Narrow Results

In 1965, Tauer produced a countably infinite family of semi-regular masas in the hyperfinite II₁ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalizers and, for each n ∈ ℕ, finding masas for which this procedure terminates at the nth stage. Such masas are said to have length n. In this paper, we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite II₁ factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalizing tower.

For a positive integer n, we denote by SUB (respectively, SUB_n) the class of all lattices that can be embedded into the lattice Co(P) of all order-convex subsets of a partially ordered set P (respectively, P of length at most n). We prove the following results:

(1) SUB_n is a finitely based variety, for any n≥1.

(2) SUB₂ is locally finite.

(3) A finite atomistic lattice L without D-cycles belongs to SUB if and only if it belongs to SUB₂; this result does not extend to the nonatomistic case.

(4) SUB_n is not locally finite for n≥3.

The first author defined a well-order in the set of links by embedding it into a canonical well-ordered set of (integral) lattice points. He also gave elementary transformations among lattice points to enumerate the prime links in terms of lattice points under this order. In this paper, we add some new elementary transformations and explain how to enumerate the prime links. We show a table of the first 443 prime links arising from the lattice points of lengths up to 10 under this order. Our argument enables us to find 7 omissions and 1 overlap in Conway's table of prime links of 10 crossings.

The most common ways used to generate indistinguishability operators, namely as transitive closure of reflexive and symmetric fuzzy relation, via the Representation Theorem and as decomposable relations, are related for archimedean t-norms introducing the notion of length of indistinguishability operators.

Let be the class of abelian p-groups. A non-empty proper subclass is bounded if it is closed under subgroups, additively bounded if it is also closed under direct sums and perfectly bounded if it is additively bounded and closed under filtrations. If , we call the partition of given by a B/U-pair. We state most of our results not in terms of bounded classes, but rather the corresponding B/U-pairs. Any additively bounded class contains a unique maximal perfectly bounded subclass. The idea of the length of a reduced group is generalized to the notion of the length of an additively bounded class. If λ is an ordinal or the symbol ∞, then there is a natural largest and smallest additively bounded class of length λ, as well as a largest and smallest perfectly bounded class of length λ. If λ ≤ ω, then there is a unique perfectly bounded class of length λ, namely the p^λ-bounded groups that are direct sums of cyclics; however, this fails when λ > ω. This parallels results of Dugas, Fay and Shelah (1987) and Keef (1995) on the behavior of classes of abelian p-groups with elements of infinite height. It also simplifies, clarifies and generalizes a result of Cutler, Mader and Megibben (1989) which states that the p^{ω + 1}-projectives cannot be characterized using filtrations.

We characterize all algebraic numbers which are roots of integer polynomials with a coefficient whose modulus is greater than or equal to the sum of moduli of all the remaining coefficients. It turns out that these numbers are zero, roots of unity and those algebraic numbers β whose conjugates over ℚ (including β itself) do not lie on the circle |z| = 1. We also describe all algebraic integers with norm B which are roots of an integer polynomial with constant coefficient B and the sum of moduli of all other coefficients at most |B|. In contrast to the above, the set of such algebraic integers is "quite small". These results are motivated by a recent paper of Frougny and Steiner on the so-called minimal weight β-expansions and are also related to some work on canonical number systems and tilings.

We study some geometric properties of helices of proper order 4 on a complex projective space which are of type 2 in the sense of submanifolds in a Euclidean space through the first standard embedding and which are generated by some Killing vector fields.

A well-order was introduced on the set of links by A. Kawauchi [3]. This well-order also naturally induces a well-order on the set of prime link exteriors and eventually induces a well-order on the set of closed connected orientable 3-manifolds. With respect to this order, we enumerated the prime links with lengths up to 10 and the prime link exteriors with lengths up to 9. Our present plan is to enumerate the 3–manifolds by using the enumeration of the prime link exteriors. In this paper, we show our latest result in this plan and as an application, give a new proof of the identification of Perko's pair.

Let be a field and let A be a finite-dimensional -algebra. We define the length of a finite generating set of this algebra as the smallest number k such that words of the length not greater than k generate A as a vector space, and the length of the algebra is the maximum of lengths of its generating sets. In this paper we study the connection between the length of an algebra and the lengths of its subalgebras. It turns out that the length of an algebra can be smaller than the length of its subalgebra. To investigate, how different the length of an algebra and the length of its subalgebra can be, we evaluate the difference and the ratio of the lengths of an algebra and its subalgebra for several representative families of algebras. Also we give examples of length computation of two and three block upper triangular matrix algebras.