Properties of Casson–Gordon’s rectangle condition

For a Heegaard splitting of a [Formula: see text]-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of [Formula: see text]-bridge decompositions of knots admitting a diagram with eight types of rectangles, which are not strongly irreducible. This says that the rectangle condition is sharp. Moreover, we define a variation of the rectangle condition so-called the sewing rectangle condition that also can guarantee the strong irreducibility of [Formula: see text]-bridge decompositions of knots. The new condition needs six types of rectangles but more complicated than nine types of rectangles for the rectangle condition.

THE STRONG IRREDUCIBILITY OF A CLASS OF COWEN–DOUGLAS OPERATORS ON BANACH SPACES

Let ${\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ be the set of Cowen–Douglas operators of index $n$ on a nonempty bounded connected open subset $\unicode[STIX]{x1D6FA}$ of $\mathbb{C}$. We consider the strong irreducibility of a class of Cowen–Douglas operators ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ on Banach spaces. We show ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})\subseteq {\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ and give some conditions under which an operator $T\in {\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ is strongly irreducible. All these results generalise similar results on Hilbert spaces.

ON THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL

This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.

RECTANGLE CONDITION FOR COMPRESSION BODIES AND 2-FOLD BRANCHED COVERINGS

We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of 2-fold covering of S3 branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of width holds for a composite knot satisfying the condition.

Finite Projective Planes that Admit a Strongly Irreducible Collineation Group

This paper studies how coding theory and group theory can be used to produce information about a finite projective plane π and a collineation group G of π.A new proof for Hering's bound on |G| is given in 2.5. Using the idea of coding theory developed in [9], a relation between two rows of the incidence matrix of π with respect to a tactical decomposition is obtained in 2.1. This result yields, among other things, some techniques in calculating |G|, and generalizes a result of Roth [16], [see 2.4 and 2.5].Hering [7] introduced the notion of strong irreducibility of G, that is, G does not leave invariant any point, line, triangle or proper subplane. He showed that if in addition G contains a non-trivial perspectivity, then there is a unique minimal normal subgroup of G. This subgroup is either non-abelian simple or isomorphic to the elementary abelian group Z3 × Z3 of order 9.