Subdirectly Irreducible Algebra - 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 ab 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.

Read more about this topic:  Subdirectly Irreducible Algebra

Other articles related to "example, examples":

Free Good
... For example, a shop might give away its stock in its promotion, but producing these goods would still have required the use of scarce resources ... Examples include ideas and works that are reproducible at zero cost, or almost zero cost ... For example, if someone invents a new device, many people could copy this invention, with no danger of this "resource" running out ...
English Orthography - Spelling Irregularities - "Ough" Words
... though /oʊ/ as in toe (other examples dough) tough /ʌf/ as in cuff (other examples rough, enough) cough /ɒf/ as in off (other examples Gough (name, some pronunciations ...
Proverbs in Advertising
... Tatira has given a number of examples of proverbs used in advertising in Zimbabwe ... However, unlike the examples given above in English, all of which are anti-proverbs, Tatira's examples are standard proverbs ... meant to make a potential customer smile, in one of the Zimbabwean examples "both the content of the proverb and the fact that it is phrased as a proverb secure the idea of a secure time-hono ...

Famous quotes containing the word examples:

    There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.
    Bernard Mandeville (1670–1733)

    In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.
    Michel de Montaigne (1533–1592)

    No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.
    André Breton (1896–1966)