Pointlike sets with respect to R and J

Abstract

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety R of all finite R-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 J-pointlike sets, where J denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for R, the natural adaptation of Henckell's algorithm to J computes pointlike sets, but not all of them.

Separating Regular Languages by Piecewise Testable and Unambiguous Languages

“…An ℓ-template is said to be unambiguous if all pairs t i , t i+1 are either two letters, two incomparable sets or a set and a letter that is not included in the set. We now use p-implementations to prove (1) =⇒ (2). The proof is divided in two steps.…”

“…In the case that disjoint elements of C cannot always be separated by an element of Sep, several natural questions arise: (1) given elements L 1 , L 2 in C, can we decide whether a separator exists in Sep? (2) if so, what is the complexity of this decision problem? (3) can we, in addition, compute a separator, and what is the complexity?…”

Efficient Separability of Regular Languages by Subsequences and Suffixes

“…Almeida [3] established a connection between a number of separation problems and properties of families of monoids called pseudovarieties. Almeida shows, e.g., that deciding whether two given regular languages can be separated by a language with its syntactic monoid lying in pseudovariety V is algorithmically equivalent to computing two-pointlike sets for a monoid in pseudovariety V. It is then shown by Almeida et al [4] how to compute these two-pointlike sets in the pseudovariety J corresponding to piecewise testable languages. Henckell et al [12] and Steinberg [25] show that the two-pointlike sets can be computed for pseudovarieties corresponding to languages definable in first order logic and languages of dot depth at most one, respectively.…”

Pointlike sets for varieties determined by groups

“…so that q 2 m L R q 2 m L b, and by Lemma 4.5 (5), q 2 m L b = q 2 m L . Thus, choosing t := m L gives q 1 t = q 1 b and q 2 t = q 2 m L b.…”

“…There has been a great deal of work on pointlike sets and there are many decidability results. A by no means exhaustive list of examples include: J [30,9], R (the variety of R-trivial semigroups) [5], any decidable variety of finite abelian groups [13], the variety of finite p-groups for a prime p [31] and the variety of finite nilpotent groups [1]. Also if V is a variety of finite monoids with decidable pointlikes, then the second author showed that V * D has decidable pointlikes, where * is the semidirect product of varieties and D is the variety of finite semigroups whose idempotents are right zeroes.…”