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alleles from continued testing against the environment. Dominance provides just such shielding. |
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To introduce dominance, we must extend the method of representation once again. Pairs of alleles will be used for each detector, so that a representation involves a pair of homologous l-tuples. The object is to let some of the extra alleles be carried along with the others in an unexpressed form, forming a kind of reservoir of protected alleles. Precisely, then, the set of representations will be extended to the set of all permutations of homologous pairs  . Since there is now a pair of alleles at each position there is no longer a direct correspondence between the detector values for a structure A and the representation of A. |
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Let (A', A") be a homologous pair of l-tuples drawn from  and let i(A', A") = df((h, v'), (h, v")) where  , designate the pair of alleles occurring at the ith position of the l-tuples. The most direct way to relate this pair of l-tuples to a structure is to designate either v' or else v" as the value of detector h, ignoring the other allele. The allele so designated will be called dominant, the other recessive. For each position i, this designation should be completely determined by information available in the pair (A', A''). Formally, for each i there should be a dominance map  such that, for each  is either the first allele or the second allele of i(A', A"). It should be emphasized that in this general form, the determination of the dominant allele in i(A', A") may depend upon the whole context (i.e., the other alleles in (A', A")). (This corresponds closely with Fisher's [1937, Chapter III] theory of dominance.) A simpler approach makes the determination dependent only upon the pair i(A', A") itself. Thus, for each h, |
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Accordingly (A', A") represents the structure  for which |
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where i(h) is the index of the pair of alleles in (A', A") for detector h. |
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A particularly interesting example of the simpler dominance map, useful for binary (two allele) codings (see chapter 4), can be constructed as follows. Let Vh = {1, 10, 0}, where 10 is to be recessive whenever it is paired with 0, and let the mapping  be given by the following table: |
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