"Unclosure" on a set with binary operation

Question

I was wondering if there is any usefulness to having a set that has no closure under a particular operation. For example, the set of prime numbers, $\mathbb{P}$ along with multiplication of integers generates every composite number. Is there a name for this property and does any branch of mathematics study this kind of thing? I was just lying in bed thinking about this, tried to find something somewhere on-line but had no luck...

Answer 1

Just to give a contrasting answer, in sympathy with Ilya's comments:

If $S$ is some algebraic structure, like the natural numbers under multiplication, or integers under addition, or [insert huge number of other examples here], and $X$ is a subset of $S$ which generates the elements of $S$ under the the relevant operation(s) on $S$, then we say that $X$ is a generating set.

The study of generating sets of algebraic objects is a big part of algebra and related areas.

E.g. in ring theory, studying small, or easy-to-work-with, generating sets of ideals is important; the theory of Grobner bases is part of this.

E.g in group theory, the geometric approach via Cayley graphs is related to trying to find good generating sets (whose "goodness" is encoded by the geometric properties of the Cayley graph that they give rise to).

E.g. to cite one of your examples, the fact that the primes generate the positive integers under multiplication, together with the uniqueness of prime factorizations, and the countable infinitude of primes, says that, under multiplication, the positive integers are a free commutative monoid on countably many generators.

There are many other situations where one has monoids on countably many generators, e.g. if you have a countable number of axioms, and you join these axioms together to make proofs of theorems, the "joining" process is an operation which (roughly) allows one to model the collection of theorems by (certain of) the elements of the free monoid on countably many generators. (Certain here means that only the elements corresponding to valid proofs are allowed.) Going back to the encoding of this free monoid as the set of positive integers, via primes and prime factorizations, one can then label these proofs by positive integers. This is the Godel numbering process that allowed Godel to encode meta-mathematics (proofs) in terms of mathematics itself (positive integers).

So it is certainly useful to know that the positive integers have this simple algebraic structure under multiplication!

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A somewhat different idea is that of category theory. A category is something like a monoid --- it has an associative multiplication law --- but only certain objects can be multiplied together. Every element has a certain "source" and "target" as auxiliary data attached to it, and a multiplication $ab$ is only possible if the source of $a$ equals the target of $b$.

If we implement the same idea, but replacing monoid by group, we generalize the concept of group to obtain the concept of groupoid. Groupoids are useful in many contexts. They appear in algebraic topology (the fundamental groupoid) and also in lots of contexts as a generalization of the concept of equivalence relation (so-called stacks).

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Actually, the way that in logic, only certain concatenations of axioms are valid --- if you try to concatenate two syllogisms to get a valid inference, the conclusion of the first must agree with the premise of the second --- and the way that in category theory, only certain products are allowed, are not totally unrelated.

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In any case, there are lots of instances of such "unclosed" structures in mathematics, both in the theory of generating sets of usual structures, and in various contexts (proofs in logic, categories, groupoids, ... ) where those structures are generalized.

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One more thing: these concepts can be combined: one can, and people do, consider generating sets of categories.

Often it happens that the usual, closed, structure $S$ is big, and hard to hold in your head all at once, while the generating set $X$ might be much smaller and more manageable (in some sense). So one makes a trade-off: one focuses on the generating set, taking advantage of its smallness to make computations, or arguments, or whatever, more manageable, but knowing that, since it is not closed under the relevant operations, it is only a shadow of the whole structure $S$, the rest of whose elements may be a somewhat hidden beneath the surface.

Answer 2

There are lots of examples:

• The odd integers under addition: the sum of two odd number is even.
• The products of matrices with $\det(M) = -1$. They will always have $\det(MN)=+1$.

The only thing that we seem to get from this is the clue that we've missed something.