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

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March 19

Divisibility lattice

Is the lattice $(\mathbb {Z} _{\geq 0},|)$ 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)