2 Varieties
2.1 Filtered varieties
A variety is an additive submonoid of \(\mathtt{Chromosome}\).
Given a predicate \(p\) on genes, the filtered variety \(V_p\) consists of all chromosomes whose support satisfies \(p\).
If \(p\) is lift-stable (i.e. \(p(g)\) iff \(p(\mathrm{lift}(g))\)), then the prime of a filtered variety is itself: \(V_p' = V_p\).
2.2 The polarized variety Pi
A chromosome \(X\) is polarized if every gene in its support has type \(\ne \mathtt{NonPolarized}\).
The variety \(\mathrm{Pi}\) is the filtered variety of all polarized chromosomes.
\(\mathrm{Pi}' = \mathrm{Pi}\).
If \(X \in \mathrm{Pi}\), the signature components of \(X\) are natural numbers.
If \(X, Y \in \mathrm{Pi}\) satisfy \(\mathrm{sig}(X^{(k)}) = \mathrm{sig}(Y^{(k)})\) for all \(k \ge 0\), then \(X = Y\).
For \(X, Y \in \mathrm{Pi}\), if \(X \le Y\) and \(Y \le X\), then \(X = Y\). Hence \(\mathrm{Pi}\) carries a partial order.
2.3 The non-polarized variety Lambda
A chromosome is non-polarized if every gene in its support has type \(\mathtt{NonPolarized}\).
The variety \(\mathrm{Lambda}\) is the filtered variety of all non-polarized chromosomes.
\(\mathrm{Lambda}' = \mathrm{Lambda}\).
2.4 Mixed varieties and labels
Given a pair of varieties \((V_1, V_2)\), the mixed variety \(\mathrm{Mix}(V_1, V_2)\) consists of all chromosomes whose even part lies in \(V_1\) and odd part lies in \(V_2\).
If the two component varieties are closed under taking odd and even parts, then prime swaps the two entries of a mixed variety: \(\mathrm{Mix}(V_1,V_2)'=\mathrm{Mix}(V_2',V_1')\).
The five labeled varieties \(V_0, \ldots , V_4\) indexed by \(\mathrm{Fin}\, 5\) are:
\(V_0 = \mathrm{Pi}\)
\(V_1 = \mathrm{Mix}(\mathrm{Lambda}, \mathrm{Pi})\)
\(V_2 = \mathrm{Mix}(\mathrm{Pi}, \mathrm{Lambda})\)
\(V_3 = \mathrm{Mix}(2\mathrm{Lambda}, \mathrm{Pi})\)
\(V_4 = \mathrm{Mix}(\mathrm{Pi}, 2\mathrm{Lambda})\)
The prime operation on varieties induces a permutation of the five labels: \(V_i' = V_{\pi (i)}\).
For every label \(i\), the variety prime of \(V_i\) is exactly \(V_{\pi (i)}\). Iterating this equality tracks membership after iterated prime operations.