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Following this question I was asking myself if (in a cancellative Abelian Monoid $M$) given three elements $a,\, b,\, c$ for which there exists the least common multiple $m$, it will also exists the greatest common divisor.

Is it maybe possible to find a Cancellative Abelian monoid in which there are three elements with a lcm but without a gcd?

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W4cc0
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  • Note that (think of the naturals) for three elements $a,b,c \in M$ we cannot hope that from $abc = dm$, $d$ is the greatest common divisor, instead we want something like $abc = d^2m$. – martini May 06 '15 at 07:26
  • I see. Do you think some result could be obtained? I will change the question. – W4cc0 May 06 '15 at 07:41

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