POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

Let G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

COMPLETELY REDUCIBLE SETS

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

MEMBERSHIP AND FINITENESS PROBLEMS FOR RATIONAL SETS OF REGULAR LANGUAGES

Let Σ be a finite alphabet. A set [Formula: see text] of regular languages over Σ is called rational if there exists a finite set [Formula: see text] of regular languages over Σ such that [Formula: see text] is a rational subset of the finitely generated semigroup [Formula: see text] with [Formula: see text] as the set of generators and language concatenation as a product. We prove that for any rational set [Formula: see text] and any regular language R ⊆ Σ* it is decidable (1) whether [Formula: see text] or not, and (2) whether [Formula: see text] is finite or not. Possible applications to semistructured databases query processing are discussed.

THE MISSING LINK FOR ω-RATIONAL SETS, AUTOMATA, AND SEMIGROUPS

In 1997, following the works of Klaus W. Wagner on ω-regular sets, Olivier Carton and Dominique Perrin introduced the notions of chains and superchains for ω-semigroups. There is a clear correspondence between the algebraic representation of each of these operations and the automata-theoretical one. Unfortunately, chains and superchains do not suffice to describe the whole Wagner hierarchy. We introduce a third notion that completes the task undertaken by these two authors.