I want to define an algebra, where there are three elements: 0, 1 and $\infty$ and two operations, addition and multiplication defined, both commutative:
$$0+0=0$$ $$0+1=1$$ $$0+\infty=\infty$$ $$1+1=1$$ $$1+\infty=\infty$$ $$\infty+\infty=\infty$$ $$0\times0=0$$ $$1\times0=0$$ $$1\times1=1$$ $$0\times\infty=1$$ $$1\times\infty=\infty$$ $$\infty\times\infty=\infty$$$$\begin{align*} 0+0&=0\\ 0+1&=1\\ 0+\infty&=\infty\\ 1+1&=1\\ 1+\infty&=\infty\\ \infty+\infty&=\infty\\ 0\times0&=0\\ 1\times0&=0\\ 1\times1&=1\\ 0\times\infty&=1\\ 1\times\infty&=\infty\\ \infty\times\infty&=\infty \end{align*}$$
I want Mathematica to simplify expressions in this system.
