We study the uniqueness of minimal liftings of cut generating functions obtained from maximal lattice-free polytopes. We prove a basic invariance property of unique minimal liftings for general maximal lattice-free polytopes. This generalizes a previous result by Basu, Cornuéjols and Köppe [3] for simplicial maximal lattice-free polytopes, thus completely settling this fundamental question abou...

We prove lifting results for DG modules that are akin to Auslander, Ding, and Solberg’s famous lifting results for modules. Introduction Convention. Throughout this paper, let R be a commutative noetherian ring. Hochster famously wrote that “life is really worth living” in a Cohen-Macaulay ring [7]. For instance, if R is Cohen-Macaulay and local with maximal regular sequence t, then R/(t) is ar...

In this article, a sequel to Global Frobenius Liftability I (math:1708:03777v2), we continue the development of comprehensive theory liftings modulo $p^2$. We study compatibility divisors and closed subschemes with liftings, blow-ups, descent under quotients by some group actions, stability base change, properties associated F-splittings. Consequently, characterise liftable surfaces Fano threef...

This paper studies the extension of the Generalization Complexity (GC) measure to real valued input problems. The GC measure, defined in Boolean space, was proposed as a simple tool to estimate the generalization ability that can be obtained when a data set is learnt by a neural network. Using two different discretization methods, the real valued inputs are transformed into binary values, from ...

We investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characte...

We compute liftings of the Nichols algebra of a Yetter-Drinfeld module of Cartan type B2 subject to the small restriction that the diagonal elements of the braiding matrix are primitive nth roots of 1 with odd n 6= 5. As well, we compute the liftings of a Nichols algebra of Cartan type A2 if the diagonal elements of the braiding matrix are cube roots of 1; this case was not completely covered i...

Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Logics with modal operators obtained from so-called predicate liftings have been shown to be invariant under behavioral equivalence. Expressivity results stating that, conversely, logically indistinguishable states are behaviorally equivalent depend on the existence ...

In this article we investigate the notion and basic properties of Boolean algebras and prove the Stone’s representation theorem. The relations of Boolean algebras to logic and to set theory will be studied and, in particular, a neat proof of completeness theorem in propositional logic will be given using Stone’s theorem from Boolean algebra. We mention here that the method we used can also be e...

We prove a representation theorem for non-monotonic inference relations that are defined over elements of some Boolean algebras and obey the rules of System P. An inference relation is represented by a closure operator on the Stone space of the Boolean algebra. This representation theorem generalizes and gives new insights into existing completeness theorems for System P. Let A be a Boolean alg...