We show that for a commutative quantale V every functor Set −→ V -cat has an enriched leftKan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V -cat. Moreover, one can build functors on V -cat by equipping Set-functors with a metric. 1998 ACM Subject Classification F.4.1 Mathematical Logic

In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module Rμ is n!, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for Rμ which would give a simple proof of the dimension. Using the theory...

The target of this study is to observe some of the algebraic structures of a single valued neutrosophic set. So, we introduce the concept of a neutrosophic submodule of a given classical module and investigate some of the crucial properties and characterizations of the proposed concept.

For each pointed abelian group (A, c), there is an associated Galkin quandle G(A, c) which is an algebraic structure defined on Z3 ×A that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number o...

Let G be a semisimple algebraic Qgroup. let. F be an arithmetic subgroup of G . and let, T be an E-split torus in G . We prove that if there is a divergent Tn-orbit in r \ G R , and Q-rank G < 2, t,hen dim T < Q-rankG. This provides a partial answer to a question of G . Tomanov and B. Weiss.

For an arbitrary-type functor F, the notion of split coalgebras, that is, coalgebras for which the canonical projections onto the simple factor split, generalizes the well-known notion of simple coalgebras. In case F weakly preserves kernels, the passage from a coalgebra to its simple factor is functorial. This is the simplification functor. It is left adjoint to the inclusion of the subcategor...

A result of Steinberg's on the existence of rational regular unipotent elements in quasi-split simple algebraic groups over fields of arbitrary characteristic is partially extended to the case of non-quasi-split groups.

In this article we describe the G×G-equivariant K-ring of X, where X is a regular compactification of a connected complex reductive algebraic group G. Furthermore, in the case when G is a semisimple group of adjoint type, and X its wonderful compactification, we describe its ordinary K-ring K(X). More precisely, we prove that K(X) is a free module over K(G/B) of rank the cardinality of the Weyl...

Various notions of observing and testing processes are placed in a uniform algebraic framework in which observations are taken as constituting a quantale. General completeness criteria are stated, and proved in our applications. 2 67("/89:&",;&<+*91(8=&>)",#"3185&.781(2"#+.,"3&3.?+*&",;&0(.*188&81/",#+*8

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends.