complemented zero divisor graphs and boolean rings abstract for a commutative ring r the zero divisor graph of r is the graph whose vertices are the nonzero zero divisors of r such that the vertices x and y are adjacent if and only if x y 0 in this paper we classify the zero divisor graphs of boolean rings as well as those of boolean rings that are rationally complete we also provide a complete list of those rings whose zero divisor graphs have the property that every vertex is either an end or adjacent to an end

the zero divisor graph of a commutative ring for each commutative ring r we associate a simple graph r we investigate the interplay between the ring theoretic properties of r and the graph theoretic properties of r

planar zero divisor graphs abstract this paper answers the question of anderson frazier lauve and livingston for which finite commutative rings r is the zero divisor graph r planar we build upon and extend work of akbari maimani and yassemi who proved that if r is any local ring with more than 32 elements and r is not a field then r is not planar they left open the question is it true that for any local ring r of cardinality 32 which is not a field r is not planar in this paper we answer this question in the affirmative we prove that if r is any local ring with more than 27 elements and r is not a field then r is not planar moreover we determine all finite commutative local rings whose zero divisor graph is planar

boolean rings and reciprocal eigenvalue properties abstract let a be the adjacency matrix of the zero divisor graph r of a finite commutative ring r containing nonzero zero divisors in this paper it is shown that r is the zero divisor graph of a boolean ring if and only if det a 1 also a is similar to plus or minus its inverse whenever r is a boolean ring as a consequence it is proved that r is the zero divisor graph of a boolean ring if and only if the set of eigenvalues including multiplicities of r can be partitioned into 2 element subsets of the form 1 furthermore any finite boolean ring r is characterized by the degree and coefficients of the characteristic polynomial of a

the cuntz semigroup and comparison of open projections abstract we show that a number of naturally occurring comparison relations on positive elements in a c algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the c algebra in particular we show that cuntz comparison of positive elements corresponds to a comparison relation on open projections that we call cuntz comparison and which is defined in terms of and is weaker than a comparison notion defined by peligrad and zsido the latter corresponds to a well known comparison relation on positive elements defined by blackadar we show that murray von neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the c algebra we use these findings to give a new picture of the cuntz semigroup

beck s coloring of a commutative ring a commutative ring r can be considered as a simple graph whose vertices are the elements of r and two different elements x and y of r are adjacent if and only if xy 0 beck conjectured that r cl r we give a counterexample where r is a finite local ring with cl r 5 but r 6 we show that if a is a regular noetherian ring with maximal ideals n1 ns such that each a ni is finite then for r a nn11 nnss r cl r finally we give a complete listing of all finite rings r with r 4

coloring of commutative rings the purpose of this article is to present the idea of coloring of a commutative ring this idea establishes a connection between graph theory and commutative ring theory which hopefully will turn out to be mutually beneficial for these two branches of mathemathics in this introductory paper we shall mainly be interested in characterizing and discussing the rings which are finitely colorable leaving aside for the moment possible applications to graph theory let r be a commutative ring we consider r as a simple graph whose vertices are the elements of r such that two different elements x and y are adjacent iff xy 0 we let x r denote the chromatic number of the graph i e the minimal number of colors which can be assigned to the elements of r in such a way that every two adjacent elements have different colors a subset c xi x is called a clique provided xixj 0 for all i j if r contains a clique with n elements and every clique has at most y1 elements we say that the clique number of r is n and write clique r n if the sizes of the cliques in r are not bounded we define clique r co we shall show that clique r co actually entails the existence of an infinite clique obviously x r 3 clique r and for general graph g we certainly may have x g clique g however in the case of commutative rings we have not found any example where x r clique r the lack of such counterexamples together with the fact that we have been able to establish the equality x r clique r for certain rather wide classes of rings like reduced and principal ideal rings motivates the following conjecture

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

rings whose zero divisor graphs have positive genus it is shown that for a fixed positive integer g there are finitely many isomorphism classes of rings whose zero divisor graph has genus g the proof can then be modified to yield an analogous result for nonorientable genus

the total graph of a commutative ring abstract let r be a commutative ring with nil r its ideal of nilpotent elements z r its set of zero divisors and reg r its set of regular elements in this paper we introduce and investigate the total graph of r denoted by t r it is the undirected graph with all elements of r as vertices and for distinct x y r the vertices x and y are adjacent if and only if x y z r we also study the three induced subgraphs nil r z r and reg r of t r with vertices nil r z r and reg r respectively

non commuting graph of a group abstract let g be a non abelian group and let z g be the center of g associate a graph g called non commuting graph of g with g as follows take g z g as the vertices of g and join two distinct vertices x and y whenever x y y x we want to explore how the graph theoretical properties of g can effect on the group theoretical properties of g we conjecture that if g and h are two non abelian finite groups such that g h then g h among other results we show that if g is a finite non abelian nilpotent group and h is a group such that g h and g h then h is nilpotent

