This is an old revision of the document!

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 $h:Aarrow B$ 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 $\mathbf{A}=\langle A,\vee,0,\wedge,1,\cdot,arrow\rangle$ that satisfies

MV: $x\vee y=(xarrow y)arrow y$

Definition

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

$1arrow x=x$

$(xarrow y)arrow ((yarrow z) arrow (xarrow z) = 1$

$(xarrow y)arrow y = (yarrow x)arrow x$

$(\neg xarrow \neg y)arrow(yarrow x)=1$

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

Definition

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

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

$0arrow x=1$

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

Example 1:

Properties

Classtype variety decidable no unbounded yes yes yes, $n=2$ yes, $e=1$ yes no yes 3)

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ \end{array}$

References

2) W. J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. III, Algebra Universalis, 32 1994, 545–608 MRreview [(COM2000> 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 MRreview [(Mu1987> Daniele Mundici, Bounded commutative BCK-algebras have the amalgamation property, Math. Japon., 32 1987, 279–282 MRreview