3 deleted 1 characters in body

First failed attempt (this poset is not a lattice)

triangle inequality fails for the three .5 nodes in the middle

Second attempt Our lattice consists of sets, with intersection and union. I show them by Venn diagrams here... The same three middle sets fail the triangle inequality for the same reasons as before. But now it is surely a lattice, right?

2 second attempt

First failed attempt (this poset is not a lattice)

triangle inequality fails for the three .5 nodes in the middle

Second attempt Our lattice consists of sets, with intersection and union. I show them by Venn diagrams here... The same three middle sets fail the triangle inequality for the same reasons as before. But now it is surely a lattice, right?

1

triangle inequality fails for the three .5 nodes in the middle