Properties
The subdirect representation theorem of universal algebra states that every algebra is subdirectly representable by its subdirectly irreducible quotients. An equivalent definition of "subdirect irreducible" therefore is any algebra A that is not subdirectly representable by those of its quotients not isomorphic to A. (This is not quite the same thing as "by its proper quotients" because a proper quotient of A may be isomorphic to A, for example the quotient of the semilattice (Z, min) obtained by identifying just the two elements 3 and 4.)
An immediate corollary is that any variety, as a class closed under homomorphisms, subalgebras, and direct products, is determined by its subdirectly irreducible members, since every algebra A in the variety can be constructed as a subalgebra of a suitable direct product of the subdirectly irreducible quotients of A, all of which belong to the variety because A does. For this reason one often studies not the variety itself but just its subdirect irreducibles.
An algebra A is subdirectly irreducible if and only if it contains two elements that are identified by every proper quotient, equivalently, if and only if its lattice Con A of congruences has a least nonidentity element. That is, any subdirect irreducible must contain a specific pair of elements witnessing its irreducibility in this way. Given such a witness (a,b) to subdirect irreducibility we say that the subdirect irreducible is (a,b)-irreducible.
Given any class C of similar algebras, Jónsson's Lemma states that the subdirect irreducibles of the variety HSP(C) generated by C lie in HS(CSI) where CSI denotes the class of subdirectly irreducible quotients of the members of C. That is, whereas one must close C under all three of homomorphisms, subalgebras, and direct products to obtain the whole variety, it suffices to close the subdirect irreducibles of C under just homomorphic images (quotients) and subalgebras to obtain the subdirect irreducibles of the variety.
Read more about this topic: Subdirectly Irreducible Algebra
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)