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1. CJM Online first

Fernández Bretón, David J.
Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property
We answer two questions of Hindman, Steprāns and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\operatorname{cov}(\mathcal{M})=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.

Keywords:ultrafilter, Stone-Cech compactification, sparse ultrafilter, strongly summable ultrafilter, union ultrafilter, finite sum, additive isomorphism, trivial sums property, Boolean group, abelian group
Categories:03E75, 54D35, 54D80, 05D10, 05A18, 20K99

2. CJM 1999 (vol 51 pp. 49)

Ndombol, Bitjong; El haouari, M.
Algèbres quasi-commutatives et carrés de Steenrod
Soit $k$ un corps de caract\'eristique $p$ quelconque. Nous d\'efinissons la cat\'egorie des $k$-alg\`ebres de cocha\^{\i}nes fortement quasi-commutatives et nous donnons une condition n\'ecessaire et suffisante pour que l'alg\`ebre de cohomologie \`a coefficients dans $\mathbb{Z}_2$ d'un objet de cette cat\'egorie soit un module instable sur l'alg\`ebre de Steenrod \`a coefficients dans $\mathbb{Z}_2$. A tout c.w.\ complexe simplement connexe de type fini $X$ on associe une $k$-alg\`ebre de cocha\^{\i}nes fortement quasi-commutative; la structure de module sur l'alg\`ebre de Steenrod d\'efinie sur l'alg\`ebre de cohomologie de celle-ci co\"\i ncide avec celle de $H^*(X; \mathbb{Z}_2)$. We define the category of strongly quasi-commutative cochain $k$-algebras, where $k$ is a field of any characteristic $p$. We give a necessary and sufficient condition which enables the cohomology algebra with $\mathbb{Z}_2$-coefficients of an object in this category to be an unstable module on the $\mathbb{Z}_2$-Steenrod algebra. To each simply connected c.w.\ complex of finite type $X$ is associated a strongly quasi-commutative model and the module structure over the $\mathbb{Z}_2$-Steenrod algebra defined on the cohomology of this model is the usual structure on $H^*(X; \mathbb{Z}_2)$.

Keywords:algèbres de cocha\^{\i}nes (fortement) quasi-commutatives, $T (V)$-modèle, carrés de Steenrod, quasi-isomorphisme
Categories:55P62, 55S05

3. CJM 1998 (vol 50 pp. 210)

Zhao, Kaiming
Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$
In this paper, we determine when two simple generalized Cartan type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type $W$ Lie algebras.

Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjecture
Categories:17B40, 17B65, 17B56, 17B68

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