Solution to an open problem: A characterization of conditionally cancellative t-subnorms

In this work we solve an open problem of U. Höhle (Klement et al. Fuzzy Sets Syst 145:471-479, 2004, Problem 11). We show that the solution gives a characterization of all conditionally cancellative t-subnorms. Further, we give an equivalence condition under which a conditionally cancellative t-subnorm has 1 as its neutral element and hence show that conditionally cancellative t-subnorms whose natural negations are strong are, in fact, t-norms

Commutators in the Two-Weight Setting

Let $R$ be the vector of Riesz transforms on $\mathbb{R}^n$, and let $\mu,\lambda \in A_p$ be two weights on $\mathbb{R}^n$, $1 < p < \infty$. The two-weight norm inequality for the commutator $[b, R] : L^p(\mathbb{R}^n;\mu) \to L^p(\mathbb{R}^n;\lambda)$ is shown to be equivalent to the function $b$ being in a BMO space adapted to $\mu$ and $\lambda$. This is a common extension of a result of Coifman-Rochberg-Weiss in the case of both $\lambda$ and $\mu$ being Lebesgue measure, and Bloom in the case of dimension one.Comment: v3: suggestions from two referees incorporate

The variety generated by all the ordinal sums of perfect MV-chains

We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek - Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras form the smallest variety containing all the ordinal sums of perfect MV-chains. We will analyze this logic and the corresponding algebraic semantics in the propositional and in the first-order case. As we will see, moreover, the variety of BL_Chang-algebras will be strictly connected to the one generated by Chang's MV-algebra (that is, the variety generated by all the perfect MV-algebras): we will also give some new results concerning these last structures and their logic.Comment: This is a revised version of the previous paper: the modifications concern essentially the presentation. The scientific content is substantially unchanged. The major variations are: Definition 2.7 has been improved. Section 3.1 has been made more compact. A new reference, [Bus04], has been added. There is some minor modification in Section 3.

Factorization theory: From commutative to noncommutative settings

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several notions of factorizations as well as distances between them are introduced. In addition, arithmetical invariants characterizing the non-uniqueness of factorizations such as the catenary degree, the $\omega$-invariant, and the tame degree, are extended from commutative to noncommutative settings. We introduce the concept of a cancellative semigroup being permutably factorial, and characterize this property by means of corresponding catenary and tame degrees. Also, we give necessary and sufficient conditions for there to be a weak transfer homomorphism from a cancellative semigroup to its reduced abelianization. Applying the abstract machinery we develop, we determine various catenary degrees for classical maximal orders in central simple algebras over global fields by using a natural transfer homomorphism to a monoid of zero-sum sequences over a ray class group. We also determine catenary degrees and the permutable tame degree for the semigroup of non zero-divisors of the ring of $n \times n$ upper triangular matrices over a commutative domain using a weak transfer homomorphism to a commutative semigroup.Comment: 45 page