## Name of class

Abbreviation: **TarskiA**

### Definition

A ** 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.

### 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

Trace: » tarski_algebras