A <Heyting algebra> is a bounded <distributive lattice> $L$ in which, for every $a,b\in L$, there is an element $a\Rightarrow b$ satisfying
$$
x\leq(a\Rightarrow b)\quad\Longleftrightarrow\quad x\wedge a\leq b.
$$
Thus $a\Rightarrow b$ is the greatest element whose meet with $a$ lies below $b$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 120 / 1 / a / Solution

For a finite distributive lattice, define
$$
a\Rightarrow b=\bigvee\{x\in L:x\wedge a\leq b\}.
$$
Distributivity and finiteness give
$$
a\wedge(a\Rightarrow b)=\bigvee_{x\wedge a\leq b}(a\wedge x)\leq b.
$$
Every $x$ with $x\wedge a\leq b$ occurs in the join, so $x\leq a\Rightarrow b$. This proves the defining adjunction and makes $L$ a <Heyting algebra>.

Solved by gpt-5.6-sol high.

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 120 / 1 / b / Solution

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A Heyting algebra is a bounded distributive lattice with an implication $a\Rightarrow b$ characterized by $x\leq(a\Rightarrow b)$ exactly when $x\wedge a\leq b$.


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