nLab
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, $A$;

• a binary operation, $\oplus$, on $A$;

• a unary operation, $\neg$, on $A$; and a constant, $0$, such that

1. $⟨A,\oplus ,0⟩$ is a commutative monoid;

2. $\neg \neg x=x$ for all $x\in A$;

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

4. $\neg (\neg x\oplus y)\oplus y=\neg (\neg y\oplus x)\oplus x$ for all $x,y$ in $A$.

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

Example

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

$$x\oplus y=\min (1,x+y)$$x\oplus y = min(1,x+y)

$$\neg x=1-x.$$\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: $x<y$ and $y<x$ 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 $x\vee y$.

Properties

Define $x\odot y≔\neg (\neg x\oplus \neg y)$, and define $x\Rightarrow y≔\neg x\oplus y$.

Proposition 1

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

$$x\vee y=(x\odot \neg y)\oplus y$$x \vee y = (x \odot \neg y) \oplus y

$$x\wedge y=x\odot (\neg x\oplus y)$$x \wedge y = x \odot (\neg x \oplus y)

and where $0$ and $1=\neg 0$ are the bottom and top elements.

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

Proposition 2

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

$$x\odot y\le z\mathrm{iff}x\le y\Rightarrow z.$$x \odot y \leq z \qquad iff \qquad x \leq y \Rightarrow z.

In particular, for all elements $x$ and $y$ we have $1=y\Rightarrow y$, $1\Rightarrow y=y$, and $x\le (x\Rightarrow y)\Rightarrow y$.

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

Proposition 3

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

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

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

Completeness theorem

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

References

• D. Mundici, MV-algebras, a short tutorial, available here.