Halaš and Jukl associated the zero-divisor graph G to a poset (X,≤) with zero by declaring two distinct elements x and y of X to be adjacent if and only if there is no non-zero lower bound for {x, y}. We characterize all the graphs that can be realized as the zero-divisor graph of a poset. Using this, we classify posets whose zero-divisor graphs are the same. In particular we show that if V is an n-element set, then there exist $\begin{array}{} \sum\limits_{\log_2(n+1)\leq k\leq n}^{}\binom{n}{k}\binom{2^k-k-1}{n-k} \end{array} $ reduced zero-divisor graphs whose vertex sets are V.

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Abstract:In this paper, we study the symmetry classes of tensors associated with some Frobenius groups of order pq, where q|p-1as a subgroups of the full symmetric group on p letters. We calculate the dimension of the symmetry classes of tensor associated with some Frobenius groups and some irreducible complex characters and we obtain two useful corollary with an example.

In this paper, we investigate connections between some algebraic properties of commutative rings and topological properties of their minimal and maximal prime spectrum with respect to the flat topology. We show that for a commutative ring [Formula: see text], the ascending chain condition on principal annihilator ideals of [Formula: see text] holds if and only if [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology and we give a characterization for a topological space [Formula: see text] for which [Formula: see text] is a Noetherian topological space as a subspace of [Formula: see text] with respect to the flat topology. Also, we give a characterization for rings whose maximal prime spectrum is a compact topological space with respect to the flat topology. Some other results are obtained too.

Abstract:The purpose of this paper is making a construction and generalization of Molaei's generalized groups by using construction of the Rees matrix semigroup over a polygroup H and a matrix with entries in H. We call it "Molaei's generalized hypergroups" and we give some examples.