We prove some results related to the problem of blowing up the power set of the least measurable cardinal. Our forcing results improve those of "Forcing the least measurable to violate GCH" by using the optimal hypothesis.

It is proved that free dynamic algebras superamalgamate. Craig interpolation for proposi- tional dynamic logic and superamalgamation for the variety of dynamic algebras follow.

We consider separably closed fields of characteristic $p > 0$ and fixed imperfection degree as modules over a skew polynomial ring. We axiomatize the corresponding theory and we show that it is complete and that it admits quantifier elimination in the usual module language augmented with additive functions which are the analog of the $p$-component functions.

This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting. Without padding, they remain in polynomial time but appear not to be in choiceless polynomial time plus counting. Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded. Finally, we describe another possible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting.

It is known that in differentially closed fields of characteristic zero, the ranks of stability $RU$, $RM$ and the topological rank $RH$ need not to be equal. Pillay and Pong have just shown however that the ranks $RU$ and $RM$ coincide in a group definable in this theory. At the opposite, we will show in this paper that the ranks $RM$ and $RH$ of a definable group can also be different, and even lead to non-equivalent notions of generic type.

Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.