This was mentioned in [S4], and in fact in [S2]. We shall first sketch the proof and then point out some applications of the theorem and the method. We generalize the notion of indiscernibility used in Ehrenfeucht-Mostowski models (from [EM]). Let I be an (index) model, M EL model and, for each sel,äs is a (finite) sequence from M. For s = , s(l) e 7,let ä-s = äs{Q) A ••• A...

A cluster obtained in high yield from the reduction of a silver-thiolate precursor, Ag-SCH(2)CH(2)Ph, exhibited a single sharp peak near 25 kDa in the matrix-assisted laser desorption mass spectrum (MALDI MS) and a well-defined metal core of ~2 nm measured with transmission electron microscopy (TEM). The cluster yields a single fraction in high-performance liquid chromatography (HPLC). Increase...

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for differential algebraic groups. The theorem establishes a sufficient criterion for the subgroup of a differential algebraic group generated by an infinite family of...

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements. Theorem 0.1 (Superstability from categoricity). Let K be a (< κ)-tame AEC with amalgamation. If κ = iκ > ...

The theme of the first two sections, is to prepare the framework of how from a " complicated " family of index models I ∈ K 1 we build many and/or complicated structures in a class K 2. The index models are characteristically linear orders, trees with κ + 1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graph, for this we phrase relevant complicate...

We give an example of a pure group that does not have the independence property, whose Fitting subgroup is neither nilpotent nor definable andwhose soluble radical is neither soluble nor definable. This answers a question asked by E. Jaligot in May 2013. The Fitting subgroupof a stable group is nilpotent and definable (F.Wagner [11]). More generally, the Fitting subgroup of a group that satisfi...

We observe simple links between preorders, semi-groups, rings and categories (and between equivalence relations, groups, fields and groupoids), which are type-definable in an arbitrary structure, and apply these observations to small structures. Recall that a structure is small if it has countably many pure n-types for each integer n. A ∅-type-definable group of finite arity in a small structur...

According to Belegradek, a first order structure is weakly small if there are countably many 1-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic 2 is finite or algebraically closed. A weakly small division ring of positive characteristic ...