Summary: We define a lattice order on a set \(F\) of binary functions. We then provide necessary and sufficient conditions for the resulting algebra \({\mathfrak L}_F\) to be a distributive lattice or a Boolean algebra. We also prove a ‘Cayley theorem’ for distributive lattices by showing that for every distributive lattice \({\mathfrak L}\), there is an algebra \({\mathfrak L}_F\) of binary functions such that \({\mathfrak L}\) is isomorphic to \({\mathfrak L}_F\), and we show that \({\mathfrak L}_F\) is a distributive lattice iff the operations \(\vee\) and \(\wedge\) are idempotent and commutative, showing that this result cannot be generalized to non-distributive lattices or quasilattices without changing the definitions of \(\vee\) and \(\wedge\). We also examine the equational properties of an Algebra \({\mathfrak U}\) for which \({\mathfrak L}_{\mathfrak U}\), now defined on the set of binary \({\mathfrak U}\)-polynomials, is a lattice or Boolean algebra.