Problem List

Hotzel asks this for semigroups and right congruences; his and the above version are equivalent. Left congruences on M are equivalence relations closed under all left translations; these are thus equivalent to congruences on the free left M -set on one generator. When M is a group, any left congruence is the decomposition of M into the left cosets of some subgroup, so the lattice of left congruences is isomorphic to the subgroup lattice. Thus, finite generation of a group is equivalent to the compactness of the greatest element in the lattice of left congruences, while ACC on that lattice is the stronger condition that all its elements be compact. On a general monoid, there are several interesting sorts of left congruences. If a left congruence C is generated by pairs of the form (a, 1), then C is determined by MC = {a ∈ M | (a, 1) ∈ C}. The subsets MC ⊆ M that arise in this way are precisely the submonoids closed under left division, i.e., such that if ab and a belong to the submonoid, so does b. Hence compactness of the greatest element of the left congruence lattice is equivalent to finite generation of M as a left-divisionclosed submonoid of itself. This can hold without M being finitely generated as a monoid. E.g., if M is the additive monoid of nonnegative elements of Z× Z under lexicographic order, it is generated in this sense by (0, 1) and (1, 0) , but requires (0, 1) and infinitely many of the elements (1,−n) to generate it as a monoid. A Rees left congruence C is the relation one gets by taking a nonempty left ideal I ⊆ M (a subset closed under all left translations) and making it one congruence class (a “zero element” of the left M -set X/C), while making all other congruence