Name of class

Abbreviation: TarskiA

Definition

A \emph{Tarski algebra} is a structure $\mathbf{A}=\langle A,\to\rangle$ of type $\langle 2\rangle$ such that $\to$ satisfies the following identities:

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

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

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

Morphisms

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

Examples

Example 1: $\langle\{0,1\},\to\rangle$ where $x\to y=0$ iff $x=1$ and $y=0$.

Basic results

Tarski algebras are the implication subreducts of Boolean algebras.

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


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