## MV-algebras

Abbreviation: MV

### Definition

An \emph{MV-algebra} (short for \emph{multivalued logic algebra}) is a structure $\mathbf{A}=\langle A, +, 0, \neg\rangle$ such that

$\langle A, +, 0\rangle$ is a commutative monoid

$\neg \neg x=x$

$x + \neg 0 = \neg 0$

$\neg(\neg x+y)+y = \neg(\neg y+x)+x$

Remark: This is the definition from 1)

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be MV-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:

$h(x+y)=h(x)+h(y)$, $h(\neg x)=\neg h(x)$, $h(0)=0$

### Definition

An \emph{MV-algebra} is a structure $\mathbf{A}=\langle A, +, 0, \cdot, 1, \neg\rangle$ such that

$\langle A, \cdot, 1\rangle$ is a commutative monoid

$\neg$ is a DeMorgan involution for $+,\cdot$: $\neg \neg x=x$, $x+y=\neg ( \neg x\cdot \neg y)$

$\neg 0=1$, $0\cdot x=0$, $\neg ( \neg x+y) +y=\neg ( \neg y+x) +x$

### Definition

An \emph{MV-algebra} is a basic logic algebra $\mathbf{A}=\langle A,\vee,0,\wedge,1,\cdot,\to\rangle$ that satisfies

MV: $x\vee y=(x\to y)\to y$

### Definition

A \emph{Wajsberg algebra} is an algebra $\mathbf{A}=\langle A, \to, \neg, 1\rangle$ such that

$1\to x=x$

$(x\to y)\to((y\to z)\to(x\to z) = 1$

$(x\to y)\to y = (y\to x)\to x$

$(\neg x\to\neg y)\to(y\to x)=1$

Remark: Wajsberg algebras are term-equivalent to MV-algebras via $x\to y=\neg x+y$, $1=\neg 0$ and $x + y=\neg x\to y$, $0=\neg 1$.

### Definition

A \emph{bounded Wajsberg hoop} is an algebra $\mathbf{A}=\langle A, \cdot, \to, 0, 1\rangle$ such that

$\langle A, \cdot, \to, 1\rangle$ is a hoop

$(x\to y)\to y = (y\to x)\to x$

$0\to x=1$

Remark: Bounded Wajsberg hoops are term-equivalent to Wajsberg algebras via $x\cdot y=\neg(x\to\neg y)$, $0=\neg 1$, and $\neg x=x\to 0$. See 2) for details.

### Definition

A \emph{lattice implication algebra} is an algebra $\mathbf{A}=\langle A, \to, -, 1\rangle$ such that

$x\to (y\to z) = y\to (x\to z)$

$1\to x = x$

$x\to 1 = 1$

$x\to y = {-}y\to {-}x$

$(x\to y)\to y = (y\to x)\to x$

Remark: Lattice implication algebras are term-equivalent to MV-algebras via $x + y = -x\to y$, $0 = -1$, and $\neg x= - x$.

Example 1:

### Properties

Classtype variety decidable decidable (FEP3)) no unbounded yes yes yes, $n=2$ yes, $e=1$ yes no yes 4)

### Finite members

 $n$ # of algs # of si's 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 1 1 2 1 2 1 3 2 2 1 4 1 2 2 5 1 4 1 4 2 2 1 7 2 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

The number of algebras with $n$ elements is given by the number of ways of factoring $n$ into a product with nontrivial factors, see http://oeis.org/A001055

### References

1) Roberto L. O. Cignoli, Itala M. L. D'Ottaviano, Daniele Mundici, \emph{Algebraic foundations of many-valued reasoning}, Trends in Logic—Studia Logica Library \textbf{7} Kluwer Academic Publishers 2000, x+231
2) W. J. Blok, D. Pigozzi, \emph{On the structure of varieties with equationally definable principal congruences. III}, Algebra Universalis, \textbf{32} 1994, 545–608
3) W. J. Blok, I. M. A. Ferreirim, \emph{On the structure of hoops}, Algebra Universalis, \textbf{43} 2000, 233–257
4) Daniele Mundici, \emph{Bounded commutative BCK-algebras have the amalgamation property}, Math. Japon., \textbf{32} 1987, 279–282