Luís Sequeira

We show that every finite affine algebra A admits a full duality. In the process, we prove that A also allows a strong duality, and that the duality may be induced by a dualizing structure $\underset{\sim }{A}$ of finite type. We give an explicit bound on the arities of the partial and total operations appearing in $\underset{\sim }{A}$. In addition, we show that the enriched partial hom-clone of A is finitely generated as a clone.

The "Modularity Conjecture" is the assertion that the join of two nonmodular varieties is nonmodular. We establish the veracity of this conjecture for the case of linear idempotent varieties. We also establish analogous results concerning $n$-permutability for some $n$, and the satisfaction of nontrivial congruence identities. Our theorems require a technical result about the equational theory of linear varieties, which might be of independent interest.