nLab MV-algebra

Idea

MV-algebras are the algebraic semantics for propositional Łukasiewicz logic.

Definition

An MV-algebra is a commutative monoid $(S, 0, \oplus)$ with an involution $x \mapsto \neg x:S \to S$ such that for all $x \in S$ and $y \in S$, $x \oplus \neg 0 = \neg 0$ and $\neg (\neg x \oplus y) \oplus y = \neg (\neg y \oplus x) \oplus x$.

An MV-algebra homomorphism between MV-algebras $A$ and $B$ is a function $f:A \to B$ such that $f(0) = 0$, for all $x \in A$, $\neg f(x) = f(\neg x)$, and for all $x \in A$ and $y \in A$, $f(x) \oplus f(y) = f(x \oplus y)$.

Examples

• The unit interval $[0, 1]$ is an MV-algebra where $x \oplus y$ is defined as $\max(x + y, 1)$ and $\neg x$ is defined as $1 - x$.

• Every Boolean algebra is a MV-algebra whose monoid operation is idempotent.

• Any singleton $1$ is a trivial MV-algebra where $x \oplus y$ is defined as the unique binary function on $1$ and $\neg x$ defined as the unique unary function on $1$, the identity function on $1$. It is also the terminal MV-algebra in the category of MV-algebras and MV-algebra homomorphisms.

Related concepts

• Łukasiewicz logic

References

• Wikipedia, MV-algebra