Conditions Equivalent to De Morgan's Law for M-Sets

De Morgan has two logical laws: (1) (sic)(p V q)= (sic)p Lambda (sic)q (2) (sic)(p Lambda q) = (sic)p V (sic) q In intuitionistic logic the second is more interesting and it is well known that (1) is true in any Heyting algebra and hence in the subobject classifier of any topos, but (2) is not generally true. Johnstone gives some equivalent or related conditions to (2), some of which are well-known theorems of classical mathematics for distributive lattices, Boolean algebras, and commutative rings. Here, we consider the second De Morgan's law in the topos of 34-sets, that is for the Heyting algebra of all left ideals of a monoid M, and prove sonic more equivalent conditions to it.