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A distributive residuated lattice is a residuated lattice $\mathbf{L}=\left\langle L,\vee ,\wedge ,\cdot ,e,\backslash ,/\right\rangle $ such that $\vee ,\wedge $ are distributive: $x\wedge \left( y\vee z\right) =\left( x\wedge y\right) \vee \left( x\wedge z\right) $ |
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A distributive residuated lattice is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that |
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$\vee, \wedge$ are distributive: $x\wedge(y\vee z) =(x\wedge y) \vee (x\wedge z)$ |
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$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)\$, $ h(e)=e$ |
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$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$ |
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