International Journal of Algebra and Computation

In the framework of two Schützenberger’s conjectures on codes, we characterize the degree of finite maximal codes over the alphabet {a, b} with at most 3 occurrences of the letter b in each word; it is strongly related to the number of factorizations of the code. We also describe all the indecomposable codes inside this family.

Two (synchronous, asynchronous, or non-deterministic asynchronous) automatic structures on a group G are “equivalent” if their union is a non-deterministic asynchronous automatic structure. We discuss this relation, giving a classification of structures up to equivalence for abelian groups and partial results in some other cases. We also discuss a “boundary” of an asynchronous automatic structure. We show that it is an invariant of the equivalence class of the structure, and describe other properties. We describe a “rehabilitated boundary” which yields Sⁿ⁻¹ for any automatic structure on ℤⁿ.

An existence variety (or e-variety) of regular semigroups is a class of regular semigroups which is closed under , and ℍ. This concept was introduced by T.E. Hall and independently for orthodox semigroups by J. Kadourek and M.B. Szendrei who called them bivarieties. In this paper we prove the existence of e-free objects in each e-variety of E-solid regular semigroups and in each e-variety of locally inverse regular semigroups. By contrast, we show that there is no e-free object in other e-varieties.