On the Use of Algebra in Genetics II: Shannon’s Genetic Algebra, from Population to Sample Studies

Even before the discovery of DNA, Claude Shannon developed a mathematical model of Mendelian inheritance. Unlike the widely recognized Hardy–Weinberg equilibrium, Shannon’s genetic algebra has received little scholarly attention. Here, we revisit Shannon’s algebra and develop two complementary extensions for modern population genetics. First, we formulate a finite population version of Shannon’s framework, moving beyond the idealized infinite population setting to propagate allelic and genotypic frequencies under realistic sampling conditions. Second, we combine Shannon’s algebra with an analytical genotype–phenotype mapping framework to characterize genotypic configurations compatible with observed phenotypic frequencies, using auxiliary variables to express the inherent degeneracy of the genotype–phenotype relationship. Together, these extensions provide a unified framework in which phenotypic observations constrain the underlying genetic structure, and these constraints are propagated through inheritance via Shannon’s algebra. The resulting analytical expressions for offspring phenotypic distributions apply to both complete and incomplete penetrance and extend naturally to multilocus systems. This work highlights Shannon’s algebra as a flexible analytical tool for two complementary problems: (i) forward propagation of genetic information in finite populations, and (ii) analytical description of phenotypic inheritance from inferred genotypic information.