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## 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

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

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

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

##### 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 $a\to b=\left\{\begin{array}{ll}1&\text{ if$a\le b}\\b&\text{ otherwise}\end{array}\right.$. Then$\langle A,\to,1\rangle$is a Hilbert algebra. ====Basic results==== Hilbert algebras are the algebraic models of the implicational fragment of [[wp>intuitionistic logic]], i.e., they are$(\to,1)$-subreducts of [[Heyting algebras]]. The variety of Hilbert algebras is not generated as a quasivariety by any of its finite members [(CelaniCabrer2005)]. ====Properties==== ^[[Classtype]] |variety [(Diego1966)] | ^[[Equational theory]] | | ^[[Quasiequational theory]] | | ^[[First-order theory]] | | ^[[Locally finite]] | | ^[[Residual size]] | | ^[[Congruence distributive]] | | ^[[Congruence modular]] | | ^[[Congruence$n$-permutable]] | | ^[[Congruence regular]] | | ^[[Congruence uniform]] | | ^[[Congruence extension property]] | | ^[[Definable principal congruences]] | | ^[[Equationally def. pr. cong.]] | | ^[[Amalgamation property]] | | ^[[Strong amalgamation property]] | | ^[[Epimorphisms are surjective]] | | ====Finite members====$\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}\$

... subvariety

... expansion

... supervariety

... subreduct