Subdirectly Irreducible Algebra - Examples

Examples

  • The two-element chain, as either a Boolean algebra, a Heyting algebra, a lattice, or a semilattice, is subdirectly irreducible. In fact, a distributive lattice is subdirectly irreducible if and only if it has exactly two elements.
  • Any finite chain with two or more elements, as a Heyting algebra, is subdirectly irreducible. (This is not the case for chains of three or more elements as either lattices or semilattices, which are subdirectly reducible to the two-element chain. The difference with Heyting algebras is that a → b need not be comparable with a under the lattice order even when b is.)
  • Any finite cyclic group of order a power of a prime (i.e. any finite p-group) is subdirectly irreducible. (One weakness of the analogy between subdirect irreducibles and prime numbers is that the integers are subdirectly representable by any infinite family of nonisomorphic prime-power cyclic groups, e.g. just those of order a Mersenne prime assuming there are infinitely many.) In fact, an abelian group is subdirectly irreducible if and only if it is isomorphic to a finite p-group or isomorphic to a Prüfer group (an infinite but countable p-group, which is the direct limit of its finite p-subgroups).
  • A vector space is subdirectly irreducible if and only if it has dimension one.

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