nLab
MV algebras

MV algebras

Idea

An MV-algebra is an algebraic structure which models Luzasiewicz multivalued logic, and the fragment of that calculus which deals with the basic logical connectives “and”, “or”, and “not”, but in a multivalued context.

Definitions

An MV-algebra consists of

  • a non-empty set, AA;

  • a binary operation, \oplus, on AA;

  • a unary operation, ¬\neg, on AA; and a constant, 00, such that

  1. A,,0\langle A, \oplus, 0\rangle is a commutative monoid;

  2. ¬¬x=x\neg\neg x = x for all xAx \in A;

  3. x¬0=¬0x\oplus \neg 0 = \neg 0 for all xAx\in A; and

  4. ¬(¬xy)y=¬(¬yx)x\neg(\neg x\oplus y)\oplus y = \neg(\neg y\oplus x)\oplus x for all x,yx, y in AA.

The last axiom is more difficult to interpret but is clarified by some examples.

Example

Let A=[0,1]A = [0,1] be the unit interval. Define

xy=min(1,x+y)x\oplus y = min(1,x+y)
¬x=1x.\neg x = 1-x.

This gives a commutative monoid easily enough and the double negation and absorption axioms are easy to check. Finally the last axiom divides into two cases: x<yx \lt y and y<xy \lt x and only one of these needs checking!

Example

Any Boolean algebra defines a MV-algebra with =\oplus = \vee, and ¬\neg being the complement operation. The expressions in the last axiom evaluate to xyx\vee y.

Properties

Define xy¬(¬x¬y)x \odot y \coloneqq \neg (\neg x \oplus \neg y), and define xy¬xyx \Rightarrow y \coloneqq \neg x \oplus y.

Proposition 1

Each MV algebra carries a lattice structure, where the join and meet operations are defined by

xy=(x¬y)yx \vee y = (x \odot \neg y) \oplus y
xy=x(¬xy)x \wedge y = x \odot (\neg x \oplus y)

and where 00 and 1=¬01 = \neg 0 are the bottom and top elements.

Let \leq be the partial ordering for the lattice structure, where xyx \leq y if x=xyx = x \wedge y.

Proposition 2

Each MV algebra is a residuated lattice?, i.e., a closed monoidal poset, where the monoidal product is \odot and

xyziffxyz.x \odot y \leq z \qquad iff \qquad x \leq y \Rightarrow z.

In particular, for all elements xx and yy we have 1=yy1 = y \Rightarrow y, 1y=y1 \Rightarrow y = y, and x(xy)yx \leq (x \Rightarrow y) \Rightarrow y.

In fact, an MV algebra is a *\ast-autonomous poset, where ¬x=x0\neg x = x \Rightarrow 0.

Proposition 3

The equational variety of MV algebras is a Mal'cev variety, where the Mal’cev operation is defined by

t(x,y,z)=((xy)z)((zy)x).t(x, y, z) = ((x \Rightarrow y) \Rightarrow z) \wedge ((z \Rightarrow y) \Rightarrow x).

In other words, t(x,y,y)=x=t(y,y,x)t(x, y, y) = x = t(y, y, x).

Completeness theorem

The variety of MV algebras is generated by [0,1][0, 1] with its standard MV algebra structure. Consequently, an identity holds for all MV algebras if and only if it holds in [0,1][0, 1].

References

  • D. Mundici, MV-algebras, a short tutorial, available here.
Revised on February 14, 2013 20:37:46 by Anonymous Coward (151.40.32.252)