Algebraic structures and their applications
Year:2024 | Volume:11 | Issue:2|Papers Count:6

‎‎‎In this paper first we introduce right(resp‎. ‎left) induced hyperlattices and investigate some of their properties‎. ‎Especially‎ ‎a characterization of the smallest strongly regular relation for the class of distributive right/left induced hyperlattice is investigated‎. ‎Next we propose and study the generated hyperlattices from hyperlattices‎. ‎Finally‎, ‎the right induced hyperlattices of two Boolean lattices of four DNA bases and physico-chemical properties of amino acids of four DNA bases are investigated‎.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎The purpose of this paper is to introduce and investigate the submodules of an $R$-module $M$ which satisfy the dual of Property $\mathcal{A}$‎, ‎the dual of strong Property $\mathcal{A}$‎, ‎and the dual of proper strong Property $\mathcal{A}$‎. ‎Moreover‎, ‎a submodule $N$ of $M$ which satisfy Property $\mathcal{S_J(N)}$ and Property $\mathcal{I^M_J(N)}$ will be introduced and investigated‎.

Let $KG$ be the modular group algebra of a group $G$ over a field $K$ of characteristic $p>0$. Recently, we have seen the classification of group algebras $KG$ with upper Lie nilpotency index $t^{L}(KG)$ up to $10p-8$. In this paper, our aim is to classify the modular group algebra $KG$ with upper Lie nilpotency index $11p-9$, for $G'= \gamma_{2}(G)$ as an abelian group.

This paper studies $\lambda$-constacyclic codes and skew $\lambda$-constacyclic codes over the finite commutative non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$ for $\lambda= (1+2u+2u^2)$ and $(3+2u+2u^2)$. We introduce distinct Gray maps and show that the Gray images of $\lambda$-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. It is also shown that the Gray images of skew $\lambda$-constacyclic codes are quasi-cyclic codes of length $2n$ and index 2 over $\mathbb{Z}_4$. Moreover, the structure of $\lambda$-constacyclic codes of odd length $n$ over the ring $R$ is determined and give some suitable examples.

Breakable semihypergroups, defined by a simple property: every non-empty subset of them is a subsemihypergroup. In this paper, we introduce a class of semihypergroups, in which every hyperproduct of $n$ elements is equal to a subset of the factors, called $\pi_n$-semihypergroups. Then, we prove that every semihypergroup of type $\pi_{2k}$, ($k\geq 2$) is breakable and every semihypergroup of type $\pi_{2k+1}$ is of type $\pi_3$. Furthermore, we obtain a decomposition of a semihypergroup of type $\pi_n$ into the cyclic group of order 2 and a breakable semihypergroup. Finally, we give a characterization of semi-symmetric semihypergroups of type $\pi_n$.

One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group $^2{}G_2(q)$, where $q\pm\sqrt{3q}+1$ is a prime number can be uniquely determined by its order and one conjugacy class size.