mathcal{P}(\\Pi)\\)
+#### The Semiring of Sets of Paths, \\(\\mathcal{P}(\\Pi)\\)
 Until now, we explicitly said that "all paths of the same length are
 equivalent". If we're giving directions, though, we might benefit
 from knowing not just that there _is_ a way, but what roads that
@@ -267,7 +267,7 @@ consisting of each path from \\(A\\) concatenated with each
 path from \\(B\\):
 
 {{< latex >}}
-A \times B = \{ a \rightarrow b\ |\ a \in A, b \in B \}
+A \times B \triangleq \{ a \rightarrow b\ |\ a \in A, b \in B \}
 {{< /latex >}}
 
 The fact that this definition of multiplication on sets is associative
@@ -325,3 +325,55 @@ Multiplying them gives:
 
 This resulting polynomial gives us all the paths from city A to city C,
 grouped by their length!
+
+#### The Tropical Semiring, \\(\\mathbb{R}\\)
+I only have one last semiring left to show you before we move on to something
+other than paths between cities. It's a fun semiring though, as even its name
+might suggest: we'll take a look at a _tropical semiring_.
+
+In this semiring, we go back to numbers; particularly, real numbers (e.g.,
+\\(1.34\\), \\(163\\), \\(e\\), that kind of thing). We even use addition --
+sort of. In the tropical semiring, addition serves as the _multiplicative_
+operation! This is even confusing to write, so I'm going to switch up notation;
+in the rest of this section, I'll use \\(\\otimes\\) to represent the
+multiplicative operation in semirings, and \\(\\oplus\\) to represent the
+additive one. The symbols \\(\\times\\) and \\(+\\) will be used to represent
+the regular operations on real numbers. With that, the operations on our
+tropical semiring over real numbers are defined as follows:
+
+{{< latex >}}
+\begin{array}{rcl}
+x \otimes y & \triangleq & x + y\\
+x \oplus y  & \triangleq & \min(x,y)
+\end{array}
+{{< /latex >}}
+
+What is this new semiring good for? How about this: suppose that in addition to
+the duration of the trip, you'd like to track the distance you must travel for
+each route (shorter routes do sometimes have more traffic!). Let's watch what
+happens when we add and multiply polynomials over this semiring.
+When we add terms with the same power but different coefficients, like
+\\(ax\oplus bx\\), we end up with a term \\(\min(a,b)x\\). In other words,
+for each trip duration, we pick the shortest length. When we multiply two
+polynomials, like \\(ax\otimes bx\\), we get \\((a+b)x\\); in other words,
+when sequencing two trips, we add up the distances to get the combined
+distance, just like we'd expect.
+
+We can, of course, come up with a polynomial to match our initial example.
+Say that the trips from A to B are represented by \\(2.0x^2\oplus1.5x\\\) (the
+shortest two-hour trip is \\(2\\) units of distance long, and the one-hour
+trip is \\(1.5\\) units long), and that the trips from B to C are represented
+by \\(4.0x^3\oplus1.0x^2\\). Multiplying the two polynomials out gives:
+
+{{< latex >}}
+\begin{array}{rcl}
+(2.0x^2\oplus1.5x)(4.0x^3\oplus1.0x^2) & = & 6.0x^5 \oplus \min(2.0+1.0, 1.5+4.0)x^4 \oplus 2.5x^3 \\
+    & = & 6.0x^5 \oplus 3.0x^4 \oplus 2.5x^3
+\end{array}
+{{< /latex >}}
+
+The only time we used the additive operation in this case was to pick between
+two trips of equal druation but different length (two-hour trip from A to B
+followed by a two-hour trip from B to C, or one-hour trip from A to C followed
+by a three-hour trip from B to C). The first trip wins out, since it requires
+only \\(3.0\\) units of distance.