arXiv subjects
Publications and source records attributed to Samuel Herman.
We show that pointlike sets are decidable for the pseudovariety of finite semigroups whose idempotent-generated subsemigroup is R-trivial. Notably, our proof is constructive: we provide an explicit relational morphism which computes the ER-pointlike subsets of a given finite semigroup.
We introduce a generalization of the product expansion of a finite semigroup. As an application, we provide an alternative proof of the decidability of pointlike sets for pseudovarieties consisting of semigroups whose subgroups all belong to a given decidable pseudovariety of groups.
We introduce a general unifying framework for the investigation of pointlike sets. The pointlike functors are considered as distinguished elements of a certain lattice of subfunctors of the power semigroup functor; in particular, we exhibit the pointlike functors as the fixed points of a closure operator induced by an antitone Galois connection between this lattice of functors and the lattice of pseudovarieties. Notably, this provides a characterization of pointlikes which does not mention relational morphisms. Along the way, we formalize various common heuristics and themes in the study of pointlike sets. As an application, we provide a general method for transferring lower bounds for pointlikes along a large class of continuous operators on the lattice of pseudovarieties.
We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. Said classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings representing subgraphs. These orbits are enumerated using Burnside's lemma. The technique used also provides an alternative proof of the formulae found by S. W. Golomb and L. R. Welch which give the number of distinct $n$-gons on fixed, regularly spaced vertices up to rotation and optionally reflection.