example of a non-lattice homomorphism

Consider the Hasse diagram of the lattice of subgroups of the quaternion group of 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 isomorphism one 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 this map. Thus it is not a lattice homomorphism.

Title	example of a non-lattice homomorphism
Canonical name	ExampleOfANonlatticeHomomorphism
Date of creation	2013-03-22 16:58:31
Last modified on	2013-03-22 16:58:31
Owner	Algeboy (12884)
Last modified by	Algeboy (12884)
Numerical id	8
Author	Algeboy (12884)
Entry type	Example
Classification	msc 06B23