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Are these just two terms for the same phenomenon, i.e., the state of being homozygous? Merriam-Webster says so, but I know dictionaries sometimes miss the nuance of scientific terms. If they are indeed synonyms, it seems homozygosity has won out, according to Ngrams
Homozygosis and homozygosity are NOT synonymous. The two terms refer to the same concept though.
Homozygote
An indivdidual or a cell is said to be homozygote for a particular gene (or any DNA sequence) when identical alleles of the gene are present on both homologous chromosomes. (Wikipedia > Zygosity).
Homozygosis
Homozygosis is the state of being homozygote (Wright 1933) for a particular gene (or any DNA sequence). According to freedictionary.com, homozygosis may also refer to the process of formation of a homozygote. I have never encountered this sense in the literature though. As you found out by yourself, Homozygosis is a term that is rarely used in the literature (in comparison to homozygote and homozygosity)
Expected Homozygosity
Expected Homozygosity is the probability of two randomly sampled alleles (in a population and at a given locus) to be identical by descent.
Expected homozygosity and Expected heterozygosity
As shown in Nei 1973, the mathematical expression of expected homozygosity $Ho$ is
$$Ho = sum_{a=1}^{n} x_a^2$$
where $n$ is the number of alleles and $x_a$ is the frequency of the allele $a$.
It is more common to use the concept of expected heterozygosity $He$ (a.k.a. gene diversity) rather than the one of expected homozygosity $Ho$. By definition, $He=1-Ho$, that is
$$He =1- sum_{a=1}^{n} x_a^2$$
In the special case of a bi-allelic locus, and replacing the two allele frequencies $x=1$ and $x=2$ by $p$ and $q=1-p$, the above formulations becomes
$$Ho = p^2 + q^2$$ $$He = 1 - (p^2 + q^2) = 2pq$$
Hardy-Weinberg equilibrium
Under the Hardy-Weinberg assumptions, the observed frequency of heterozygote (a.k.a. observed heterozygosity) is equal to the expected heterozygosity and observed frequency of homozygotes (a.k.a. observed homozygosity) is equal to the expected homozygosity.
