Number Of Isomorphism Types Research Articles

Classifying good gradings on structural matrix algebras

ABSTRACTLet k be a field and let be a partial order on the set . If G is a group, we classify the good G-gradings on the associated structural matrix algebra as the orbits of the action of the automorphism group of on a certain set of functions. We explicitly count the number of isomorphism types of good gradings in some cases where G is finite and the graph associated with has a cyclic associated undirected graph.

Spectra of computable models for Ehrenfeucht theories

We construct an example of a theory with a finite (greater than one) number of isomorphism types of countable models such that its prime and saturated models have computable presentations and there exists a model which lacks in such.

On inverse γ-systems and the number ofL∞λ-equivalent, non-isomorphic models for λ singular

AbstractSuppose λ is a singular cardinal of uncountable cofinality κ. For a modelof cardinality λ, let No() denote the number of isomorphism types of modelsof cardinality λ which areL∞λ-equivalent to. In [7] Shelah considered inverse κ-systemsof abelian groups and their certain kind of quotient limits Gr()/ Fact(). In particular Shelah proved in [7, Fact 3.10] that for every cardinal Μ there exists an inverse κ-systemsuch thatconsists of abelian groups having cardinality at most Μκand card(Gr()/ Fact()) = Μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that θκ< λ for every θ < λ): ifis an inverse κ-system of abelian groups having cardinality < λ, then there is a modelsuch that card() = λ and No() = card(Gr()/ Fact()). The following was an immediate consequence (when θκ< λ for every θ < λ): for every nonzero Μ < λ or Μ = λκthere is a model, of cardinality λ with No() = Μ. In this paper we show: for every nonzero Μ ≤ λκthere is an inverse κ-systemof abelian groups having cardinality < λ such that card(Gr()/ Fact()) = Μ (under the assumptions 2κ< λ and θ<κ< λ for all θ < λ when Μ > λ), with the obvious new consequence concerning the possible value of No. Specifically, the case No() = λ is possible when θκ> λ for every λ < λ.

Minimal Free Resolutions and (G , n )-Complexes for Finite Abelian Groups

let G be a finite abelian group and let n be a non-negative integer. A (G, n)-resolution is a partial free resolution of the ZG-module Z with n+1 intermediate finitely generated free FG-modules F 0 ,..., F n . The kernel K of the map from F n to F n−1 is called a (G, n)-module, and K is called a minimal (G, n)-module if it has the smallest possible Z-rank. We shall calculate the number of isomorphism types of minimal (G, n)-modules and various related numbers. In particular, we shall calculate the number of minimal (G, n)-complexes for all n ≥ 2. This extends work of Browning, Dyer and Sieradski who had already dealt with the case n = 2

The number of infinite substructures

AbstractGiven a relational structure M and a cardinal λ < |M|, let øλ denote the number of isomorphism types of substructures of M of size λ. It is shown that if μ < λ are cardinals, and |M| is sufficiently larger than λ, then øμ ≤ øλ. A description is also given for structures with few substructures of given infinite cardinality.