Subdirectly Irreducible Algebra

In the branch of mathematics known as universal algebra (and in its applications), a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirect product of "simpler" algebras. Subdirectly irreducible algebras play a somewhat analogous role in algebra to primes in number theory.

Read more about Subdirectly Irreducible Algebra:  Definition, Examples, Properties, Applications

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Subdirectly Irreducible Algebra - Applications
... A necessary and sufficient condition for a Heyting algebra to be subdirectly irreducible is for there to be a greatest element strictly below 1 ... chain of two or more elements as a Heyting algebra is subdirectly irreducible ... By Jónsson's Lemma the subdirect irreducibles of the variety generated by a class of subdirect irreducibles are no larger than the generating subdirect ...

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