In this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

Regular languages can be studied not just for finite words, but also for finite trees, infinite words and infinite trees. Almost all of the theory of regular languages, such as closure under boolean operations, works also for these extensions, but the constructions are significantly more challenging and mathematically interesting. For instance, automata for infinite words can be determinised un...

We consider first order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO2[<,MOD], the two-variable first order logic over this signature, interpreted over finite words. We give an algebraic characterization of the corresponding regular languages...

A topological space X is said to be completely regular if whenever F is a nonempty closed set and x ∈ X \F , there is a continuous function f : X → [0, 1] such that f(x) = 0 and f(F ) = {1}. A completely regular space need not be Hausdorff. For example, ifX is any set with more than one point, then the trivial topology, in which the only closed sets are ∅ and X, is vacuously completely regular,...

We utilize recent results of André and Gabber on the existence weakly functorial, integral perfectoid big Cohen–Macaulay (BCM) algebras to study singularities local rings in mixed characteristic. In particular, we introduce a characteristic BCM-variant rational/F-rational singularities, log terminal/F-regular multiplier/test ideals divisor pairs. prove number about these objects including restr...