A family of pseudovarieties of solvable groups is constructed, each of which has decidable membership and undecidable extension problem for partial permutations. Included are a pseudovariety U satisfying no non-trivial group identity and a metabelian pseudovariety Q. For each of these pseudovarieties V, the inverse monoid pseudovariety Sl∗V has undecidable membership problem. As a consequence, it is proved that the pseudovariety operators ∗, ∗∗, m○, ♦, ♦n, andP do not preserve decidability. In addition, several joins, including A ∨ U, are shown to be undecidable.

. Suppose H is a pseudovariety of groups. This paper studies the pseudovariety of monoids corresponding to the variety of languages generated by the polynomial closure of H- languages (which is shown to be J H via a short syntactic argument) and the pseudovariety of ordered monoids corresponding to the polynomial closure of the H-languages. We also explore alternative descriptions of these pseudovarieties under additional hypotheses. Decidability results are given. In addition, we study the positive varieties of pro-V open and closed recognizable sets for a pseudovariety V of monoids. In particular, we obtain a basis of ordered pseudoidentities for such positive varieties. For pseudovarieties of groups H, we relate these positive varieties to the polynomial closure of the H-languages. Again, decidability results are obtained. 1. Introduction Concatenation hierarchies have played an important role in the study of rational languages, both because they give a natural way in w...

This paper gives an elementary, self-contained proof that a finite product of finitely generated subgroups of a free group is closed in the profinite topology. The proof uses inverse automata (immersions) and inverse monoid theory. Generalizations are given to other topologies. In particular, we obtain the new result that for arborescent pseudovarieties, the product of two closed finitely generated subgroups is again closed. An application to monoid theory is given.

This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = hAjRi be a group presentation and BA;R its standard 2complex. Suppose X is a 2-complex with a morphism to BA;R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A;R) to the presentation hAjRi with the property that pointed subgraphs of covers of BA;R are classi ed by closed inverse submonoids of M(A;R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language...

This note shows that PH = \SigmaH = J H = J fl m H holds for any non-trivial extension closed pseudovariety of groups. It is known that BH 6= J fl m H if H 6= G. Hence, we generalize as much as possible of the equation PG = BG. 1. Introduction In this note, we generalize one of the most difficult results of finite monoid theory, that PG = BG, or said differently, the pseudovariety of block groups is generated by power groups. However, the true result is that PG = \SigmaG = J G = J fl m G = BG: Our goal is to show that all these equalities, except the last, hold when G is replaced by a (nontrivial) extension closed pseudovariety H. Since the power operator is a difficult operator to deal with in finite monoid theory, while the structure of J fl m H has been studied in detail by the author in [7], such a result is quite useful. As with the original result, the most difficult part is that J H = J fl m H. This was proved by the author in [7]. The rest is merely a minor modifica...

In this paper we compute the abelian kernels of the monoids POI n and POPI n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI n (n 2 N ) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI n (n 2 N ) with the pseudovariety Ab of all finite abelian groups.

We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition. We define a pseudovariety of groups H to be arboreous if all finitely generated free pro-H groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties H, apro-H analog of the Ribes and Zalesskiĭ product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions H to the much studied pseudovariety equation J ∗ H = J m ○ H.

This paper gives decidable conditions for when a finitely generated subgroup of a free group is the fundamental group of a Schützenberger automaton corresponding to a monoid presentation of an inverse monoid. Also, generalizations are given to specific types of inverse monoids as well as to monoids which are "nearly inverse." This result has applications to computing membership for inverse monoids in a Mal'cev product of the pseudovariety of semilattices with a pseudovariety of groups. This paper also shows that there is a bijection between strongly connected inverse automata and subgroups of a free group, generated by positive words. Hence, we also obtain that it is decidable whether a finite strongly connected inverse automaton is a Schutzenberger automaton corresponding to a monoid presentation of an inverse monoid. Again, we have generalizations to other types of inverse monoids and to "nearly inverse" monoids. We show that it is undecidable whether a finite strongly connected in...

It has become a classical technique to turn to theoretical computer science to provide computational tools for algebraic geometry. A more recent transformation is that now we also get logical tools, and these too should be useful in the study of algebraic varieties. The purpose of this note is to consider a very small part of this picture, and try to motivate the study of computer theorem-proving techniques by looking at how they might be relevant to a particular class of problems in algebraic geometry. This is only an informal discussion, based more on questions and possible research directions than on actual results. This note amplifies the themes discussed in my talk at the “Arithmetic and Differential Galois Groups ” conference (March 2004, Luminy), although many specific points in the discussion were only finished more recently. I would like to thank: André Hirschowitz and Marco Maggesi, for their invaluable insights about computer-formalized mathematics as it relates

If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of nite Rees quotients of E-unitary inverse semigroups with maximal group image in C. The converse is shown if C is a pseudovariety. When C is a pseudovariety, the above problems are shown to be equivalent to the problem of embedding a nite labeled graph in the Cayley graph of a group in C. This latter problem is shown to be equivalent to deciding whether a nite labeled graph is a Schutzenberger graph of an E-unitary inverse semigroup with maximal group image in C. 1.