on some graphs which satisfy reciprocal eigenvalue properties abstract we consider only simple graphs consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contracting all matching edges is also bipartite on the class h g of such graphs g the equivalence of the following statements are known i the reciprocal of the spectral radius of the adjacency matrix a g is the least positive eigenvalue of the adjacency matrix ii the graph is isomorphic to its inverse where the inverse of a graph g is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g via a diagonal matrix of 1s iii the graph has the reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g iv the graph has the strong reciprocal eigenvalue property that is the reciprocal of each eigenvalue of the adjacency matrix a g is also an eigenvalue of a g and they both have the same multiplicities v the graph is a corona graph that is it is obtained by taking a bipartite graph and then by inserting a new adjacent vertex of degree one at each vertex it is known that the statements iii and iv are not equivalent if we allow nonbipartite graphs however the equivalence of these two is not yet known for the class of bipartite graphs with unique perfect matchings where we relax the condition of bipartiteness of the contracted graph in this article we establish the equivalence of the first four statements for a class larger than h g we then supply a constructive structural description of those graphs

basic observables for a calculus for global computing we develop the semantic theory of a foundational language for modelling applications over global computers whose interconnection structure can be explicitly manipulated together with process distribution process mobility and remote asynchronous communication through distributed data repositories the language has primitives for explicitly modelling inter node connections and for dynamically activating and deactivating them for the proposed language we define natural notions of extensional observations and study their closure under operational reductions and or language contexts to obtain barbed congruence and may testing equivalence we then focus on barbed congruence and provide an alternative characterisation in terms of a labelled bisimulation to test practical usability of the semantic theory we model a system of communicating mobile devices and use the introduced proof techniques to verify one of its key properties

the diameter of a zero divisor graph let r be a commutative ring and let z r be its set of nonzero zero divisors the set z r makes up the vertices of the corresponding zero divisor graph r with two distinct vertices forming an edge if the product of the two elements is zero the distance between vertices a and b not necessarily distinct from a is the length of the shortest path connecting them and the diameter of the graph diam r is the sup of these distances for a reduced ring r with nonzero zero divisors 1 diam r diam r x diam r x 3 a complete characterization for the possible diameters is given exclusively in terms of the ideals of r a similar characterization is given for diam r and diam r x when r is nonreduced various examples are provided to illustrate the difficulty in dealing with the power series ring over a nonreduced ring

some properties on the tensor product of graphs obtained by monogenic semigroups abstract in das et al 2013 8 a new graph s m on monogenic semigroups s m with zero having elements 0 x x 2 x 3 x n has been recently defined the vertices are the non zero elements x x 2 x 3 x n and for 1 i j n any two distinct vertices x i and x j are adjacent if x i x j 0 in s m as a continuing study in akgunes et al 2014 3 it has been investigated some well known indices first zagreb index second zagreb index randic index geometric arithmetic index atom bond connectivity index wiener index harary index first and second zagreb eccentricity indices eccentric connectivity index the degree distance over s m in the light of above references our main aim in this paper is to extend these studies over s m to the tensor product in detail we will investigate the diameter radius girth maximum and minimum degree chromatic number clique number and domination number for the tensor product of any two not necessarily different graphs s m 1 and s m 2

a network conscious π calculus and its coalgebraic semantics traditional process calculi usually abstract away from network details modeling only communication over shared channels they however seem inadequate to describe new network architectures such as software defined networks where programs are allowed to manipulate the infrastructure in this paper we present the network conscious p calculus ncpi a proper extension of the p calculus with an explicit notion of network network links and nodes are represented as names in full analogy with ordinary p calculus names and observations are routing paths through which data is transported however restricted links do not appear in the observations which thus can possibly be as abstract as in the p calculus then we construct a presheaf based coalgebraic semantics for ncpi along the lines of turi plotkin s approach by indexing processes with the network resources they use we give a model for observational equivalence in this context and we prove that it admits an equivalent nominal automaton hd automaton suitable for verification finally we give a concurrent semantics for ncpi where observations are multisets of routing paths we show that bisimilarity for this semantics is a congruence and this property holds also for the concurrent version of the p calculus

pullbacks c x algebras and their cuntz semigroup abstract in this paper we analyse the structure of the cuntz semigroup of certain c x algebras for compact spaces of low dimension that have no k 1 obstruction in their fibres in a strong sense the techniques developed yield computations of the cuntz semigroup of some surjective pullbacks of c algebras as a consequence this allows us to give a complete description in terms of semigroup valued lower semicontinuous functions of the cuntz semigroup of c x a where a is a not necessarily simple c algebra of stable rank one and vanishing k 1 for each closed two sided ideal we apply our results to study a variety of examples

