We give a new definition of Scott rank motivated by our main theorem: For every countable structure A and ordinal α < ω1, we have that: every automorphism orbit is Σ α -definable without parameters if and only if A has a Π α+1 Scott sentence, if and only if A is uniformly boldface ∆α-categorical. As a corollary, we show that a structure is computably categorical on a cone if and only if it is t...

New results in the representation theory of “semisimple” algebraic monoids are obtained, based on Renner’s monoid version of Chevalley’s big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as “polynomial” representations of the associated reductive group of units in the monoid, and this representation cate...

In this work, we introduce $H^*$-condition on the set of submodules of a module. Let $M$ be a module. We say $M$ satisfies $H^*$ provided that for every submodule $N$ of $M$, there is a direct summand$D$ of $M$ such that $(N+D)/N$ and $(N+D)/D$ are cosingular. We show that over a right perfect right $GV$-ring,a homomorphic image of a $H^*$ duo module satisfies $H^*$.