Narrow Results

Let $R$ be a commutative ring with identity and $I$ be an ideal of $R$. The zero-divisor graph of $R$ with respect to $I$, denoted by ${\mathrm{\Gamma }}_I(R)$, is the graph whose vertices are the set $\{x\in R\setminus I|xy\in I\mathrm{\text{for}}\mathrm{\text{some}}y\in R\setminus I\}$ with distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. The cozero-divisor graph ${\mathrm{\Gamma }}^{\prime }(R)$ of $R$ is the graph whose vertices are precisely the non-zero, non-unit elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x\notin yR$ and $y\notin xR$. In this paper, we introduced and investigated a new generalization of the cozero-divisor graph ${\mathrm{\Gamma }}^{\prime }(R)$ of $R$ denoted by ${\mathrm{\Gamma }}_I^{″}(R)$. In fact, ${\mathrm{\Gamma }}_I^{″}(R)$ is a dual notion of ${\mathrm{\Gamma }}_I(R)$.

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.

In 2006, Nelson and Wong introduced that every finite quandle can be decomposed into a disjoint union of some quandles and for given $n$ quandles satisfying certain conditions, there is a quandle which is a disjoint union of the $n$ quandles. In 2008, Ehrman, Gurpinar, Thibault and Yetter also introduced similar results about the decomposition of quandles. In this paper, we observe an operation table $Q$ which consists of four sub-operation tables, where diagonal tables are quandle operations. We study which conditions make $Q$ a quandle operation table.

We study problems related to indecomposability of modules over certain local finite-dimensional trivial extension algebras. We do this by purely combinatorial methods. We introduce the concepts of graph of cyclic modules, of combinatorial dimension, and of fundamental combinatorial dimension of a module. We use these concepts to establish, under favorable conditions, criteria for the indecomposability of a module. We present categorified versions of these constructions and we use this categorical framework to establish criteria for the indecomposability of modules of infinite rank.

This study aims to investigate $L^B$-valued GFA from algebraic and topological perspectives, where $L$ stands for residuated lattice and B is a set of propositions about the general fuzzy automata, in which its underlying structure is a complete infinitely distributive lattice. Further, the concepts of $L^B$-valued general fuzzy automata (or simply $L^B$-valued GFA) contractible spaces, $L^B$-valued GFA path homotopy, $L^B$-valued GFA retraction, $L^B$-valued GFA deformation retraction, $L^B$-valued GFA path connected space and $L^B$-valued GFA homotopy equivalent space are introduced and explicated. In addition, $L^B$-valued GFA fundamental groups are proposed and studied. Regarding these issues, some properties are also established and explained.

Over the past four decades an extensive literature covered the properties of $S$-acts. However, only few studies had generalized some known properties of $S$-acts to the $S$-posets. The reversible, and indecomposable properties in $S$-posets have been addressed previously but connectivity has not been defined in $S$-posets yet. Connectivity property was found to be related to those of reversibility and flatness in the category of $S$-acts. The primary objective of this paper is to define connectivity in the category of $S$-posets for both versions: ordered “poconnected” and unordered “connected”. Examples are presented to show the difference between the two versions. The relationship between connectivity with other properties such as reversibility, and indecomposability had also been investigated. We show that the poconnected in $S$-posets is always indecomposable, but the inverse is not true. We also find that the weakly reversible is always connected and indecomposable. These relations among these properties in $S$-posets are different from their corresponding relations in $S$-acts.

A Kähler graph consists of two kinds of graphs having the same set of vertices. As candidates of model Kähler graphs which correspond to homogeneous spaces admitting magnetic fields, vertex transitive normal Kähler graphs are proposed in [5]. To give many examples of such Kähler graphs, we define some product operations. In this note, we restrict ourselves to the product Kähler graphs whose first kind graphs are of tensor product type, and study whether they are connected, are bipartite, are vertex-transitive and are normal.