equivariant homomorphisms rokhlin constraints and equivariant uhf absorption abstract we classify equivariant homomorphisms between c dynamical systems associated to actions of finite groups with the rokhlin property in addition the given actions are classified an obstruction is obtained for the cuntz semigroup of a c algebra allowing such an action we also obtain an equivariant uhf absorption result

some remarks on the compressed zero divisor graph abstract let r be a commutative ring with 1 0 the zero divisor graph r of r is the undirected graph with vertices the nonzero zero divisors of r and distinct vertices r and s are adjacent if and only if r s 0 the relation on r given by r s if and only if ann r r ann r s is an equivalence relation the compressed zero divisor graph e r of r is the undirected graph with vertices the equivalence classes induced by other than 0 and 1 and distinct vertices r and s are adjacent if and only if r s 0 let r e be the set of equivalence classes for on r then r e is a commutative monoid with multiplication r s r s in this paper we continue our study of the monoid r e and the compressed zero divisor graph e r we consider several equivalence relations on r and their corresponding graph theoretic translations to r we also show that the girth of e r is three if it contains a cycle and determine the structure of e r when it is acyclic and the monoids r e when e r is a star graph

hilbert c modules over a commutative c algebra this paper studies the problems of embedding and isomorphism for countably generated hilbert c modules over commutative c algebras when the fiber dimensions differ sufficiently relative to the dimension of the spectrum we show that there is an embedding between the modules this result continues to hold over recursive subhomogeneous c algebras for certain modules including all modules over c0 x w hen dimx 3 isomorphism and embedding are determined by the restrictions to the sets where the fiber dimensions are constant these considerations yield results for the cuntz semigroup including a computation of the cuntz semigroup for c0 x w hen dimx 3 in terms of cohomological data about x

the equivariant cuntz semigroup we introduce an equivariant version of the cuntz semigroup that takes an action of a compact group into account the equivariant cuntz semigroup is naturally a semimodule over the representation semiring of the given group moreover this semimodule satisfies a number of additional structural properties we show that the equivariant cuntz semigroup as a functor is continuous and stable moreover cocycle conjugate actions have isomorphic associated equivariant cuntz semigroups one of our main results is an analog of julg s theorem the equivariant cuntz semigroup is canonically isomorphic to the cuntz semigroup of the crossed product we compute the induced semimodule structure on the crossed product which in the abelian case is given by the dual action as an application of our results we show that freeness of a compact lie group action on a compact hausdorff space can be characterized in terms of a canonically defined map into the equivariant cuntz semigroup extending results of atiyah and segal for equivariant k theory finally we use the equivariant cuntz semigroup to classify locally representable actions on direct limits of one dimensional nccw complexes generalizing work of handelman and rossmann

a combinatorial development of fibonacci numbers in graph spectra abstract consider a graph whose vertices are the distinguishable submultisets of a finite multiset such that two vertices r and s are adjacent if and only if r s in this paper the walks on these graphs are linked to square and domino tilings of rectangular and circular boards among the results provided the number of walks of any given length is shown to be a product of generalized fibonacci numbers as an application the eigenvalues are computed for the unrestricted zero divisor graphs of finite commutative rings without nonzero nilpotents

isomorphism of hilbert modules over stably finite c algebras abstract it is shown that if a is a stably finite c algebra and e is a countably generated hilbert a module then e gives rise to a compact element of the cuntz semigroup if and only if e is algebraically finitely generated and projective it follows that if e and f are equivalent in the sense of coward elliott and ivanescu cei and e is algebraically finitely generated and projective then e and f are isomorphic in contrast to this we exhibit two cei equivalent hilbert modules over a stably finite c algebra that are not isomorphic

a bivariant theory for the cuntz semigroup abstract we introduce a bivariant version of the cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary cuntz semigroup the theory has many features formally analogous to kk theory including a composition product we establish basic properties like additivity stability and continuity and study categorical aspects in the setting of local c algebras we determine the bivariant cuntz semigroup for numerous examples such as when the second algebra is a kirchberg algebra and cuntz homology for compact hausdorff spaces which provides a complete invariant moreover we establish identities when tensoring with strongly self absorbing c algebras finally we show how to use the bivariant cuntz semigroup of the present work to classify unital and stably finite c algebras

reactive systems semi saturated semantics and coalgebras on presheaves the semantics of process calculi has traditionally been specified by labelled transition systems ltss but with the development of name calculi it turned out that reaction rules i e unlabelled transition rules are often more natural this leads to the question of how behavioral equivalences bisimilarity trace equivalence etc defined for lts can be transferred to unlabelled transition systems recently in order to answer this question several proposals have been made with the aim of automatically deriving an lts from reaction rules in such a way that the resulting equivalences are congruences furthermore these equivalences should agree with the standard semantics whenever one exists in this paper we propose saturated semantics based on a weaker notion of observation and orthogonal to all the previous proposals and we demonstrate the appropriateness of our semantics by means of two examples logic programming and open petri nets we also show that saturated semantics can be efficiently characterized through the so called semi saturated games finally we provide coalgebraic models relying on presheaves

