261 research outputs found
Infinite combinatorial issues raised by lifting problems in universal algebra
The critical point between varieties A and B of algebras is defined as the
least cardinality of the semilattice of compact congruences of a member of A
but of no member of B, if it exists. The study of critical points gives rise to
a whole array of problems, often involving lifting problems of either diagrams
or objects, with respect to functors. These, in turn, involve problems that
belong to infinite combinatorics. We survey some of the combinatorial problems
and results thus encountered. The corresponding problematic is articulated
around the notion of a k-ladder (for proving that a critical point is large),
large free set theorems and the classical notation (k,r,l){\to}m (for proving
that a critical point is small). In the middle, we find l-lifters of posets and
the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea
A Categorical View on Algebraic Lattices in Formal Concept Analysis
Formal concept analysis has grown from a new branch of the mathematical field
of lattice theory to a widely recognized tool in Computer Science and
elsewhere. In order to fully benefit from this theory, we believe that it can
be enriched with notions such as approximation by computation or
representability. The latter are commonly studied in denotational semantics and
domain theory and captured most prominently by the notion of algebraicity, e.g.
of lattices. In this paper, we explore the notion of algebraicity in formal
concept analysis from a category-theoretical perspective. To this end, we build
on the the notion of approximable concept with a suitable category and show
that the latter is equivalent to the category of algebraic lattices. At the
same time, the paper provides a relatively comprehensive account of the
representation theory of algebraic lattices in the framework of Stone duality,
relating well-known structures such as Scott information systems with further
formalisms from logic, topology, domains and lattice theory.Comment: 36 page
Lifting retracted diagrams with respect to projectable functors
We prove a general categorical theorem that enables us to state that under
certain conditions, the range of a functor is large. As an application, we
prove various results of which the following is a prototype: If every diagram,
indexed by a lattice, of finite Boolean (v,0)-semilattices with
(v,0)-embeddings, can be lifted with respect to the \Conc functor on
lattices, then so can every diagram, indexed by a lattice, of finite
distributive (v,0)-semilattices with (v,0-embeddings. If the premise of this
statement held, this would solve in turn the (still open) problem whether every
distributive algebraic lattice is isomorphic to the congruence lattice of a
lattice. We also outline potential applications of the method to other
functors, such as the functor on von Neumann regular rings
A survey of recent results on congruence lattices of lattices
We review recent results on congruence lattices of (infinite) lattices.
We discuss results obtained with box products, as well as categorical,
ring-theoretical, and topological results
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