On Further Properties of Fully Zero-Simple Semihypergroups

Let $mfF_0$ the class of fully zero-simple semihypergroups. In this paper we study the main properties of residual semihypergroup $(H_+, star)$ of a semihypergroup $(H, circ)$ in $mfF_0$. We prove that the quotient semigroup $H_+/eta^*_{H_+}$ is a completely simple and periodic semigroup. Moreover, we find the necessary and sufficient conditions for $(H_+, star)$ to be a torsion group and, in particular, an Abelian $2$-group

On hypercyclic fully zero-simple semihypergroups

Let I be the class of fully zero-simple semihypergroups generated by a hyperproduct. In this paper we study some properties of residual semihypergroup (H_+; star) of a semihypergroup (H; \u25e6)in I. Moreover, we find sufficient conditions for (H; \u25e6) and (H_+; star) to be cyclic

A family of 0-simple semihypergroups related to sequence A000070

For any integer n 65 2, let R_0(n + 1) be the class of 0-semihypergroups H of size n + 1 such that {y} 86 xy 86 {0, y} for all x, y 08 H - {0}, all subsemihypergroups K 86 H are 0-simple and, when |K| 65 3, the fundamental relation \u3b2_K is not transitive. We determine a transversal of isomorphism classes of semihypergroups in R0(n + 1) and we prove that its cardinality is the (n + 1)-th term of sequence A000070 in [21], namely, 11 _{k=0}^n p(k), where p(k) denotes the number of non-increasing partitions of integer k

n−n-absorbing I−I-prime hyperideals in multiplicative hyperrings

In this paper, we define the concept $I-$prime hyperideal in a multiplicative hyperring $R$. A proper hyperideal $P$ of $R$ is an $I-$prime hyperideal if for $a, b \in R$ with $ab \subseteq P-IP$ implies $a \in P$ or $b \in P$. We provide some characterizations of $I-$prime hyperideals. Also we conceptualize and study the notions $2-$absorbing $I-$prime and $n-$absorbing $I-$prime hyperideals into multiplicative hyperrings as generalizations of prime ideals. A proper hyperideal $P$ of a hyperring $R$ is an $n-$absorbing $I-$prime hyperideal if for $x_1, \cdots,x_{n+1} \in R$ such that $x_1 \cdots x_{n+1} \subseteq P-IP$, then $x_1 \cdots x_{i-1} x_{i+1} \cdots x_{n+1} \subseteq P$ for some $i \in \{1, \cdots ,n+1\}$. We study some properties of such generalizations. We prove that if $P$ is an $I-$prime hyperideal of a hyperring $R$, then each of $\frac{P}{J}$, $S^{-1} P$, $f(P)$, $f^{-1}(P)$, $\sqrt{P}$ and $P[x]$ are $I-$prime hyperideals under suitable conditions and suitable hyperideal $I$, where $J$ is a hyperideal contains in $P$. Also, we characterize $I-$prime hyperideals in the decomposite hyperrings. Moreover, we show that the hyperring with finite number of maximal hyperideals in which every proper hyperideal is $n-$absorbing $I-$prime is a finite product of hyperfields.Comment: Journal of algebraic system

1-hypergroups of small sizes

In this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation \u3b2. Many of these hypergroups can be obtained using the aforesaid hypergroup construction