Karl Auinger

On the extension problem for partial permutations

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 *, **, ?, =, = n , and P do not preserve decidability. In addition, several joins, including A V U, are shown to be undecidable.

The geometry of profinite graphs with applications to free groups and finite monoids

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, a pro-H analog of the Ribes and Zalesskii 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? H.

Congruences on the Lattice of Pseudovarieties of Finite Semigroups

As a step in a study of the lattice ℒ(ℱ) of pseudovarieties of finite semigroups that attempts to take full advantage of the underlying lattice structure, a family of complete congruences is introduced on ℒ(ℱ). Such congruences provide a framework from which to study ℒ(ℱ) both locally and globally. Each is associated with a mapping of the form

Pseudovarieties generated by Brauer type monoids

{\cal U} \to {\cal U} \cap {\cal A}

Matrix identities involving multiplication and transposition

for some special class

On power groups and embedding theorems for relatively free profinite monoids

{\cal A} of finite semigroups. In some instances the class is itself a pseudovariety while in others the class will be defined in terms of certain congruences associated with Greens relations. The basic properties of these complete congruences are presented and some relations between certain operators associated with these congruences are obtained.

Constructing divisions into power groups

Abstract. It is proved that the series of all Brauer monoids generates the pseudovariety of all finite monoids while the series of their aperiodic analogues, the Jones monoids (also called Temperly–Lieb monoids), generates the pseudovariety of all finite aperiodic monoids. The proof is based on the analysis of wreath product decomposition and Krohn–Rhodes theory. The fact that the Jones monoids form a generating series for the pseudovariety of all finite aperiodic monoids can be viewed as solution of an old problem popularized by J.-É. Pin. For the latter, the relationship between the Jones monoids and the monoids of order preserving mappings of a chain of length n is investigated.

Semigroups with complemented congruence lattices

We study matrix identities involving multiplication and unary operations such as transposition or Moore-Penrose inversion. We prove that in many cases such identities admit no finite basis.

Unary enhancements of inherently non-finitely based semigroups

We determine those pseudovarieties of groups

An application of a Theorem of Ash to finite covers

{\bf H}