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Each finite algebra $\mathbf A$ induces a lattice~$\mathbf L_{\mathbf A}$ via the quasi-order~$\to$ on the finite members of the variety generated by~$\mathbf A$, where $\mathbf B \to \mathbf C$ if there exists a homomorphism from $\mathbf B$ to~$\mathbf C$. In this paper, we introduce the question: `Which lattices arise as the homomorphism lattice $\mathbf L_{\mathbf A}$ induced by a finite algebra $\mathbf A$?' Our main result is that each finite distributive lattice arises as~$\mathbf L_{\mathbf Q}$, for some quasi-primal algebra~$\mathbf Q$. We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form $\mathbf L\oplus \mathbf 1$, where $\mathbf L$ is an interval in the subgroup lattice of a finite group.

We clarify what it means for two full dualities based on the same algebra to be different. Our main theorem gives conditions on two different alter egos of a finite algebra under which, if one yields a full duality, then the other does too. We use this theorem to obtain a better understanding of several important examples from the theory of natural dualities. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual classes.

We find all finite Ockham algebras that admit only finitely many compatible relations (modulo a natural equivalence). Up to isomorphism and symmetry, these Ockham algebras form two countably infinite families: one family consists of the quasi-primal Ockham algebras, and the other family is a sequence of generalised Stone algebras., Comment: 29 pages

We make a start on one of George McNulty's Dozen Easy Problems: "Which finite automatic algebras are dualizable?" We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain $\mathbf A_1 \le \mathbf A_2 \le \mathbf A_3 \le ...b$ of finite automatic algebras that are alternately dualizable and non-dualizable.