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Abul Basar
Open AccessArticle

On characterization of regular ordered ternary semihypergroups by relative

Annals of Communications in Mathematics 2021

, 4 (1)

, 73-88

DOI: https://doi.org/10.62072/acm.2021.040108

AbstractIn the present paper, we introduce the relative left, right, lateral, two-sided hyperideal, relative quasi-hyperideal, relative bi-hyperideal, relative sub-idempotent ordered bi-hyperideal, relative generalized quasi-hyperideal, relative generalized bi-hyperideal, relative regularity of ordered ternary semihypergroups and relative left (right, lateral) simple ordered ternary semihypergroups. We characterize relative regular ordered ternary semihypergroups through relative quasi-hyperideals and relative bi-hyperideals. We also obtain some results based on relative simple ordered ternary semihypergroups, and other results connecting these relative hyperideal-theoretic notions.
Open AccessArticle

On some relative weakly hyperideals and relative prime bi-hyperideals in ordered hypersemigroups and in involution ordered hypersemigroups

Annals of Communications in Mathematics 2020

, 3 (1)

, 63-79

DOI: https://doi.org/10.62072/acm.2020.030107

AbstractThe aim of the present paper is to define and bring together the fundamental definitions such as relative hyperideals, relative bi-hyperideals, relative quasi-hyperideals, relative prime hyperideals, relative weakly prime hyperideals, relative semiprime hyperideals, relative prime and relative semiprime bi-hyperideals, and hyper relative regularity of dynamic algebraic character to develop the theory of hypersemigroups, and obtain the results relating to and connecting these hyperideal-theoretic definitions of this vast theory to the larger framework of the algebraic area of ordered hypersemigroups as well as of involution ordered hypersemigroups.
Open AccessArticle

On Relative (2, 2)-Γ-hyperideals of 2-duo Ordered Γ-semihypergroups

Annals of Communications in Mathematics 2024

, 7 (1)

, 47-56

DOI: https://doi.org/10.62072/acm.2024.070105

AbstractIn this paper, first we obtain the necessary and sufficient condition for an ordered Γ-semihypergroup H to be a relative completely regular 2-duo ordered Γ-semihypergroup for any relative (2, 2)-Γ-hyperideal as well as for any relative (2, 2)-quasi-Γ-hyperideal of H. Then, we find the necessary and sufficient condition that Q = (Q2]S for every relative (2, 2)-Γ-hyperideal Q of H to be a relative quasi-Γ-prime with S ⊆ H. Finally, we prove the necessary and sufficient condition for relative (2, 2)-Γ-hyperideal to be a relative quasi-Γ-prime for a relative completely regular and relative (2, 2)-Γ-hyperideal of H making a chain with inclusive relation.
Open AccessArticle

Ordered Γ-semihypergroup of the Associated Γ-semihypergroup with All Relative Bi-Γ-hyperideals

Annals of Communications in Mathematics 2024

, 7 (1)

, 71-79

DOI: https://doi.org/10.62072/acm.2024.070107

AbstractIn this paper, the main goal is to study an ordered Γ-semihypergroup H in the context of the characterizations of the associated Γ-semihypergroup B(H) of all bi-Γ-hyperideals of H. We show that an ordered Γ-semihypergroup H is a Clifford ordered Γ-semihypergroup if and only if B(H) is a semilattice. We also show that a Γsemihypergroup B(H) is a normal band if and only if the ordered Γ-semihypergroup H is simultaneously regular and intra regular. Furthermore, for each subclass S with many bands, we prove that for an ordered Γ-semihypergroup H, the conditional inclusion B(H) ∈ S holds true.