## MV-algebras

Abbreviation: MV

### Definition

An MV-algebra (short for 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 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 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 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 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 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, Algebraic foundations of many-valued reasoning, Trends in Logic—Studia Logica Library 7 Kluwer Academic Publishers 2000, x+231
2) W. J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. III, Algebra Universalis, 32 1994, 545–608
3) W. J. Blok, I. M. A. Ferreirim, On the structure of hoops, Algebra Universalis, 43 2000, 233–257
4) Daniele Mundici, Bounded commutative BCK-algebras have the amalgamation property, Math. Japon., 32 1987, 279–282