Einblick - Mathematics - # SemigroupTheory

On the Structure and Topology of Simple Inverse ω-Semigroups with Compact Maximal Subgroups

Q: How do the findings of this paper generalize to other classes of semigroups, such as those with non-compact maximal subgroups or those that are not simple?

The findings of this paper heavily rely on the properties of simple inverse ω-semigroups with compact maximal subgroups. This specific structure allows for a neat characterization using Bruck-Reilly extensions and a clear understanding of their locally compact topologies. Generalizing these results to broader classes of semigroups poses significant challenges: Non-compact maximal subgroups: The compactness of maximal subgroups is crucial for applying tools like the sum direct topology and deriving the dichotomy between compact and discrete topologies. With non-compact maximal subgroups, the topological structure becomes much more complex, and alternative approaches would be needed. For instance, Example 2.10 demonstrates that a Hausdorff locally compact simple inverse ω-semigroup can admit topologies other than compact or discrete when a maximal subgroup is non-compact. Non-simple semigroups: Simple semigroups, by definition, lack non-trivial ideals, which simplifies their structure. For non-simple semigroups, the presence of ideals introduces additional complexity. The interplay between the topology on the semigroup and the topologies on its ideals would need careful consideration. The techniques used in the paper, focusing on the Bruck-Reilly extension of the entire semigroup, might not be directly applicable. Therefore, extending these findings would require new techniques and potentially different topological representations. Investigating semigroups with specific properties, such as local compactness or restrictions on the structure of ideals, could be promising directions for future research.

Q: Could there be alternative topological representations of these semigroups beyond Bruck-Reilly extensions?

While Bruck-Reilly extensions provide a powerful tool for representing simple inverse ω-semigroups, exploring alternative topological representations is an intriguing question. Some potential avenues include: Semilattices of topological spaces: Since these semigroups can be represented as a semilattice of groups, one could explore representing them as a semilattice of topological spaces, where each space corresponds to a maximal subgroup. This approach might offer more flexibility in handling cases with non-compact maximal subgroups. Actions on topological spaces: Representing semigroups through their actions on suitable topological spaces is a common technique. Investigating actions that capture the specific properties of these semigroups, such as the ω-chain of idempotents, could lead to alternative representations. Inverse semigroup actions on C-algebras:* This approach, inspired by the theory of C*-dynamical systems, could provide a richer framework for studying the interplay between the algebraic and topological structures of these semigroups. It's important to note that the existence and usefulness of alternative representations would depend on the specific goals and the class of semigroups under consideration.

Q: What are the potential applications of these findings in other areas of mathematics or theoretical computer science, such as automata theory or formal language theory?

The findings of this paper, focusing on the topological and algebraic structure of specific semigroups, could potentially contribute to other areas like automata theory and formal language theory: Automata with infinite states: ω-semigroups, with their inherent notion of infinity, could be relevant in modeling automata with infinitely many states. The topological aspects studied in the paper might provide insights into the behavior and properties of such automata. Formal languages of infinite words: The structure of simple inverse ω-semigroups could be leveraged to study formal languages consisting of infinite words (ω-languages). The topological properties of these semigroups might translate into interesting characteristics of the corresponding languages. Symbolic dynamics: The interplay between topology and algebraic structure in semigroups could find applications in symbolic dynamics, particularly in studying systems with infinitely many configurations. Geometric group theory: The techniques used to analyze the topological structure of these semigroups might inspire new approaches in geometric group theory, especially for groups with interesting semigroup actions. While these are potential areas of application, further research is needed to establish concrete connections and explore the full extent of these findings' implications in other fields.

Kernkonzepte

This paper characterizes the structure and topology of simple inverse ω-semigroups with compact maximal subgroups, showing they are topologically isomorphic to specific Bruck-Reilly extensions and examining the topological implications of adjoining a zero element.

