Subdirect irreducible
Encyclopedia
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
.
A is said to be subdirectly irreducible when A has more than on element, and when any subdirect representation
of A includes (as a factor) an algebra isomorphic to A.
states that every algebra is subdirectly representable by its subdirectly irreducible quotient
s. 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 congruence
s 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.
By Jónsson's Lemma the subdirect irreducibles of the variety generated by a class of subdirect irreducibles are no larger than the generating subdirect irreducibles, since the quotients and subalgebras of an algebra A are never larger than A itself. Hence the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra H must be just the nondegenerate quotients of H, namely all smaller linearly ordered nondegenerate Heyting algebras.
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
(and in its applications), a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirect product
Subdirect product
In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product...
of "simpler" algebras. Subdirectly irreducible algebras play a somewhat analogous role in algebra to primes
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
.
Definition
A universal algebraUniversal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
A is said to be subdirectly irreducible when A has more than on element, and when any subdirect representation
Subdirect product
In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product...
of A includes (as a factor) an algebra isomorphic to A.
Examples
- The two-element chain, as either a Boolean algebra, a Heyting algebraHeyting algebraIn mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
, a latticeLattice (order)In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, or a semilatticeSemilatticeIn mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
, is subdirectly irreducible. In fact, a distributive latticeDistributive latticeIn mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
is subdirectly irreducible if and only if it has exactly two elements. - Any finite chain with two or more elements, as a Heyting algebraHeyting algebraIn mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
, 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 groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order a power of a prime (i.e. any finite p-groupP-groupIn mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
) 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 groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
is subdirectly irreducible if and only if it is isomorphic to a finite p-group or isomorphic to a Prüfer groupPrüfer groupIn mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...
(an infinite but countable p-group, which is the direct limitDirect limitIn mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
of its finite p-subgroups). - A vector space is subdirectly irreducible if and only if it has dimension one.
Properties
The subdirect representation theorem of universal algebraUniversal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
states that every algebra is subdirectly representable by its subdirectly irreducible quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
s. 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
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
, 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 congruence
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...
s 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.
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. The witnessing pair is that element and 1, and identifying any other pair a, b of elements identifies both a→b and b→a with 1 thereby collapsing everything above those two implications to 1. Hence every finite 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 irreducibles, since the quotients and subalgebras of an algebra A are never larger than A itself. Hence the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra H must be just the nondegenerate quotients of H, namely all smaller linearly ordered nondegenerate Heyting algebras.