1 Genes and Chromosomes
1.1 Gene types and genes
A gene type is one of three polarization types: \(\mathtt{NonPolarized}\), \(\mathtt{Positive}\), or \(\mathtt{Negative}\). There is a natural involution \(\varepsilon \mapsto -\varepsilon \) that swaps \(\mathtt{Positive}\) and \(\mathtt{Negative}\) and fixes \(\mathtt{NonPolarized}\).
A gene \(g = (n, \varepsilon )\) consists of a rank \(n \in \mathbb {N}\) and a type \(\varepsilon \in \mathtt{GeneType}\).
The signature of a gene \(g = (n, \varepsilon )\) is a pair \((a, b) \in \mathbb {Q} \times \mathbb {Q}\) defined by:
If \(\varepsilon = \mathtt{NonPolarized}\): \((a,b) = (n/2,\, n/2)\).
If \(\varepsilon = \mathtt{Positive}\): \((a,b) = (\lceil n/2 \rceil ,\, \lfloor n/2 \rfloor )\).
If \(\varepsilon = \mathtt{Negative}\): \((a,b) = (\lfloor n/2 \rfloor ,\, \lceil n/2 \rceil )\).
For any gene \(g\), we have \(a(g) + b(g) = \mathrm{rank}(g)\).
1.2 Chromosomes
A chromosome is a finitely supported function \(X \colon \mathtt{Gene} \to \mathbb {N}\), i.e. a formal sum of genes with non-negative integer multiplicities. We write \(\mathtt{Chromosome} = \mathtt{Gene} \to _0 \mathbb {N}\).
The signature of a chromosome \(X\) is the additive extension: \(\mathrm{sig}(X) = \sum _{g} X(g) \cdot \mathrm{sig}(g) \in \mathbb {Q} \times \mathbb {Q}\).
The rank of a chromosome \(X\) is \(\mathrm{rank}(X) = \sum _g X(g) \cdot \mathrm{rank}(g)\).
For any chromosome \(X\), the sum of the two signature components equals the rank: \(\mathrm{sig}(X).1 + \mathrm{sig}(X).2 = \mathrm{rank}(X)\).
For chromosomes \(X\) and \(Y\), we say \(X\) dominates \(Y\) (written \(Y \le X\)) if \(\mathrm{sig}(X^{(k)}) \le \mathrm{sig}(Y^{(k)})\) componentwise for all \(k \ge 0\), where \(X^{(k)}\) denotes the \(k\)-th derivative.
1.3 Prime and lift operations
The prime (derivative) operation \(X \mapsto X'\) is the additive homomorphism that maps a gene \((n, \varepsilon )\) with \(n \ge 1\) to \((n-1, -\varepsilon )\), and sends genes of rank \(0\) to \(0\).
The lift operation is an additive homomorphism mapping a gene \((n, \varepsilon )\) to \((n+1, -\varepsilon )\).
The prime operation is a left inverse to lift: \((\mathrm{lift}\, X)' = X\) for all \(X\).
If \(X \ne 0\), then \(\mathrm{maxRank}(X') {\lt} \mathrm{maxRank}(X)\).
1.4 Decompositions
For \(k \in \mathbb {N}\), \(X_{\le k}\) (below) filters genes with rank \(\le k\), and \(X_{{\gt} k}\) (above) filters genes with rank \({\gt} k\). We have \(X = X_{\le k} + X_{{\gt} k}\).
Every chromosome decomposes as \(X = X_{\mathrm{odd}} + X_{\mathrm{even}}\) where \(X_{\mathrm{odd}}\) (resp. \(X_{\mathrm{even}}\)) collects genes of odd (resp. even) rank.
\((X')_{\mathrm{even}} = (X_{\mathrm{odd}})'\) and \((X')_{\mathrm{odd}} = (X_{\mathrm{even}})'\).