Mathematical Structures: History of Distributive residuated lattices

History of Distributive residuated lattices

 Revision 5 . . July 8, 2004 2:22 pm by Jipsen Revision 4 . . July 8, 2004 2:20 pm by Jipsen Revision 3 . . (edit) April 19, 2003 10:21 pm by Peter Jipsen

Difference (from prior major revision) (author diff)

Changed: 32,36c32
 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: $x\wedge \left( y\vee z\right) =\left( x\wedge y\right) \vee \left( x\wedge z\right)$
 A distributive residuated lattice is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

 $\vee, \wedge$ are distributive: $x\wedge(y\vee z) =(x\wedge y) \vee (x\wedge z)$
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)\$, $h(e)=e$
 $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(e)=e$