We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property.

We extend the definition of orbifold (Chen-Ruan) cohomology to tame Deligne-Mumford stacks over fields of positive characteristic and prove that a modified version of the Frobenius action preserves the product. We conjecture a relationship between the arithmetic information contained by this action and the zeta function of a crepant resolution of the coarse moduli space (when each exist).

During the Notre Dame workshop on Vaught’s Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for models of a first order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types. I am grateful to the logic group at Notre...

A henselian valued field K is called a tame field if its separable-algebraic closure Ksep is a tame extension, that is, Ksep is equal to the ramification field of the normal extension Ksep|K. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to ...

We prove a downward transfer from categoricity in a successor in tame abstract elementary classes (AECs). This complements the upward transfer of Grossberg and VanDieren and improves the Hanf number in Shelah’s downward transfer (provided the class is tame). Theorem 0.1. Let K be an AEC with amalgamation. If K is LS(K)-weakly tame and categorical in a successor λ ≥ i(2LS(K))+ , then K is catego...

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this new context. We assume that K is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include: Theorem 0.1. Suppose ...

Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the BasicWild-Rank Conjecture. A covering criterion on the rank of a basic wild algebr...

This paper is a survey of recent results in the theory of nite semigroups using syntactic techniques. This involves manipulating pseudoidentities and the study of relatively free proonite semigroups, whose structure encodes algebraic and combinatorial properties common to the members of a pseudovariety of semigroups. These techniques are illustrated with applications in the study of semigroups ...

Let C = F [x1, x2, . . . , xn] be the polynomial ring in the variables x1, x2, . . . , xn over a field F , and let AutC be the group of automorphisms of C as an algebra over F . An automorphism τ ∈ AutC is called elementary if it has a form τ : (x1, . . . , xi−1, xi, xi+1, . . . , xn) 7→ (x1, . . . , xi−1, αxi + f, xi+1, . . . , xn), where 0 6= α ∈ F, f ∈ F [x1, . . . , xi−1, xi+1, . . . , xn]....

A word u ∈ A+ is an expletive for a fuzzy language λ if λ(xuy) = λ(xy) for all x, y ∈ A+. A full expletive fuzzy language is a fuzzy language in which the set of expletives is A+. Here we prove that the set of full expletive fuzzy languages is a variety and the associated pseudovariety of semigroups is that of midunit semigroups. Subject Classification: 20M35, 20M07, 68Q45