Cardinal Product Research Articles

Successors of singulars, cofinalities of reduced products of cardinals and productivity of chain conditions

We continue here our investigation of cofinalities of reduced products of regular cardinals and give some applications, such as the non-productiveness ofλ +-c.c. whenλ>2cfλ;.

An extension of an inequality by Ahlswede, El Gamal and pang for pairs of binary codes

Let A and B be two binary codes of length n such that all values of the Hamming distance between codewords of A and B belong to a single residue class modulo 4. It is shown that the product of the cardinalities of A and B does not exceed 2 n when n is even and 2 n − 1 when n is odd. This generalizes a recent result by Ahlswede, El Gamal and Pang. The proof mainly consists in showing that translated versions of A and B are mutually orthogonal.

Kronecker Products and Local Joins of Graphs

When studying the category raph of finite graphs and their morphisms, Ave can exploit the fact that this category has products, [we define these ideas in detail in § 2]. This categorical product of graphs is usually called their Kronecker product, though it has been approached by various authors in various ways and under various names, including tensor product, cardinal product, conjunction and of course categorical product (see for example [6; 7; 11 ; 14; 17 and 23]).

Eine Bemerkung zum Kardinalprodukt

Eine Bemerkung zum Kardinalprodukt

A topology for a lattice-ordered group

Let G be an arbitrary lattice-ordered group. We define a topology on G, called the J \mathcal {J} -topology, which is a group and lattice topology for G and which is preserved by cardinal products. The J \mathcal {J} -topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the J \mathcal {J} -topology to be Hausdorff, and we investigate J \mathcal {J} -topology convergence.

The hulls of representablel-groups andf-rings

A lattice-ordered group (“l-group”)Gwill be calledaP-groupifG=g″ ⊕g′ for eachg∈G(projectable)anSP-groupifG=C⊕C′ for each polarCofG(strongly projectable)anL-groupif each disjoint subset has a 1. u. b. (laterally complete)anO groupif it is both anL-groupand aP-group (orthocomplete).Gisrepresentableif it is anl-subgroup of a cardinal product of totally ordered groups. It follows that aP-group must be representable and henceSP-groups andO-groups are also representable.

On free abelian 𝑙-groups

Let F F denote the free abelian lattice ordered group over an unordered torsion free group G G . Necessary and sufficient conditions are given on G G in order for F F to be an l l -subgroup of a cardinal product of integers. The result encompasses Weinberg’s theorem that the freeness of G G is sufficient. The corresponding embedding theorem for F F is also established whenever G G is completely decomposable and homogeneous.

Completely irreduciblem-ideals in a cardinal product

Completely irreduciblem-ideals in a cardinal product

The Categorical Product of Graphs

Undirected graphs and graph homomorphisms as introduced by Sabidussi (6, p. 386), form a category that admits a categorical product. For the category of graphs and full graph homomorphisms, the categorical product was introduced by Čulik (1) under the name cardinal product. It was independently defined by Weichsel (8) who called it the Kronecker product and investigated the connectedness of products of finitely many factors. Hedetniemi (4) was the first to make use of the fact that the cardinal product is categorical.

On the closure operators of the cardinal product of partially ordered sets

On the closure operators of the cardinal product of partially ordered sets

The closure operators of the cardinal and ordinal sums and products of partially ordered sets and closed lattices

The closure operators of the cardinal and ordinal sums and products of partially ordered sets and closed lattices

Cardinal Algebras. By A. Tarski (with an Appendix on Cardinal Products of Isomorphism Types By B. Jónsson and A. Tarski). Pp. xii, 326. 50s. 1949. (Geoffrey Cumberlege, Oxford University Press)

Cardinal Algebras. By A. Tarski (with an Appendix on Cardinal Products of Isomorphism Types By B. Jónsson and A. Tarski). Pp. xii, 326. 50s. 1949. (Geoffrey Cumberlege, Oxford University Press)