zero divisor graphs of genus one abstract in this paper we find all rings of the forms z p 1 1 z p n n and z n x x m that have genus at most one

graphs associated to co maximal ideals of commutative rings abstract let r be a commutative ring r with 1 in p k sharma s m bhatwadekar a note on graphical representation of rings j algebra 176 1995 124 127 sharma and bhatwadekar defined a graph on r r with vertices as elements of r where two distinct vertices a and b are adjacent if and only if a r b r r let 2 r be the subgraph of r induced by non unit elements in this paper we derive several properties of 2 r we are able to characterize those rings r for which 2 r j r is a forest and those rings r for which 2 r j r is eulerian where j r is the jacobson radical of r we are also able to show that 2 r max r where g is the clique number of a graph g and max r is the set of maximal ideals of r moreover we find all finite rings r such that the genus of 2 r resp r is at most one

recasting the elliott conjecture let a be a simple unital finite and exact c algebra which absorbs the jiang su algebra mathcal z tensorially we prove that the cuntz semigroup of a admits a complete order embedding into an ordered semigroup which is obtained from the elliott invariant in a functorial manner we conjecture that this embedding is an isomorphism and prove the conjecture in several cases in these same cases mathcal z stable algebras all we prove that the elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker outside the class of mathcal z stable c algebras this weaker conjecture has no known counterexamples and it is plausible that none exist thus we reconcile the still intact principle of elliott s classification conjecture that mathrm k theoretic invariants will classify separable and nuclear c algebras with the recent appearance of counterexamples to its strongest concrete form

non simple purely infinite c algebras a c algebra a is defined to be purely infiniteif there are no characters on a a nd if for every pair of positive elements a b in a such that b lies in the closed two sided ideal generated by a there exists a sequence rn in a such that rn arn b this definition agrees with the usual definition by j cuntz when a is simple it is shown that the property of being purely infinite is preserved under extensions morita equivalence inductive limits and it passes to quotients and to hereditary sub c algebras it is shown that a is purely infinite for every c algebra a purely infinite c algebras admit no traces and conversely an approximately divisible exact c algebra is purely infinite if it admits no nonzero trace

classification of inductive limits of 1 dimensional nccw complexes abstract a classification result is obtained for the c algebras that are stably isomorphic to inductive limits of 1 dimensional noncommutative cw complexes with trivial k 1 group the classifying functor cu is defined in terms of the cuntz semigroup of the unitization of the algebra for the simple c algebras covered by the classification cu reduces to the ordered k 0 group the cone of traces and the pairing between them as an application of the classification it is shown that the crossed products by a quasi free action o 2 r are all isomorphic for a dense set of positive irrational numbers

zero divisor graphs of semigroups the zero divisor graph of a commutative semigroup with zero is a graph whose vertices are the nonzero zero divisors of the semigroup with two distinct vertices joined by an edge in case their product in the semigroup is zero we continue the study of this construction and its extension to a simplicial complex

structured coalgebras and minimal hd automata for the π calculus the coalgebraic framework developed for the classical process algebras and in particular its advantages concerning minimal realizations does not fully apply to the calculus due to the constraints on the freshly generated names that appear in the bisimulation in this paper we propose to model the transition system of the calculus as a coalgebra on a category of name permutation algebras and to define its abstract semantics as the final coalgebra of such a category we show that permutations are sufficient to represent in an explicit way fresh name generation thus allowing for the definition of minimal realizations we also link the coalgebraic semantics with a slightly improved version of history dependent hd automata a model developed for verification purposes where states have local names and transitions are decorated with names and name relations hd automata associated with agents with a bounded number of threads in their derivatives are finite and can be actually minimized we show that the bisimulation relation in the coalgebraic context corresponds to the minimal hd antomaton

on the zero divisor graph of a commutative ring abstract let r be a commutative ring and r be its zero divisor graph in this paper it is shown that for any finite commutative ring r the edge chromatic number of r is equal to the maximum degree of r unless r is a complete graph of odd order in d f anderson a frazier a lauve p s livingston in lecture notes in pure and appl math vol 220 marcel dekker new york 2001 pp 61 72 it has been proved that if r and s are finite reduced rings which are not fields then r s if and only if r s here we generalize this result and prove that if r is a finite reduced ring which is not isomorphic to z 2 z 2 or to z 6 and s is a ring such that r s then r s