nLab MV-algebra

Contents

Idea

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

Definition

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

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

Examples

  • The unit interval [0,1][0, 1] is an MV-algebra where xyx \oplus y is defined as max(x+y,1)\max(x + y, 1) and ¬x\neg x is defined as 1x1 - x.

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

  • Any singleton 11 is a trivial MV-algebra where xyx \oplus y is defined as the unique binary function on 11 and ¬x\neg x defined as the unique unary function on 11, the identity function on 11. It is also the terminal MV-algebra in the category of MV-algebras and MV-algebra homomorphisms.

References

Last revised on June 6, 2024 at 10:23:15. See the history of this page for a list of all contributions to it.