It is known that the powers m of the maximal ideal of a local Noetherian ring share certain homological properties for all sufficiently large integers n. For example, the natural homomorphisms R → R/m are Golod, respectively, small, for all large n. We give effective bounds on the smallest integers n for which such properties begin to hold.

States of unital Abelian lattice-groups (normalised positive group homomorphisms to R) provide an abstraction expected-value operators. A well-known theorem due Mundici asserts that the category (with unit-preserving lattice-group as morphisms) is equivalent algebraic MV-algebras, and their homomorphisms. Through this equivalence, states correspond certain [0,1]-valued functionals on which are ...

we call a module  essentially retractable if homr for all essential submodules n of m. for a right fbn ring r, it is shown that: (i)  a non-zero module  is retractable (in the sense that homr for all non-zero ) if and only if certain factor modules of m are essentially retractable nonsingular modules over r modulo their annihilators. (ii)  a non-zero module  is essentially retractable if and on...

KURATOWSKI showed that the derived set operator Z), acting on the space 2^ of closed subsets of a metric space X, is a Borel map of class exactly two and posed the problem of determining the precise classes of the higher order derivatives 0. We show that the exact classes of the higher derivatives D" are unbounded in ©i. In particular, we show that D* is not of class a and that, for limit ordin...

We show in this paper that the Gentry-Szydlo algorithm for cyclotomic orders, previously revisited by Lenstra-Silverberg, can be extended to complex-multiplication (CM) orders, and even to a more general structure. This algorithm allows to test equality over the polarized ideal class group, and finds a generator of the polarized ideal in polynomial time. Also, the algorithm allows to solve the ...

We define an algebraic structure, Paired Complete Idempotent Semirings (pcis), which are appropriate for defining a denotational semantics for multiple context-free grammars of dimension 2 (2-mcfg). We demonstrate that homomorphisms of this structure will induce wellbehaved morphisms of the grammar, and generalize the syntactic concept lattice from context-free grammars to the 2-mcfg case. We s...