We consider the cyclic closure of a language, and its generalisation to the operators $C^k$ introduced by Brandstadt. We prove that the cyclic closure of an indexed language is indexed, and that if $L$ is a context-free language then $C^k(L)$ is indexed.
This note proves a generalisation to inverse semigroups of Anisimovs theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word problem -- t
he set of all pairs of words over a generating set for the semigroup which both represent the same element.
The aim of this paper is to investigate whether the class of automaton semigroups is closed under certain semigroup constructions. We prove that the free product of two automaton semigroups that contain left identities is again an automaton semigroup
. We also show that the class of automaton semigroups is closed under the combined operation of free product followed by adjoining an identity. We present an example of a free product of finite semigroups that we conjecture is not an automaton semigroup. Turning to wreath products, we consider two slight generalizations of the concept of an automaton semigroup, and show that a wreath product of an automaton monoid and a finite monoid arises as a generalized automaton semigroup in both senses. We also suggest a potential counterexample that would show that a wreath product of an automaton monoid and a finite monoid is not a necessarily an automaton monoid in the usual sense.
We consider the class of groups whose word problem is poly-context-free; that is, an intersection of finitely many context-free languages. We show that any group which is virtually a finitely generated subgroup of a direct product of free groups has
poly-context-free word problem, and conjecture that the converse also holds. We prove our conjecture for several classes of soluble groups, including metabelian groups and torsion-free soluble groups, and present progress towards resolving the conjecture for soluble groups in general. Some of the techniques introduced for proving languages not to be poly-context-free may be of independent interest.
