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

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype variety

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