Can I safely say that the set of finite and cofinite subsets of the integers equipped with operations of union and intersection is isomorphic to the direct product of countably infinitely many $\mathbb Z_2$?
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Let $\mathscr{S}$ be the set of subsets of the integers that are either finite or cofinite. $\langle\mathscr{S},\cup,\cap\rangle$ is not a ring: $\varnothing$ is an identity for $\cup$, and $\mathbb{Z}$ is an identity for $\cap$, but neither operation has inverses. Thus, $\langle\mathscr{S},\cup,\cap\rangle$ cannot be isomorphic to any ring; in particular, it cannot be isomorphic to the direct sum $$\bigoplus_{n\in\mathbb{N}}G_n\;,$$ where each $G_n$ is a copy of $\mathbb{Z}_2$. It cannot be isomorphic to the direct product $\mathbb{Z}_2^\omega$ for an even more fundamental reason: it’s countable, and $\mathbb{Z}_2^\omega$ is uncountable. |
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