Computing finite index congruences of finitely presented semigroups and monoids

In this paper, we describe an algorithm for computing the left, right, or 2-sided congruences of a finitely presented semigroup or monoid with finitely many classes, and an alternative algorithm when the finitely presented semigroup or monoid is finite. We compare the two algorithms presented with existing algorithmsand implementations. The first algorithm is a generalization of Sims’ low-index subgroup algorithm for finding the congruences of a monoid. The second algorithm involves determining the distinct principal congruences, and then finding all of their possible joins. Variations of this algorithm have been suggested in numerous contexts by numerous authors. We show how to utilize the theory of relative Green’s relations, and a version of Schreier’s Lemma for monoids, to reduce the number of principal congruences that must be generated as the first step of this approach. Both of the algorithms described in this paper are implemented in the <sc>GAP</sc> [ <italic>GAP - groups, algorithms, and programming, version 5.5.4</italic> , 2025] package <sc>Semigroups</sc> (see J. Mitchell et al. [ <italic> Semigroups package for <sc>GAP</sc> </italic> , 2025]), and the first algorithm is available in the C++ library <sc>libsemigroups</sc> (see R. Cirpons, J. Edwards, J. Mitchell, M. Tsalakou, M. Whyte [ <italic>libsemigroups c++ library for semigroups and monoids</italic> , version 1.1.0, 2025]) and in its <sc>Python</sc> bindings <sc>libsemigroups_pybind11</sc> (see J. Mitchell, C. Nagpal, and M. Tsalakou [ <italic>libsemigroups pybind11 v0.10.1</italic> , 2023]).