example of a non-lattice homomorphism

Consider the Hasse diagram of the lattice of subgroups of the quaternion groupof order $8$ , $Q_{8}$ . [The use of $Q_{8}$ is only for a concrete realization of the lattice.]

\\xy<5mm,0mm>:<0mm,5mm>::(0,3)+*{Q_{8}}="Q8";(-2,2)+*{\\langle i\\rangle}="i";(0,%2)+*{\\langle j\\rangle}="j";(2,2)+*{\\langle k\\rangle}="k";(0,1)+*{\\langle-1%\\rangle}="-1";(0,0)+*{\\langle 1\\rangle}="1";"1";"-1"**@{-};"-1";"i"**@{-};"-1"%;"j"**@{-};"-1";"k"**@{-};"i";"Q8"**@{-};"j";"Q8"**@{-};"k";"Q8"**@{-};

To establish an order-preserving map which is not a lattice isomorphismone can simply “skip” $\\langle-1\\rangle$ , which we display graphically as:

\\xy<5mm,0mm>:<0mm,5mm>::(-3,3)+*{Q_{8}}="Q81";(-5,2)+*{\\langle i\\rangle}="i1";%(-3,2)+*{\\langle j\\rangle}="j1";(-1,2)+*{\\langle k\\rangle}="k1";(-3,1)+*{%\\langle-1\\rangle}="-11";(-3,0)+*{\\langle 1\\rangle}="11";(3,2.5)+*{Q_{8}}="Q82"%;(1,1.5)+*{\\langle i\\rangle}="i2";(3,1.5)+*{\\langle j\\rangle}="j2";(5,1.5)+*{%\\langle k\\rangle}="k2";(3,0.5)+*{\\langle-1\\rangle}="-12";(3,-0.5)+*{\\langle 1%\\rangle}="12";"11";"-11"**@{-};"-11";"i1"**@{-};"-11";"j1"**@{-};"-11";"k1"**@%{-};"i1";"Q81"**@{-};"j1";"Q81"**@{-};"k1";"Q81"**@{-};"12";"-12"**@{-};"-12";%"i2"**@{-};"-12";"j2"**@{-};"-12";"k2"**@{-};"i2";"Q82"**@{-};"j2";"Q82"**@{-}%;"k2";"Q82"**@{-};"Q81";"Q82"**@{..};"i1";"i2"**@{..};"j1";"j2"**@{..};"k1";"k%2"**@{..};"-11";"12"**@{..};"11";"12"**@{..};

Since containment is still preserved the map is order-preserving. However, the intersection (meet) of $\\langle i\\rangle$ and $\\langle j\\rangle$ , which is $\\langle-1\\rangle$ , is not perserved under thismap. Thus it is not a lattice homomorphism.