Varieties of algebras when viewed as categories can be unexpectedly equivalent. For a reason explained at the end, I was looking at varieties of unital rings satisfying the equations p = 0 and x^p = x, one such variety for each prime integer p. The equivalence type of these categories is independent of p. The easiest way of establishing that is to show that each is equivalent to the category of Boolean algebras (a well-known fact when p = 2) and all the equivalences can by established by just one functor. Given a unital ring, R, define B(R) to be the boolean algebra of its central idempotents where the meet of a and b is ab and the join is a + b - ab. Then the restriction of B to the p'th variety described above is always an equivalence of categories. The fastidious will note (one would certainly hope) that B is not a functor in general (homomorphisms don't preserve centrality). But in a ring "without nilpotents" (that is, in which x^2 = 0 implies x = 0) all idempotents are central. The equations x^p = x, of course, imply the absence of nilpotents. (Given p the inverse functor to B can be described as follows: for a Boolean algebra C consider the set of "p-labeled partitions of unity", that is, the set of functions f:Z_p -> C whose values are pairwise disjoint and have unity as their join. Given two such, f and g, define their sum by setting (f+g)i to be the join of the set { fj ^ gk | j+k = i } and their product by setting (fg)i to be the join of { fj ^ gk | jk = i }.) I was looking for examples of equational theories with unique maximal consistent equational extensions. The best known example is the theory of lattices: every equation consistent with the theory of lattices is a consequence of distributivity. (Inconsistent in the equational setting means that all equations can be proved, or equivalently, the one equation x = y can be proved.) That is, the unique maximal consistent extension of the theory of lattices is the theory of distributive lattices (fortunately this is independent of your choice of whether top and/or bottom are considered to be part of the theory of lattices). A less-well-known example is the theory of Heyting algebras: every equation consistent with the theory of Heyting algebras is a consequence of the law of double-negation: (x -> 0) -> 0 = x. That is, the unique maximal consistent extension of the theory of Heyting algebras is the theory of Boolean algebras. This search for examples was sparked by what I consider a great example -- not to be described here -- in "algebraic real analysis". The only other examples I've found are the theories of unital rings of characteristic p, one such example for each prime p. To shift to the traditional language here, any polynomial identity consistent with characteristic p is a consequence of characteristic p and the identity x^p = x. A lot of examples. But, then again, maybe just one example.