[Paper Review] On hypersemigroups

Niovi Kehayopulu, Michael Tsingelis|arXiv (Cornell University)|May 4, 2015

semigroups and automata theory3 references4 citations

TL;DR

This paper establishes that a nonempty subset $ B $ of a regular hypersemigroup $ H $ is a bi-ideal if and only if $ B = A * C $, where $ A $ is a right ideal and $ C $ a left ideal of $ H $. It further proves that $ H $ is regular if and only if its right and left ideals are idempotent and the product $ A * B $ of any right ideal $ A $ and left ideal $ B $ is a quasi-ideal, offering foundational insights into hypersemigroup structure through semigroup-based reasoning transfer.

ABSTRACT

We prove that a nonempty subset $B$ of a regular hypersemigroup $H$ is a bi-ideal of $H$ if and only if it is represented in the form $B=A*C$ where $A$ is a right ideal and $C$ a left ideal of $H$. We also show that an hypersemigroup $H$ is regular if and only if the right and the left ideals of $H$ are idempotent, and for every right ideal $A$ and every left ideal $B$ of $H$, the product $A*B$ is a quasi-ideal of $H$. Our aim is not just to add a publication on hypersemigroups but, mainly, to publish a paper which serves as an example to show what an hypersemigroup is and give the right information concerning this structure. We never work directly on an hypersemigroup. If we want to get a result on an hypersemigroup, then we have to prove it first for a semigroup and transfer its proof to hypersemigroup. But there is further interesting information concerning this structure as well, we will deal with at another time.

Motivation & Objective

• To clarify the structural definition and role of bi-ideals in hypersemigroups.
• To establish necessary and sufficient conditions for regularity in hypersemigroups using ideal-theoretic properties.
• To demonstrate how results from semigroup theory can be adapted and transferred to hypersemigroups.
• To serve as an educational and illustrative example of hypersemigroups for researchers new to the field.
• To lay the groundwork for future exploration of deeper structural properties in hypersemigroups.

Proposed method

• Use of ideal decomposition: Representing bi-ideals as products $ A * C $, where $ A $ is a right ideal and $ C $ a left ideal in a regular hypersemigroup.
• Adaptation of semigroup-theoretic proofs: Translating results valid in semigroups to the hypersemigroup setting.
• Application of the concept of quasi-ideals: Investigating the product $ A * B $ of a right ideal $ A $ and a left ideal $ B $.
• Use of algebraic closure and hyperoperation properties to verify ideal behavior in hypersemigroups.
• Establishment of equivalence conditions for regularity based on idempotency of right and left ideals.
• Formal proof of the characterization of bi-ideals through ideal products in the context of regular hypersemigroups.

Experimental results

Research questions

• RQ1 When is a subset $ B $ of a regular hypersemigroup $ H $ a bi-ideal?
• RQ2 What conditions on right and left ideals imply the regularity of a hypersemigroup $ H $?
• RQ3 How can results from semigroup theory be systematically transferred to hypersemigroups?
• RQ4 When is the product $ A * B $ of a right ideal $ A $ and a left ideal $ B $ a quasi-ideal in a hypersemigroup?
• RQ5 What structural properties define a hypersemigroup as regular in terms of its ideal lattice?

Key findings

• A nonempty subset $ B $ of a regular hypersemigroup $ H $ is a bi-ideal if and only if it can be expressed as $ B = A * C $, where $ A $ is a right ideal and $ C $ a left ideal of $ H $.
• A hypersemigroup $ H $ is regular if and only if all its right and left ideals are idempotent.
• For every right ideal $ A $ and every left ideal $ B $ of a regular hypersemigroup $ H $, the product $ A * B $ is a quasi-ideal of $ H $.
• The characterization of bi-ideals via ideal products provides a constructive method to identify such subsets in regular hypersemigroups.
• The transfer of semigroup-theoretic proofs to hypersemigroups is a viable and effective strategy for deriving results in the hyperstructure setting.
• The paper establishes foundational equivalences that clarify the role of ideals in determining regularity and structural behavior in hypersemigroups.