Adjoint algebraic entropy

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove an Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact abelian group, having finite positive algebraic entropy, is discontinuous.Comment: 27 page

Native American Women in Academia

This essay is meant as a guide to help Indian women understand aspects of the race- and gender-based educational and social problems created by the American system to subdue their ascension. I present real-life examples of women who have flourished, despite the intentional obstacles laid before them. I hope to do justice for the women of my ethnicity who have contributed to my life in ways they may not have known affected me, including as matriarchs of the family, scholastic mentors, inspirational figures, and supporters of my past attempts to understand gender differences while retaining the fact that they were my equal. I came upon this idea during research of Native views of men and women and how they have had roles conferred upon them—some the same, others not—by the Great Spirit and must work in unison, relying on one another to achieve aspirations; only when the European colonizer disrupted this time-honored cycle did Native people refrain from the natural ways of humane treatment.https://repository.usfca.edu/listening_to_the_voices/1007/thumbnail.jp

When the Intrinsic Algebraic Entropy is not Really Intrinsic

The intrinsic algebraic entropy ent(ɸ) of an endomorphism ɸ of an Abelian group G can be computed using fully inert subgroups of ɸ-invariant sections of G, instead of the whole family of ɸ-inert subgroups. For a class of groups containing the groups of finite rank, aswell as those groupswhich are trajectories of finitely generated subgroups, it is proved that only fully inert subgroups of the group itself are needed to comput ent(ɸ). Examples show how the situation may be quite different outside of this class

An Independence Result on Cotorsion Theories over Valuation Domains

AbstractIt is shown that, over suitable valuation domains R with field of quotients Q, the cotorsion theory GK generated by K=Q/R coincides with the cotorsion theory C∂ cogenerated by the Fuchs' divisible module ∂, provided that Gödel's Axiom of Constructibility V=L is assumed. On the other hand, assuming Martin's Axiom and the negation of the Continuum Hypothesis, it is proved that the cotorsion theory GK is strictly smaller than C∂ by exhibiting a strongly (ℵ1−K)-free divisible module M of projective dimension 2 such that Ext1R(M,K)=0. Applications to Whitehead modules are derived

Abelian p-groups with minimal full inertia

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert

The Pierce Decomposition and Pierce Embedding of Endomorphism Rings of Abelian p-Groups

We prove that more classes of groups than those described by Pierce have the property that the map Ψ is surjective, and we furnish examples of groups which do not have this property. Several results connecting the Pierce decomposition and the Pierce embedding of End(G) are obtained that allow one to derive general conditions on a group G which ensure that the Pierce embedding of End(G) is not surjective