# Differences

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mv-algebras [2011/07/17 13:10] jipsen |
mv-algebras [2016/11/23 14:17] (current) jipsen |
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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==== |

Trace: