Heyting algebras

Abbreviation: HA


A \emph{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 $\wedge$: $x\wedge y\leq z\Longleftrightarrow y\leq x\to z$


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

$x\wedge y=x\cdot y$


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:

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


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.


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

Values known up to size 49 1)




1) Marcel Ern\'e;, Jobst Heitzig and J\“urgen Reinhold,\emph{On the number of distributive lattices}, Electron. J. Combin., \textbf{9}2002,Research Paper 24, 23 pp. (electronic)MRreview

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