 Distributive lattices can be arbitrarily “thin.” At the extreme, any chain (linear or total order, \(e.g\). the reals standardly ordered) under the usual operations of max and min forms a distributive lattice. Since we have omitted the constants this includes the empty lattice, which we have not excluded here as an algebra. (Some authors disallow the empty set as an algebra but this proscription spoils many good theorems without gaining any useful ones.) Hence there exist distributive lattices of every possible cardinality.