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  "body": "\n## Statement\n\nOne of the two De Morgan laws is satisfied in every Heyting algebra, namely\n\n$$\\forall x,y \\in H: \\qquad \\lnot(x \\vee y)=\\lnot x \\wedge \\lnot y.$$\n\nHowever, the other De Morgan law does not always hold. We have instead a weak de Morgan law:\n\n$$\\forall x,y \\in H: \\qquad \\lnot(x \\wedge y)= \\lnot \\lnot (\\lnot x \\vee \\lnot y).$$\n\nThe following statements are equivalent for all Heyting algebras *H*:\n\n1. *H* satisfies both De Morgan laws,\n1. $$\\lnot(x \\wedge y)=\\lnot x \\vee \\lnot y \\mbox{ for all } x, y \\in H,$$\n1. $$\\lnot(x \\wedge y)=\\lnot x \\vee \\lnot y \\mbox{ for all regular } x, y \\in H,$$\n1. $$\\lnot\\lnot (x \\vee y) = \\lnot\\lnot x \\vee \\lnot\\lnot y \\mbox{ for all } x, y \\in H,$$\n1. $$\\lnot\\lnot (x \\vee y) = x \\vee y \\mbox{ for all regular } x, y \\in H,$$\n1. $$\\lnot(\\lnot x \\wedge \\lnot y) = x \\vee y \\mbox{ for all regular } x, y \\in H,$$\n1. $$\\lnot x \\vee \\lnot\\lnot x = 1 \\mbox{ for all } x \\in H.$$\n\nCondition 2 is the other De Morgan law. Condition 6 says that the join operation ∨\u003csub\u003ereg\u003c/sub\u003e on the Boolean algebra *H*\u003csub\u003ereg\u003c/sub\u003e of regular elements of *H* coincides with the operation ∨ of *H*. Condition 7 states that every regular element is complemented, i.e., *H*\u003csub\u003ereg\u003c/sub\u003e = *H*\u003csub\u003ecomp\u003c/sub\u003e.\n\nWe prove the equivalence. Clearly the metaimplications 1 ⇒ 2, 2 ⇒ 3 and 4 ⇒ 5 are trivial. Furthermore, 3 ⇔ 4 and 5 ⇔ 6 result simply from the first De Morgan law and the definition of regular elements. We show that 6 ⇒ 7 by taking ¬*x* and ¬¬*x* in place of *x* and *y* in 6 and using the identity *a* ∧ ¬*a* = 0. Notice that 2 ⇒ 1 follows from the first De Morgan law, and 7 ⇒ 6 results from the fact that the join operation ∨ on the subalgebra *H*\u003csub\u003ecomp\u003c/sub\u003e is just the restriction of ∨ to *H*\u003csub\u003ecomp\u003c/sub\u003e, taking into account the characterizations we have given of conditions 6 and 7. The metaimplication 5 ⇒ 2 is a trivial consequence of the weak De Morgan law, taking ¬*x* and ¬*y* in place of *x* and *y* in 5.\n\nHeyting algebras satisfying the above properties are related to De Morgan logic in the same way Heyting algebras in general are related to intuitionist logic.\n",
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