Page 16 - Textos de Matemática Vol. 41

2 Chapter 1. Preliminaries
1.2.1. left (resp. right) adjoints preserve suprema (resp. infima) and if the posets in question are complete lattices, each map preserving all suprema is a left ad- joint and similarly the maps preserving infima are right adjoints.
1.3. Heyting algebras. A Heyting algebra is a lattice with an extra operation → satisfying
a∧b≤c iff a≤b→c. (H) Thus, each map (−) ∧ b is a Galois left adjoint and hence a Heyting algebra is
a distributive lattice. Moreover, whenever i∈J ai exists we have ( ai) ∧ b = (ai ∧ b),
i∈J i∈J and the right adjoint a→(−) results in
a→ bi = (a→bi). i∈J i∈J
From (H) we can immediately infer the contravariant adjunction a≤b→c iff b≤a→c
which yields the equation
(f-distr)
(-distr) (H*)
( ai)→b = (ai →b). i∈J i∈J
(-distr) 1.3.1. A few more Heyting formulas. In this paragraph, the use of the formula
(H) is mostly automatic.
Proposition. In any Heyting algebra L we have
(H1) a→(b ∧ c) = (a→b) ∧ (a→c),
(H2) there is a largest element 1, and a→a = 1 and 1→a = a for all a, (H3) a≤b iff a→b=1,
(H4) a ≤ b→a,
(H5) a→b = a→(a ∧ b),
(H6) a∧(a→b)=a∧b,
(H7) a∧b=a∧c iff a→b=a→c,
(H8) (a ∧ b)→c = a→(b→c) and similarly a→(b→c) = b→(a→c),