We believe that the study of the notions of universal algebra modelled in an arbitarry topos rather than in the category of sets provides a deeper understanding of the real features of the algebraic notions. [2], [3], [4], [S], [6], [7], [13], [14] are some examples of this approach. The lattice Id(L) of ideals of a lattice L (in the category of sets) is an important ingredient of the categ...

Let M be the interior of a compact 3–manifold with non–empty boundary, and T be an ideal (topological) triangulation of M . This paper describes necessary and sufficient conditions for the existence of angle structures, semi–angle structures and generalised angle structures on (M ; T ) respectively in terms of a generalised Euler characteristic function on the solution space of normal surface t...

Let M be the interior of a compact 3–manifold with boundary, and let T be an ideal triangulation of M. This paper describes necessary and sufficient conditions for the existence of angle structures, semi–angle structures and generalised angle structures on (M ;T ) respectively in terms of a generalised Euler characteristic function on the solution space of the normal surface theory of (M ; T )....

The diagonal in a product of projective spaces is cut out by the ideal of 2×2-minors of a matrix of unknowns. The multigraded Hilbert scheme which classifies its degenerations has a unique Borel-fixed ideal. This Hilbert scheme is generally reducible, and its main component is a compactification of PGL(d)/PGL(d). For n = 2 we recover the manifold of complete collineations. For projective lines ...

Working under large cardinal assumptions, we study the Borel-reducibility between equivalence relations modulo restrictions of the non-stationary ideal on some fixed cardinal κ. We show the consistency of E ++,λ++ λ-club , the relation of equivalence modulo the non-stationary ideal restricted to Sλ ++ λ in the space (λ++)λ ++ , being continuously reducible to E ++ λ+-club , the relation of equi...

No surgery on a non-torus 2-bridge knot yields a reducible 3-manifold as shown in Theorem 2(a) in [7] by A. Hatcher and W. Thurston. Dehn surgeries on 2-bridge knots are already well studied by M. Brittenham and Y.-Q. Wu in [2]. See also [11]. On 2-bridge links of 2-components, Wu showed in [13, Theorem 5.1 and Remark 5.5] the following theorem. The universal covering space of a laminar 3-manif...

We study homogeneous Lorentzian manifolds M=G/L of a connected reductive Lie group G modulo subgroup L, under the assumption that M is (almost) G-effective and isotropy representation totally reducible. show description such reduces to case semisimple groups G. Moreover, we prove space reductive. describe all reducible subgroups Lorentz divide them into three types. The Type I are compact, whil...

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a charac...