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We are given some (finite) set of primes $P=\{p_1, p_2, \ldots, p_n\}$. Define the following two sets:

  1. $S$ is the set of all integers that can be generated from $P$ by multiplying members of any subset of $P$, possibly by repeating some of them. That is: $$S=\{z|z=\prod^{n}_{i=1}p_i^{x_i}\wedge {x_i}\mbox{ is a non-negative integer}\}$$

  2. $T$ is the set of all rational numbers that can be generated from $P$ by multiplying and dividing members of any subset of $P$, possibly by repeating some of them. That is: $$T=\{z|z=\prod^{n}_{i=1}p_i^{x_i}\wedge {x_i}\mbox{ is an integer}\}$$

Is there any name given to $S$ and $T$?

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2 Answers 2

up vote 5 down vote accepted

$S$ is the submonoid of $(\mathbb Z^*, \cdot)$ generated by $P$.

$T$ is the subgroup of $(\mathbb Q^*, \cdot)$ generated by $P$.

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Don't they both require an identity element? –  Jus12 Jan 15 '13 at 6:00
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in your definition they both contain 1.... –  N. S. Jan 15 '13 at 6:01
    
oh yes, my mistake. :) –  Jus12 Jan 15 '13 at 6:02

A set with a binary operation (like multiplication) which is associative (parentheses can be dropped between multiplications) and closed under multiplication is generally called a semigroup. If it also contains an identity element (here it is $1$), then the semigroup is a monoid. If, furthermore, every element has an inverse (the inverse of $a$ is $\frac{1}{a}$), then the monoid becomes a group.

In particular, $\mathbb{Z}$ with the usual multiplication is a monoid, and $\mathbb{Q}$ with the usual multiplication is a group.

The key word here is "generated."

If $M$ is a monoid, the set $B$ is generated by a subset $A\subseteq M$ if the elements in $B$ are precisely the products of those elements in $A$ (including the empty product, which is taken to be the identity). It's not difficult to show that $B$ is itself a monoid, and so is called a submonoid of $M$.

Similarly, f $G$ is a group, the set $B$ is generated by a subset $A\subseteq G$ if the elements in $B$ are precisely the products of those elements in $A$ and their inverses (including the empty product). As before, $B$ is a group and so a subgroup of $G$.

Thus, we apply these definitions to give us the names:

$S$ is "the submonoid (of $\mathbb{Z}$) generated by $P$".

$T$ is "the subgroup (of $\mathbb{Q}$) generated by $P$."

As far as I have seen, we don't have better names for these.

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