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. $\langle A, \oplus, 0\rangle$ 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

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

$$\neg x = 1-x.$$

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: $x \lt y$ and $y \lt x$ 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 $x\vee y$.

Properties

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

Proposition 1

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

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

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

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

Let $\leq$ be the partial ordering for the lattice structure, where $x \leq y$ 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

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

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

In fact, an MV algebra is a $\ast$-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

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

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.

A recent preprint is

• Mark V. Lawson, Philip Scott, AF inverse monoids and the structure of countable MV-algebras, arXiv:1408.1231