Wikipedia:Reference desk/Archives/Mathematics/2018 March 19

= March 19 =

Divisibility lattice
Is the lattice $$(\mathbb{Z}_{\ge 0}, |)$$ a Heyting algebra? Is the dual lattice a Heyting algebra? GeoffreyT2000 (talk) 03:54, 19 March 2018 (UTC)
 * No and no, because neither has a good negation. The negation must satisfy for every $$a$$ that $$a \wedge \neg a = 0$$, that the only element disjoint from $$a\vee \neg a$$ is 0, and that $$a \to b = \neg a \vee b$$ (here 0 refers to the 0 of the lattice, which is the number 1 for the divisibility lattice and the number 0 for the dual).
 * In the divisibility lattice, being disjoint is equivalent to being coprime. For an intermediate $$a$$, if $$ \neg a = 1$$ (the lattice 1, which is the number 0), then the first requirement fails, and if $$ \neg a \neq 1$$ then the second fails because of the infinitude of primes.
 * In the dual lattice, no two nonzero elements are disjoint, again because of the infinitude of primes. So to meet the first requirement, we would need $$\neg a = 0$$ for all nonzero $$a$$.  But then since join is g.c.d., it follows that $$a \to b = b$$.  But it's easy to see that this operation doesn't satisfy the definition of a Heyting algebra.--129.74.238.54 (talk) 16:59, 19 March 2018 (UTC)