Some results are obtained concerning n(T), the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if n(T) = 3 and T has a tight prime model, then T is unstable. Secondly, it is proved that if n(T) is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links...

Countable Lawvere theories model computational effects such as exceptions, side-effects, interactive input/output, nondeterminism and probabilistic nondeterminism. The category of countable Lawvere theories has sums, tensors, and distributive tensors, modelling natural combinations of such effects. It is also closed under taking images. Enrichment in a category such as Cpo allows one to extend ...

We introduce a notion of L-Betti numbers for locally compact, second countable, unimodular groups. We study the relation to the standard notion of L-Betti numbers of countable discrete groups for lattices. In this way, several new computations are obtained for countable groups, including lattices in algebraic groups over local elds, and Kac-Moody lattices. We also extend the vanishing of reduce...

No countable C,-free graph contains every countable C,-free graph as a subgraph, for n 2 4. For n = 4, this was proved earlier by J. Pach.

In this paper, spaces with σ-locally countable weak-bases are characterized as the weakly open msss-images of metric spaces (or g-first countable spaces with σ-locally countable cs-networks). To find the internal characterizations of certain images of metric spaces is an interesting research topic on general topology. Recently, S. Xia introduced the concept of weakly open mappings, by using it,...

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

We give, for each countable ordinal ξ ≥ 1, an example of a ∆2 countable union of Borel rectangles that cannot be decomposed into countably many Πξ rectangles. In fact, we provide a graph of a partial injection with disjoint domain and range, which is a difference of two closed sets, and which has no ∆ξ-measurable countable coloring. 2010 Mathematics Subject Classification. Primary: 03E15, Secon...

We consider the homeomorphism problem for countable topological spaces and investigate its descriptive complexity as an equivalence relation. It is shown that even for countable metric spaces the homeomorphism problem is strictly more complicated than the isomorphism problem for countable graphs and indeed it is not Borel reducible to any orbit equivalence relation induced by a Borel action of ...

Spanning trees are a representative example of linear matroid bases that efficiently countable. Perfect matchings Pfaffian bipartite graphs countable common two matrices. Generalizing these two, Webb (2004) introduced the notion pairs as pair matrices for which counting their is tractable via Cauchy–Binet formula.