TY - JOUR

T1 - Varieties of abelian topological groups with coproducts

AU - Gabriyelyan, Saak S.

AU - Leiderman, Arkady G.

AU - Morris, Sidney A.

N1 - Funding Information:
This study has been supported by grants from the John Ran-dolph and Dora Haynes Foundation, the California Department of Transportation (through the University of California Transportation Center), and the UCLA Academic Senate.
Publisher Copyright:
© 2015, Springer Basel.

PY - 2015/11/25

Y1 - 2015/11/25

N2 - Varieties of groups, introduced in the 1930s by Garret Birkhoff and B.H. Neumann, are defined as classes of groups satisfying certain laws or equivalently as classes of groups closed under the formation of subgroups, quotient groups, and arbitrary cartesian products. In the 1960s the third author introduced varieties of topological groups as classes of (not necessarily Hausdorff) topological groups closed under subgroups, quotient groups and cartesian products with the Tychonoff topology. While there is only a countable number of varieties of abelian groups, there is a proper class of varieties of abelian topological groups. We observe that while every variety of abelian groups is closed under abelian coproducts, varieties of abelian topological groups are in general not closed under abelian coproducts with the coproduct topology. So this paper studies varieties of abelian topological groups which are also closed under abelian coproducts with the coproduct topology. Noting that the variety of all abelian groups is singly generated, that is, it is the smallest variety containing some particular group, but that the variety of all abelian topological groups is not singly generated, it is proved here that the variety of all abelian topological groups with coproducts is indeed singly generated. There is much literature describing varieties of topological groups generated by various classical topological groups, and the study of varieties with coproducts generated by particular classical topological groups is begun here. Some nice results are obtained about those varieties of abelian topological groups with coproducts which are also closed with regard to forming Pontryagin dual groups.

AB - Varieties of groups, introduced in the 1930s by Garret Birkhoff and B.H. Neumann, are defined as classes of groups satisfying certain laws or equivalently as classes of groups closed under the formation of subgroups, quotient groups, and arbitrary cartesian products. In the 1960s the third author introduced varieties of topological groups as classes of (not necessarily Hausdorff) topological groups closed under subgroups, quotient groups and cartesian products with the Tychonoff topology. While there is only a countable number of varieties of abelian groups, there is a proper class of varieties of abelian topological groups. We observe that while every variety of abelian groups is closed under abelian coproducts, varieties of abelian topological groups are in general not closed under abelian coproducts with the coproduct topology. So this paper studies varieties of abelian topological groups which are also closed under abelian coproducts with the coproduct topology. Noting that the variety of all abelian groups is singly generated, that is, it is the smallest variety containing some particular group, but that the variety of all abelian topological groups is not singly generated, it is proved here that the variety of all abelian topological groups with coproducts is indeed singly generated. There is much literature describing varieties of topological groups generated by various classical topological groups, and the study of varieties with coproducts generated by particular classical topological groups is begun here. Some nice results are obtained about those varieties of abelian topological groups with coproducts which are also closed with regard to forming Pontryagin dual groups.

KW - abelian groups

KW - abelian topological groups

KW - coproducts

KW - free abelian topological group

KW - varieties of groups

KW - varieties of topological groups

UR - http://www.scopus.com/inward/record.url?scp=84942196435&partnerID=8YFLogxK

U2 - 10.1007/s00012-015-0351-2

DO - 10.1007/s00012-015-0351-2

M3 - Article

AN - SCOPUS:84942196435

SN - 0002-5240

VL - 74

SP - 241

EP - 251

JO - Algebra Universalis

JF - Algebra Universalis

IS - 3-4

ER -