On the number of finite algebras

We understand an algebra as a set A together with a set of finitary operations on A. If A is finite of size at least 2, then we can choose from countably many operations to obtain continuum many distinct algebraic structures. However some of these algebras with distinct basic operations might still have similar structures: for example, from the operations of a Boolean algebra we can build the operations of a Boolean ring, and conversely. More precisely, this algebra and this ring have the same term functions (we say they are term equivalent) and consequently the same subalgebras, congruences, endomorphisms, etc. Clone theory is a part of Universal Algebra that strives to classify algebras up to term equivalence. We present classical and new descriptions of finite algebras whose term functions contain popular operations like that of a lattice or that of a group. This is joint work with Erhard Aichinger (JKU Linz) and Ralph McKenzie (Vanderbilt University).