Generalized topological ordered groups

F. J. García-Pacheco; M. A. Moreno-Frías; S. Rahbariyan; F. J. García-Pacheco; M. A. Moreno-Frías; S. Rahbariyan

doi:10.3934/era.2026147

Electronic Research Archive

2026, Volume 34, Issue 5: 3277-3288. doi: 10.3934/era.2026147

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Generalized topological ordered groups

Department of Mathematics, University of Cadiz, Puerto Real 11519, Spain

Received: 12 December 2025 Revised: 22 February 2026 Accepted: 31 March 2026 Published: 15 April 2026

A structured group is a group endowed with a binary internal relation compatible with the group operation, which can be generalized to a type of partially ordered group. In this paper, we investigated two kinds of structured groups: first, those in which the binary internal relation is an equivalence relation, where we showed that the quotient set acquires a group structure, leading to a characterization of its normal subgroups. As a consequence, we obtained that group quotients by congruencies are equivalent to group quotients by normality. Additionally, we proved that the quotient set of a topological group with a compatible equivalence relation is also a topological group whose canonical projection is an open map. The second type of structured groups, we considered, are those with a partial order relation. In this case, we identified nonrestrictive sufficient conditions for the order topology to define a monoid topology, hence, a group topology.
- topological group,
- ordered group,
- order topology
Citation: F. J. García-Pacheco, M. A. Moreno-Frías, S. Rahbariyan. Generalized topological ordered groups[J]. Electronic Research Archive, 2026, 34(5): 3277-3288. doi: 10.3934/era.2026147

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Abstract

A structured group is a group endowed with a binary internal relation compatible with the group operation, which can be generalized to a type of partially ordered group. In this paper, we investigated two kinds of structured groups: first, those in which the binary internal relation is an equivalence relation, where we showed that the quotient set acquires a group structure, leading to a characterization of its normal subgroups. As a consequence, we obtained that group quotients by congruencies are equivalent to group quotients by normality. Additionally, we proved that the quotient set of a topological group with a compatible equivalence relation is also a topological group whose canonical projection is an open map. The second type of structured groups, we considered, are those with a partial order relation. In this case, we identified nonrestrictive sufficient conditions for the order topology to define a monoid topology, hence, a group topology.

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