Subdirect product
Encyclopedia
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. The notion was introduced by Birkhoff
in 1944 and has proved to be a powerful generalization of the notion of direct product.
(in the sense of universal algebra
) A of a direct product
ΠiAi such that every induced projection (the composite pjs: A → Aj of a projection pj: ΠiAi → Aj with the subalgebra inclusion s: A → ΠiAi) is surjective.
A direct (subdirect) representation of an algebra A is a direct (subdirect) product isomorphic to A.
An algebra is called subdirectly irreducible if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the areas of abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
known as universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, and module theory, a subdirect product is a subalgebra
Subalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
of a direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
that depends fully on all its factors without however necessarily being the whole direct product. The notion was introduced by Birkhoff
Garrett Birkhoff
Garrett Birkhoff was an American mathematician. He is best known for his work in lattice theory.The mathematician George Birkhoff was his father....
in 1944 and has proved to be a powerful generalization of the notion of direct product.
Definition
A subdirect product is a subalgebraSubalgebra
In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
(in the sense of universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
) A of a direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
ΠiAi such that every induced projection (the composite pjs: A → Aj of a projection pj: ΠiAi → Aj with the subalgebra inclusion s: A → ΠiAi) is surjective.
A direct (subdirect) representation of an algebra A is a direct (subdirect) product isomorphic to A.
An algebra is called subdirectly irreducible if it is not subdirectly representable by "simpler" algebras. Subdirect irreducibles are to subdirect product of algebras roughly as primes are to multiplication of integers.
Examples
- Any 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...
L is subdirectly representable as a subalgebra of a direct power of the two-element distributive lattice. This can be viewed as an algebraic formulation of the representability of L as a set of sets closed under the binary operations of union and intersection, via the interpretation of the direct power itself as a power set. In the finite case such a representation is direct (i.e. the whole direct power) if and only if L is a complemented latticeComplemented latticeIn the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0....
, i.e. a Boolean algebra. - The same holds for any 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...
when "semilattice" is substituted for "distributive lattice" and "subsemilattice" for "sublattice" throughout the preceding example. That is, every semilattice is representable as a subdirect power of the two-element semilattice. - The chain of natural numbers together with infinity, as a Heyting algebra, is subdirectly representable as a subalgebra of the direct product of the finite linearly ordered Heyting algebras. The situation with other Heyting algebras is treated in further detail in the article on subdirect irreducibleSubdirect irreducibleIn the branch of mathematics known as universal algebra , a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirect product of "simpler" algebras...
s. - The group of integers under addition is subdirectly representable by any (necessarily infinite) family of arbitrarily large 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...
s. In this representation, 0 is the sequence of identity elements of the representing groups, 1 is a sequence of generators chosen from the appropriate group, and integer addition and negation are the corresponding group operations in each group applied coordinate-wise. The representation is faithful (no two integers are represented by the same sequence) because of the size requirement, and the projections are onto because every coordinate eventually exhausts its group. - Every vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a given field is subdirectly representable by the one-dimensional space over that field, with the finite-dimensional spaces being directly representable in this way. (For vector spaces, as for 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...
s, direct product with finitely many factors is synonymous with direct sum with finitely many factors, whence subdirect product and subdirect sum are also synonymous for finitely many factors.) - Subdirect products are used to represent many small perfect groupPerfect groupIn mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
s in .