Author Identifier (ORCID)
Abstract
This paper develops a geometric framework for analyzing the ideal structure of the bipolar semigroup ��={(−��,��)∣��,��∈ℝ+0} under coordinate-wise addition. Subsets of B are interpreted as planar regions, allowing ideals to be described in terms of boundary behavior. In particular, we prove that the complement of a simply connected region is an ideal of the commutative additive semigroup (��,+) if and only if its boundary contains no strictly decreasing segment. This provides a direct and visually verifiable criterion for ideality, linking algebraic structure to geometric shape. Each ideal can be written as a union of translates of the form ��+��, with minimal generating sets determined by boundary structure. Potential applications to modeling bipolar system states, including cybersecurity contexts, are also discussed. These results uncover an intrinsic symmetry between algebraic closure and geometric monotonicity, offering a new perspective on semigroup ideals through spatial structure. In contrast to our previous work on the semiring (��,+,·), where ideals necessarily exhibit symmetry with respect to the line ��=−��, we show that removing the multiplicative operation leads to a fundamentally different geometric behavior: ideals in the semigroup (��,+) no longer possess symmetric shapes. This demonstrates that the multiplicative structure is the key mechanism enforcing geometric symmetry.
Keywords
bipolar set, boundary monotonicity, geometric characterization, ideal, semigroup
Document Type
Journal Article
Date of Publication
6-1-2026
Article Number
899
Volume
18
Issue
6
Publication Title
Symmetry
Publisher
MDPI
School
School of Science
RAS ID
95229
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Recommended Citation
Laipaporn, K., Maungchang, R., Cook, D. M., & Khachorncharoenkul, P. (2026). Geometric characterization of ideals in bipolar semigroups. Symmetry, 18(6). https://doi.org/10.3390/sym18060899