MV-algebra

Remarks

Example: the unit interval $[0, 1]$ with $x \oplus y = \min(1, x + y)$, $\neg x = 1 - x$ and $0$ is an MV-algebra. Clauses 1 to 3 are direct, since $\neg 0 = 1$. For clause 4, if $y \ge x$ the left side is $\neg 1 \oplus y = y$, and if $y < x$ it is $(x - y) \oplus y = x$; so both sides equal $\max(x, y)$.

Non-example: the integers $(\Z, +, -, 0)$ satisfy clauses 1 and 2, but fail clause 3, since $1 + (-0) = 1 \ne 0$.

Every Boolean algebra is an MV-algebra with $\oplus = \vee$ and $\neg$ the complement. The constant $\neg 0$ is written $1$, and $x \odot y = \neg(\neg x \oplus \neg y)$ is the dual operation.