# Differences

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mv-algebras [2011/07/17 12:12]
jipsen
mv-algebras [2018/10/20 08:48] (current)
jipsen
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====Definition==== ====Definition====
-A \emph{bounded hoop} is an algebra $\mathbf{A}=\langle A, \cdot, \to, 0, 1\rangle$ such that+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 $\langle A, \cdot, \to, 1\rangle$ is a hoop
+
+$(x\to y)\to y = (y\to x)\to x$
$0\to x=1$ $0\to x=1$
Remark: Remark:
-Bounded hoops are term-equivalent to Wajsberg algebras via $x\cdot y=\neg(x\to\neg y)$, $0=\neg 1$, and $\neg x=x\to 0$.+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. See [(BP1994)] 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 $\neg x= - x$.
====Examples==== ====Examples====
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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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^$n$       | 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 | ^$n$       | 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 |
-^# of algs | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 2 |  2 |  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
-^# 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  |+^# 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 |
+
+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
====Subclasses==== ====Subclasses====
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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>