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mv-algebras [2010/07/29 18:30]
127.0.0.1 external edit
mv-algebras [2018/10/20 08:48] (current)
jipsen
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Abbreviation: **MV** Abbreviation: **MV**
 +
====Definition==== ====Definition====
An \emph{MV-algebra} (short for \emph{multivalued logic algebra}) is a An \emph{MV-algebra} (short for \emph{multivalued logic algebra}) is a
structure $\mathbf{A}=\langle A, +, 0, \neg\rangle$ such that structure $\mathbf{A}=\langle A, +, 0, \neg\rangle$ such that
- 
$\langle A, +, 0\rangle$ is a [[commutative monoid]] $\langle A, +, 0\rangle$ is a [[commutative monoid]]
- 
$\neg \neg x=x$ $\neg \neg x=x$
- 
$x + \neg 0 = \neg 0$ $x + \neg 0 = \neg 0$
- 
$\neg(\neg x+y)+y = \neg(\neg y+x)+x$ $\neg(\neg x+y)+y = \neg(\neg y+x)+x$
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==Morphisms== ==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be MV-algebras. A morphism from $\mathbf{A} 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: +$ to $\mathbf{B}$ is a function $h:A\to B$ that is a homomorphism:
$h(x+y)=h(x)+h(y)$, $h(\neg x)=\neg h(x)$, $h(0)=0$ $h(x+y)=h(x)+h(y)$, $h(\neg x)=\neg h(x)$, $h(0)=0$
 +
====Definition==== ====Definition====
An \emph{MV-algebra} is a An \emph{MV-algebra} is a
structure $\mathbf{A}=\langle A, +, 0, \cdot, 1, \neg\rangle$ such that structure $\mathbf{A}=\langle A, +, 0, \cdot, 1, \neg\rangle$ such that
- 
$\langle A, \cdot, 1\rangle$ is a [[commutative monoid]] $\langle A, \cdot, 1\rangle$ is a [[commutative monoid]]
 +$\neg $ is a DeMorgan involution for $+,\cdot $:  $\neg \neg x=x$, $x+y=\neg ( \neg x\cdot \neg y)$
-$\neg $ is a DeMorgan involution for $+,\cdot $:  $\neg \neg x=x$, $x+y=\neg ( \neg x\cdot \neg +$\neg 0=1$, $0\cdot x=0$, $\neg ( \neg x+y) +y=\neg ( \neg y+x) +x$
-y) $ +
-$\neg 0=1$, $0\cdot x=0$, $\neg ( \neg 
-x+y) +y=\neg ( \neg y+x) +x$ 
====Definition==== ====Definition====
An \emph{MV-algebra} is a [[basic logic algebra]] $\mathbf{A}=\langle An \emph{MV-algebra} is a [[basic logic algebra]] $\mathbf{A}=\langle
-A,\vee,0,\wedge,1,\cdot,arrow\rangle$ that satisfies+A,\vee,0,\wedge,1,\cdot,\to\rangle$ that satisfies 
 + 
 +MV:  $x\vee y=(x\to y)\to y$
-MV:  $x\vee y=(xarrow y)arrow y$ 
====Definition==== ====Definition====
-A \emph{Wajsberg algebra} is an algebra $\mathbf{A}=\langle A, arrow, \neg, 1\rangle$ such that+A \emph{Wajsberg algebra} is an algebra $\mathbf{A}=\langle A, \to, \neg, 1\rangle$ such that
 +$1\to x=x$
-$1arrow x=x$+$(x\to y)\to((y\to z)\to(x\to z) = 1$
 +$(x\to y)\to y = (y\to x)\to x$
-$(xarrow y)arrow ((yarrow z) arrow (xarrow z) = 1$+$(\neg x\to\neg y)\to(y\to x)=1$
 +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
-$(xarrow y)arrow y = (yarrow x)arrow x$+$\langle A, \cdot, \to, 1\rangle$ is a hoop
 +$(x\to y)\to y = (y\to x)\to x$
-$(\neg xarrow \neg y)arrow(yarrow x)=1$+$0\to x=1$
Remark: Remark:
-Wajsberg algebras are term-equivalent to MV-algebras via $xarrow y=\neg x+y$, $1=\neg 0$ and $x + y=\neg xarrow y$, $0=\neg 1$.+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 [(BP1994)] for details.
====Definition==== ====Definition====
-A \emph{bounded hoop} is an algebra $\mathbf{A}=\langle A, \cdot, arrow, 0, 1\rangle$ such that+A \emph{lattice implication algebra} is an algebra $\mathbf{A}=\langle A, \to, -, 1\rangle$ such that
-$\langle A, \cdot, arrow, 1\rangle$ is a hoop+$x\to (y\to z) = y\to (x\to z)$
-$0arrow x=1$+$1\to x = x$
-Remark:  +$x\to 1 = 1$
-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 [(BP1994)] for details.+
 +$x\to y = {-}y\to {-}x$
 +$(x\to y)\to y = (y\to x)\to x$
 +
 +Remark:
 +Lattice implication algebras are term-equivalent to MV-algebras via $x + y = -x\to y$, $0 = -1$, and $\neg x= - x$.
====Examples==== ====Examples====
Example 1: Example 1:
 +
====Basic results==== ====Basic results====
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^[[Classtype]]  |variety | ^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable | ^[[Equational theory]]  |decidable |
-^[[Quasiequational theory]]  | |+^[[Universal theory]]  |decidable (FEP[(BF2000)])|
^[[First-order theory]]  | | ^[[First-order theory]]  | |
^[[Locally finite]]  |no | ^[[Locally finite]]  |no |
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^[[Strong amalgamation property]]  | | ^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | | ^[[Epimorphisms are surjective]]  | |
 +
 +
====Finite members==== ====Finite members====
-$\begin{array}{lr} +^$n$       | 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 | 
-f(1)= &1\\ +^# of algs | 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 | 
-f(2)= &1\\ +^# of si's | 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 | 
-f(3)= &1\\ + 
-f(4)= &2\\ +The number of algebras with $n$ elements is given by the number of ways of factoring $n$ into a product with nontrivial factors, 
-f(5)= &1\\ +see http://oeis.org/A001055
-\end{array}$+
====Subclasses==== ====Subclasses====
[[Boolean algebras]] [[Boolean algebras]]
 +
====Superclasses==== ====Superclasses====
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====References==== ====References====
 +
 +[(BF2000>
 +W. J. Blok, I. M. A. Ferreirim,
 +\emph{On the structure of hoops},
 +Algebra Universalis,
 +\textbf{43} 2000, 233--257)]
[(BP1994> [(BP1994>
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\emph{On the structure of varieties with equationally definable principal congruences. III}, \emph{On the structure of varieties with equationally definable principal congruences. III},
Algebra Universalis, Algebra Universalis,
-\textbf{32} 1994, 545--608 [[MRreview]]+\textbf{32} 1994, 545--608)]
[(COM2000> [(COM2000>
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Trends in Logic---Studia Logica Library Trends in Logic---Studia Logica Library
\textbf{7} Kluwer Academic Publishers \textbf{7} Kluwer Academic Publishers
-2000, x+231 [[MRreview]]+2000, x+231)]
[(Mu1987> [(Mu1987>
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\emph{Bounded commutative BCK-algebras have the amalgamation property}, \emph{Bounded commutative BCK-algebras have the amalgamation property},
Math. Japon., Math. Japon.,
-\textbf{32} 1987, 279--282  +\textbf{32} 1987, 279--282)]
-[[MRreview]] +
-)]+