Hilbert algebras

Abbreviation: HilA

Definition

A Hilbert algebra is a structure $\mathbf{A}=\langle A,\to,1\rangle$ of type $\langle 2, 1\rangle$ such that

$x\to(y\to x)=1$

$(x\to(y\to z))\to((x\to y)\to(x\to z))=1$

$x\to y=1\mbox{ and }y\to x=1 \Longrightarrow x=y$

Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be Hilbert algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\to y)=h(x)\to h(y)$ and $h(1)=1$.

Definition

A Hilbert algebra is a structure $\mathbf{A}=\langle A,\to,1\rangle$ of type $\langle 2, 1\rangle$ such that

$x\to x=1$

$1\to x=x$

$x\to(y\to z)=(x\to y)\to(x\to z)$

$(x\to y)\to((y\to x)\to x)=(y\to x)\to((x\to y)\to y)$

Examples

Example 1: Given any poset with top element 1, $\langle A,\le, 1\rangle$, define $a\to b=\begin{cases}1&\text{ if $a\le b$}\\ b&\text{ otherwise.}\end{cases}$ Then $\langle A,\to,1\rangle$ is a Hilbert algebra.

Basic results

Hilbert algebras are the algebraic models of the implicational fragment of intuitionistic logic, i.e., they are $(\to,1)$-subreducts of Heyting algebras.

The variety of Hilbert algebras is not generated as a quasivariety by any of its finite members 1).

Properties

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

... subvariety

... expansion

Superclasses

... supervariety

... subreduct

References


1) S. Celani and L. Cabrer: Duality for finite Hilbert algebras. Discrete Math. 305 (2005), no. 1-3, 74-–99.
2) A. Diego, Sur les algébres de Hilbert, Collection de Logique Math\'ematique, S\'er. A, 1966, 1–55