## Heyting algebras

Abbreviation: HA

### Definition

A Heyting algebra is a structure $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\to \rangle$ such that

$\langle A,\vee ,0,\wedge ,1\rangle$ is a bounded distributive lattice

$\to$ gives the residual of $\wedge$: $x\wedge y\leq z\Longleftrightarrow y\leq x\to z$

### Definition

A Heyting algebra is a FLew-algebra $\mathbf{A}=\langle A,\vee ,0,\wedge ,1,\cdot ,\to \rangle$ such that

$x\wedge y=x\cdot y$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be Heyting algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(0)=0$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\to y)=h(x)\to h(y)$

### Examples

Example 1: The open sets of any topological space $\mathbf X$ form a Heyting algebra under the operations of union $\cup$, empty set $\emptyset$, intersection $\cap$, whole space $X$, and the operation $U\to V=$ interior of $(X - U)\cup V$.

Example 2: Any frame can be expanded to a unique Heyting algebra by defining $x\to y = \bigvee\{z:x\wedge z\le y\}$.

### Basic results

Any finite distributive lattice is the reduct of a unique Heyting algebra. More generally the same result holds for any complete and completely distributive lattice.

A Heyting algebra is subdirectly irreducible if and only if it has a unique coatom.

### Properties

Classtype variety decidable decidable undecidable no unbounded yes yes yes, $n=2$ yes, $e=1$ no yes yes yes yes yes yes

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &3\\ f(6)= &5\\ f(7)= &8\\ f(8)= &15\\ f(9)= &26\\ f(10)= &47\\ f(11)= &82\\ f(12)= &151\\ f(13)= &269\\ f(14)= &494\\ f(15)= &891\\ f(16)= &1639\\ f(17)= &2978\\ f(18)= &5483\\ f(19)= &10006\\ f(20)= &18428\\ \end{array}$

Values known up to size 49 1)

### References

1) Marcel Ern\'e;, Jobst Heitzig and J\”urgen Reinhold,On the number of distributive lattices, Electron. J. Combin., 92002,Research Paper 24, 23 pp. (electronic)MRreview