# What's the meaning of $\mathbb{N}_0,\wedge$

What's the meaning of the $\wedge$ in $\mathbb{N}_0,\wedge$?

The context is an exercise about the properties of mathematical operations.

I'm aware of $\wedge$ as a logic operator, but I'm quite sure that's not meant here.

I've found lots and lots of pages about a $\wedge$- or external product on vectorspaces, but I don't think I should interpret $\mathbb{N}_0$ as a vectorspace?

I've also found $a\wedge b = \min(a,b)$ (in this question). That would make sense to me. Is there a logical explanation why this should the correct interpretation?

• As I know $\mathbb N_0$ is the set of natural number with $0$. – user63181 Dec 17 '13 at 18:45
• The logical explanation is that it just depends on the context. If you gave some context maybe someone could verify this for you. – parsiad Dec 17 '13 at 18:46
• @longtom For the french notation we denote $\mathbb N=\{0,1,2,\cdots\}$ but I saw this notation $\mathbb N_0=\{0,1,2,\cdots\}$ here. – user63181 Dec 17 '13 at 18:48
• @longtom One doesn't need to look hard for that. – Daniel Fischer Dec 17 '13 at 18:50
• @DanielFischer The nice thing is it's compatible with the partial order notation: gcd is the "min", and lcm the "max". The notation is confirmed on the french wiki: fr.wikipedia.org/wiki/Plus_grand_commun_diviseur#Notations – Jean-Claude Arbaut Dec 17 '13 at 18:55

The symbol $\wedge$ is borrowed from the theory of lattices. Normally there is also $\vee$. They are called meet and join, respectively. They are each binary, infix, operations, that satisfy certain lattice axioms. Two familiar lattices on $\mathbb{N}$ are:

1. $\wedge$ denotes gcd (greatest common divisor), while $\vee$ denotes lcm (least common multiple).

2. $\wedge$ denotes min, while $\vee$ denotes max.

Absent any context it is impossible to tell which lattice on $\mathbb{N}$ is being discussed. If $\mathbb{N}_0$ includes $0$, we must exclude the first possibility; but in either case there are many possible lattices.

• So the question, without context, is ambiguous? – long tom Dec 17 '13 at 19:02
• In Belgium $\mathbb{N}_0$ means $0$ is excluded. That might suggest the first possibility is meant? – long tom Dec 17 '13 at 19:03
• I would say the question is ambiguous, since there are many possible lattices on $\mathbb{N}_0$. For example, one could take the non-standard total order $0<2<4<6<\cdots<1<3<5<\cdots$. – vadim123 Dec 17 '13 at 19:03

In some programming languages, $\wedge$ means the bitwise xor function. That is, $a\wedge b$ means computing the sum of $a$ and $b$ in base two without carry: so $2^k \wedge 2^k = 0$, $2^k \wedge 2^l = 2^k + 2^l$ if $k \ne l$, and $\wedge$ is commutative and associative.