We extend to spiking neural P systems a notion investigated in the “standard” membrane systems: the language of the traces of a distinguished object. In our case, we distinguish a spike by “marking” it and we follow its path through the neurons of the system, thus obtaining a language. Several examples are discussed and some preliminary results about this way of associating a language with a sp...

The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic ω-semigroup. It is inherited by a subpseudovariety of a give...

In this paper we continue a classical work of Schützenberger on codes with bounded synchronization delay. He was interested to characterize those regular languages where the groups in the syntactic monoid belong to a variety H. He allowed operations on the language side which are union, intersection, concatenation and modified Kleene-star involving a mapping of a prefix code of bounded synchron...

We relate two measures of complexity of regular languages. The first is syntactic complexity, that is, the cardinality of the syntactic semigroup of the language. That semigroup is isomorphic to the semigroup of transformations of states induced by non-empty words in the minimal deterministic finite automaton accepting the language. If the language has n left quotients (its minimal automaton ha...

Let g be a Lie algebra over a field F of characteristic zero, let C be a certain tensor category of representations of g, and C a certain category of duals. In [M 4] we associated to C and C by a Tannaka reconstruction a monoid M with a coordinate ring F [M ] of matrix coefficients, as well as a Lie algebra Lie(M). We interpreted the Tannaka monoid M algebraic geometrically as a weak algebraic ...

For a regular ring R and an affine monoid M the homotheties of M act nilpotently on the Milnor unstable groups of R[M ]. This strengthens the K2 part of the main result of [G5] in two ways: the coefficient field of characteristic 0 is extended to any regular ring and the stableK2-group is substituted by the unstable ones. The proof is based on a polyhedral/combinatorial techniques, computations...

The original motivation of this paper, as presented in [15], was to compute the dual space of a lattice of subsets of some free monoid A∗. According to StonePriestley duality, the dual space of a lattice can be identified with the set of its prime filters, but it is not always the simplest way to describe it. Consider for instance the Boolean algebra generated by the sets of the form uA∗, where...

An event E is a subset of the free monoid A* generated by the finite alphabet A. E is noncounting if and only if there exists an integer k ~> 0, called the order of E, such that for any x, y, z E A*, xykz ~ E if and only if xyk+lz E E. From semigroup theory it follows that the number of noncounting events of order < 1 is finite. Each such event is regular and the finite automata accepting such ...

The goal of this presentation is to collect several open problems on words and (finite) languages. The problems are not intended to be new. On the other hand, all of these problems are very easily stated and in that respect extremely natural. Moreover, the choice of these problems reflects the interests of the author – each of those have some connections to the work of the author. In other word...

We continue our investigation on the relationship between regular languages and syntactic monoid size. In this paper we confirm the conjecture on two generator transformation semigroups. We show that for every prime n ≥ 7 there exist natural numbers k and ` with n = k + ` such that the semigroup Uk,` is maximal w.r.t. its size among all (transformation) semigroups which can be generated with tw...