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arxiv.org

Gutik, O., & Maksymyk, K. (2024, November 6). On Semitopological Simple Inverse ω-Semigroups with Compact Maximal Subgroups. arXiv. [Preprint]

This paper aims to describe the structure and topology of simple inverse Hausdorff semitopological ω-semigroups where all maximal subgroups are compact.

Wichtige Erkenntnisse aus

On semitopological simple inverse $\omega$-semigroups with compact maximal subgroups

by Oleg Gutik, ... um arxiv.org 11-08-2024

https://arxiv.org/pdf/2409.06344.pdf

$On semitopological simple inverse $\omega$-semigroups with compact maximal subgroups$

Tiefere Fragen

How do the findings of this paper generalize to other classes of semigroups, such as those with non-compact maximal subgroups or those that are not simple?

The findings of this paper heavily rely on the properties of simple inverse ω-semigroups with compact maximal subgroups. This specific structure allows for a neat characterization using Bruck-Reilly extensions and a clear understanding of their locally compact topologies. Generalizing these results to broader classes of semigroups poses significant challenges:

Non-compact maximal subgroups: The compactness of maximal subgroups is crucial for applying tools like the sum direct topology and deriving the dichotomy between compact and discrete topologies. With non-compact maximal subgroups, the topological structure becomes much more complex, and alternative approaches would be needed. For instance, Example 2.10 demonstrates that a Hausdorff locally compact simple inverse ω-semigroup can admit topologies other than compact or discrete when a maximal subgroup is non-compact.

Non-simple semigroups:  Simple semigroups, by definition, lack non-trivial ideals, which simplifies their structure.  For non-simple semigroups, the presence of ideals introduces additional complexity. The interplay between the topology on the semigroup and the topologies on its ideals would need careful consideration. The techniques used in the paper, focusing on the Bruck-Reilly extension of the entire semigroup, might not be directly applicable.
Therefore, extending these findings would require new techniques and potentially different topological representations. Investigating semigroups with specific properties, such as local compactness or restrictions on the structure of ideals, could be promising directions for future research.

Could there be alternative topological representations of these semigroups beyond Bruck-Reilly extensions?

While Bruck-Reilly extensions provide a powerful tool for representing simple inverse ω-semigroups, exploring alternative topological representations is an intriguing question. Some potential avenues include:

Semilattices of topological spaces: Since these semigroups can be represented as a semilattice of groups, one could explore representing them as a semilattice of topological spaces, where each space corresponds to a maximal subgroup. This approach might offer more flexibility in handling cases with non-compact maximal subgroups.

Actions on topological spaces:  Representing semigroups through their actions on suitable topological spaces is a common technique.  Investigating actions that capture the specific properties of these semigroups, such as the ω-chain of idempotents, could lead to alternative representations.

Inverse semigroup actions on C-algebras:*  This approach, inspired by the theory of C*-dynamical systems, could provide a richer framework for studying the interplay between the algebraic and topological structures of these semigroups.
It's important to note that the existence and usefulness of alternative representations would depend on the specific goals and the class of semigroups under consideration.

What are the potential applications of these findings in other areas of mathematics or theoretical computer science, such as automata theory or formal language theory?

The findings of this paper, focusing on the topological and algebraic structure of specific semigroups, could potentially contribute to other areas like automata theory and formal language theory:

Automata with infinite states:  ω-semigroups, with their inherent notion of infinity, could be relevant in modeling automata with infinitely many states. The topological aspects studied in the paper might provide insights into the behavior and properties of such automata.

Formal languages of infinite words:  The structure of simple inverse ω-semigroups could be leveraged to study formal languages consisting of infinite words (ω-languages).  The topological properties of these semigroups might translate into interesting characteristics of the corresponding languages.

Symbolic dynamics:  The interplay between topology and algebraic structure in semigroups could find applications in symbolic dynamics, particularly in studying systems with infinitely many configurations.

Geometric group theory:  The techniques used to analyze the topological structure of these semigroups might inspire new approaches in geometric group theory, especially for groups with interesting semigroup actions.
While these are potential areas of application, further research is needed to establish concrete connections and explore the full extent of these findings' implications in other fields.