Extending a complete lattice to get a "nice" Boolean lattice - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:44:03Z http://mathoverflow.net/feeds/question/83144 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83144/extending-a-complete-lattice-to-get-a-nice-boolean-lattice Extending a complete lattice to get a "nice" Boolean lattice CSstudent 2011-12-10T20:47:37Z 2011-12-29T11:18:45Z <p>Suppose we have a complete lattice. Which additional axioms (e.g. distributivity axioms) are needed to obtain a Boolean lattice in which complement(a) = lub{b | b /\ a = bottom} = glb{b | b \/ a = top}?</p> http://mathoverflow.net/questions/83144/extending-a-complete-lattice-to-get-a-nice-boolean-lattice/84504#84504 Answer by Goldstern for Extending a complete lattice to get a "nice" Boolean lattice Goldstern 2011-12-29T11:17:46Z 2011-12-29T11:17:46Z <p>Since you are asking about universally quantified equations, there is only the trivial answer: $x=y$. </p> <p>Theorem: <strong>Assume</strong> that $\varphi$ is a (finite or infinite) list of universally quantified equations (sometimes called "laws" or "identities") such that every complete lattice satisfying $\varphi$ will have complements. <strong>Then</strong> there is no nontrivial lattice satisfying $\varphi$, or in other words, the equations $\varphi$ imply $(\forall x,y): x=y$. </p> <p>Proof: From a nontrivial lattice satisfying $\varphi$ I will construct a complete lattice satisfying $\varphi$ which is not Boolean. </p> <p>All the following lattices will satisfy $\varphi$: </p> <p>(a) Some nontrivial lattice with at least 2 points $p &lt; q$. </p> <p>(b) The 2-point Boolean lattice containing only $p &lt; q$. (A sublattice of (a)).</p> <p>(c) <code>$\{p,q\}^2$</code>, a 4-element Boolean lattice. </p> <p>(d) <code>$\{(p,p), (p,q), (q,q)\}$</code>, a sublattice of (c). This is a 3-element chain.</p> <p>The lattice in (d) is complete, satisfies $\varphi$, but is not Boolean. </p>