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Is there an example of cancellative Abelian monoid $M$ in which we may find two elements $x$ and $y$ such that they have a least common multiple but not a greatest common divisor?

W4cc0
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2 Answers2

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Let $a,b \in M$ and $m \in M$ be a least common multiple of $a$ and $b$. As $ab$ is a common multiple of $a$ and $b$, we must have $m \mid ab$, say $md = ab$ for some $d \in M$. Now let $t \in M$ be any common divisor of $a$ and $b$, say $a = st$, $b = rt$, define $m' = rst$. Then $m'$ is a common multiple of $a$ and $b$, so for some $d' \in M$ we have $m' = d'm$. Now we have $$ d'tm = rst^2 = ab = dm$$ As $M$ is cancellative, this implies $d't = d$. So $t$ is a divisor of $d$. As $t$ was arbitrary, $d$ is a greatest common divisor of $a$ and $b$.

Addendum: To see that $d$ divides $a$ and $b$, note that $m$ is a multiple of $a$ and $b$, say $m = aa' = bb'$ for some $a', b' \in M$. Then we have $$ ab = md = aa'd, \quad ab = b'db $$ therefore by cancellation, $b = a'd$ and $a = b'd$.

martini
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Theorem $\,\ [a,b]\,$ exists $\,\Rightarrow\, (a,b) = ab/[a,b],\: $ for $\ (x,y)=\gcd(x,y),\,$ $\,[x,y] = {\rm lcm}(x,y)$

Proof $\ \ c\mid a,b \iff a,b\mid ab/c \iff [a,b]\mid ab/c\iff c\mid ab/[a,b]$

Bill Dubuque
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  • This doesn't use cancellativity? – goblin GONE May 04 '15 at 14:31
  • @goblin It does e.g. $,c\mid a,b\iff ca,cb\mid ab\iff a,b\mid a(b/c)=b(a/c),$ employs cancellation of $,c.,$ Also that the divisions are well-defined uses cancellation. – Bill Dubuque May 04 '15 at 14:49
  • This is very nice. Its also stuff I've always wanted to understand better. Can you recommend a good place to learn these kinds of things? Namely, divisibility, gcd's and lcm's in both arbitrary commutative monoids and also cancellative commutative monoids. – goblin GONE May 04 '15 at 14:56
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    @goblin The above proof uses gcd,lcm duality that arises from an involution (reflection) symmetry - see this answer and its links. Unfortunately I don't recall any places where this can be found (it is not in most textbooks). It might be presented in some textbooks on lattice theory (Birkhoff?). – Bill Dubuque May 04 '15 at 15:01
  • Thanks for the link. Really, someone should write a short guide to these kinds of basic facts about abelian monoids, perhaps aimed at those with knowledge of basic lattice theory but little knowledge of commutative algebra (hint hint)... I'll check out Birkhoff when I get the chance and let you know if I find anything. – goblin GONE May 04 '15 at 15:18
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    @goblin A prototypical commutative cancellative monoid is the multiplicative monoid of an integral domain. Any purely multiplicative domain-theoretic proof (e.g. above) automatically transfers to any commutative cancellative monoid. Many common properties of domains are purely multiplicative so can be described in terms of monoid structure. See here for some examples and literature references. – Bill Dubuque May 04 '15 at 15:28