Search

This article is concerned with bounded partially ordered sets P such that for every p ∈ P ∖ {1} there exists q ∈ P ∖ {0} such that 0 is the only lower bound of {p, q}. The posets P such that P ≅ Q if and only if P and Q have isomorphic zero-divisor graphs are completely characterized. In the case of finite posets, this result is generalized by proving that posets with isomorphic zero-divisor graphs form an interval under the partial order given by P ≲ Q if and only if there exists a bijective poset-homomorphism P → Q. In particular, the singleton intervals correspond to the posets that are completely determined by their zero-divisor graphs. These results are obtained by exploring universal and couniversal orderings with respect to posets that have isomorphic zero-divisor graphs. [ABSTRACT FROM AUTHOR]

The algebraic composition of an element x of a commutative binary structure ℬ is modeled by its divisor (pseudo)graph Γx (which can be regarded as a simple graph if ℬ is an abelian group), whose vertices are the elements of ℬ such that two vertices a and b are adjacent if and only if ab = x. With respect to the action of a group of symmetries on the set of divisor pseudographs of ℬ, the numbers of orbits are considered, and these numbers are shown to yield groupoid-isotopy invariants when ℬ is a finite abelian group. The minimum number of orbits is computed for every abelian group of order at most 11, and it is also determined for all finite cyclic groups. Moreover, systems of pseudographs that can be realized as those of finite abelian groups are completely characterized. In fact, recurrence relations are given for constructing systems of divisor pseudographs of finite cyclic groups, and all commutative binary structures that are isotopic to a finite abelian group are established. [ABSTRACT FROM AUTHOR]

In this paper, Eulerian complements of zero-divisor graphs are classified for a special class of finite pseudocomplemented posets. Also, it is proved that the complement of the zero-divisor graph of a finite pseudocomplemented poset P is Hamiltonian if and only if P has at least three atoms. These results are applied to zero-divisor graphs and intersection graphs of ideals of reduced commutative Artinian rings. [ABSTRACT FROM AUTHOR]

The notion of a zero-divisor graph is considered for commutative groupoids with zero. Moufang groupoids and certain medial groupoids with zero are shown to have connected zero-divisor graphs of diameters at most four and three, respectively. As x ranges over the elements of a commutative groupoid A (not necessarily with zero), a system of pseudographs is obtained such that the vertices of a pseudograph are the elements of A and vertices a and b are adjacent if and only if ab = x. These systems are completely characterized as being partitions of complete pseudographs Kn whose parts are indexed by the vertices of Kn. Furthermore, morphisms are defined in the class of all such systems of pseudographs making it (categorically) isomorphic to the category of commutative groupoids, thereby combinatorializing the theory of commutative groupoids. Also, concepts of "congruence" and "direct product" that are compatible with those in the category of commutative groupoids are established for these systems of pseudographs. [ABSTRACT FROM AUTHOR]

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented. [ABSTRACT FROM AUTHOR]