We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K〈x, y, z〉 over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms ...

To better understand the biology of tameness, i.e. tolerance of human presence and handling, we analyzed two lines of wild-derived rats (Rattus norvegicus) artificially selected for tameness and defensive aggression towards humans. In response to a gloved human hand, tame rats tolerated handling, whereas aggressive rats attacked. Cross-fostering showed that these behavioral differences are not ...

The wealth of information that is available on the lattice of varieties of bands, is used to illuminate the structure of the lattice of sub-pseudovarieties of DA, a natural generalization of bands which plays an important role in language theory and in logic. The main result describes a hierarchy of decidable sub-pseudovarieties of DA in terms of iterated Mal’cev products with the pseudovarieti...

We describe the inherently non-dualisable finite algebras from some semigroup related classes. The classes for which this problem is solved include the variety of bands, the pseudovariety of aperiodic monoids, commutative monoids, and (assuming a reasonable conjecture in the literature) the varieties of all finite monoids and finite inverse semigroups. The first example of an inherently non-dua...

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.

The semidirect product of pseudovarieties of semigroups with an order-computable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning the behavior of the operation with respect to various kinds of tameness properties are obtained as applicati...

We prove that if π is a recursive set of primes, then pointlike sets are decidable for the pseudovariety of semigroups whose subgroups are π-groups. In particular, when π is the empty set, we obtain Henckell’s decidability of aperiodic pointlikes. Our proof, restricted to the case of aperiodic semigroups, is simpler than the original proof.

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees (see Definition 5.1) implies that in some homogenous generic extension of V there is a transitive model M containing Ord∪R such that M AD + Θ > θ0. In particular, this implies the existence (in V ) of a non-tame mouse. The results of this paper significantly extend Steel’s earlier results from [16] for ...

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes finite algebras closed under products, subalgebras and quotients. In this article, is generalized to Eilenberg-Moore a monad T on category D: we prove class -algebras pseudovariety iff it presentable by profinit...

In this paper we study multidimensional persistence modules [5, 13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact mul-tidimensional persistence modules. To count such features we introduce the feature counting invariant and ...