Quantum computational algebra with a non-commutative generalization

We introduce a non-commutative generalization of quasi-MV algebra, called quasipseudo-MV algebra. We present some properties of quasi-pseudo-MV algebras and investigate the direct product decomposition of them. Further, we generalize quasi

Point stabilisers for the enhanced and exotic nilpotent cones

We give a semi-direct product decomposition of the point stabilisers for the enhanced and exotic nilpotent cones. In particular, we arrive at formulas for the number of points in each orbit over a finite field, which is in accordance with a recent conjecture of Achar and Henderson.

Direct product decomposition of theories of modules

This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In §1, I give a decomposition theorem which works for arbitrary direct product theories T. Given such a T, I define T-indecomposable structures and show that every model of T is elementarily equivalent to a direct product of T-indecomposable models of T. In §2, I show that if R is a commutative ring then every R-module is elementarily equivalent to ΠMM where M ranges over the maximal ideals of R and M is the localization of at M. This is applied to prove that if R is a commutative von Neumann regular ring and TR is the theory of R-modules then the TR-indecomposables are precisely the cyclic modules of the form R/M where M is a maximal ideal. In §3, I use the decomposition established in §2 to characterize the ω1-categorical and ω-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of ω-stable and superstable modules; e.g., if R is any countable ring, then an R-moduIe is ω-stable if and only if every R-module elementarily equivalent to it is equationally compact.