Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety image of all finite image-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute image-pointlike sets, where image denotes the pseudovariety of all finite image-trivial semigroups. We finally show that, in contrast with the situation for image, the natural adaptation of Henckell’s algorithm to image computes pointlike sets, but not all of them.