mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, especially in the areas of

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...

known as

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...

group theory In 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 ...

, 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.

Definition

A subdirect product is a subalgebra (in the sense of

) ''A'' of a direct product Π''_iA_i'' such that every induced projection (the composite ''p_js'': ''A'' → ''A_j'' of a projection ''p''_''j'': Π''_iA_i'' → ''A_j'' with the subalgebra inclusion ''s'': ''A'' → Π''_iA_i'') is

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

. A direct (subdirect) representation of an algebra ''A'' is a direct (subdirect) product isomorphic to ''A''. An algebra is called

subdirectly irreducible In the branch of mathematics known as universal algebra (and in its applications), a subdirectly irreducible algebra is an universal algebra, algebra that cannot be factored as a subdirect product of "simpler" algebras. Subdirectly irreducible algeb ...

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 lattice In mathematics, a distributive lattice is a lattice in 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 ...

''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 lattice In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element ''a'' has a complement, i.e. an element ''b'' satisfying ''a'' ∨ ''b''& ...

, i.e. a Boolean algebra. * The same holds for any

semilattice In mathematics, a join-semilattice (or upper semilattice) is a partially ordered set that has a join (a least upper bound) for any nonempty finite subset. Dually, a meet-semilattice (or lower semilattice) is a partially ordered set which has a ...

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 In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...

, 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 irreducibles. * The

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of integers under addition is subdirectly representable by any (necessarily infinite) family of arbitrarily large finite

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

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 space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...

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 group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...

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 group In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universa ...

s in .

References

* * {{Citation , last1=Holt , first1=Derek F. , last2=Plesken , first2=W. , title=Perfect groups , publisher=The Clarendon Press Oxford University Press , series=Oxford Mathematical Monographs , isbn=978-0-19-853559-1 , mr=1025760 , year=1989 , url-access=registration , url=https://archive.org/details/perfectgroups0000holt Universal algebra

Definition

Examples

See also

References