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Hilbert algebras

Abbreviation: HilA


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

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

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

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

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.


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 $h(1)=1$.


An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$


Example 1:

Basic results


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


1) A. Diego, Sur les algbres de Hilbert, Collection de Logique Math\'ematique, S\'er. A, 1966, 1–55 MRreview