The regular graph of a commutative ring

Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2 ⁿ , where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.     

Operads and varieties of algebras defined by polylinear identities

We show that varieties of algebras over abstract clones and over the corresponding operads are rationally equivalent. We introduce the class of operads (which we call commutative for definiteness) such that the varieties of algebras over these operads resemble in a sense categories of modules over commutative rings. In particular, the notions of a polylinear mapping and the tensor product of algebras. The categories of modules over commutative rings and the category of convexors are examples of varieties over commutative operads. By analogy with the theory of linear multioperator algebras, we develop a theory of C-linear multioperator algebras; in particular, of algebras, defined by C-polylinear identities (here C is a commutative operad). We introduce and study symmetric C-linear operads. The main result of this article is as follows: A variety of C-linear multioperator algebras is defined by C-polylinear identities if and only if it is rationally equivalent to a variety of algebras over a symmetric C-linear operad.     

Semigroups whose endomorphisms are power functions

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=x ^m is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomorphism of S preserving 1 and 0 is equal to a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. Immediate consequences of the results are, on the one hand, a characterization of commutative rings whose multiplicative endomorphisms are power functions given by Greg Oman in the paper (Semigroup Forum, 86 (2013), 272–278), and on the other hand, a partial solution of Problem 1 posed by Oman in the same paper.     

The Anderson-Badawi conjecture for commutative algebras over infinite fields

In [1], Anderson and Badawi conjecture that every n-absorbing ideal of a commutative ring is strongly n-absorbing. In this article we prove their conjecture in certain cases (in particular this is the case for commutative algebras over an infinite field). We also show that an affirmative answer to another conjecture in [1] implies the Anderson-Badawi Conjecture.     

On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the P(q,?) (q odd prime power and ?>1 odd) commutative semifields constructed by Bierbrauer (Des. Codes Cryptogr. 61:187?C196, 2011) are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth (SETA, pp.?403?C414, 2008). Also, we show that they are strongly isotopic if and only if q??1(mod?4). Consequently, for each q???1(mod?4) there exist isotopic commutative presemifields of order q ^2? (?>1 odd) defining CCZ-inequivalent planar DO polynomials.     

Unitaires multiplicatifs commutatifs

In a previous article, we proved that every commutative multiplicative unitary is a multiple of the multiplicative unitary associated to a locally compact group. In the present Note we give a simpler proof of this generalization of a theorem of Weil. To cite this article: S. Baaj, G. Skandalis, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Strong NMV-algebras, commutative basic algebras and naBL-algebras

The aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that

1. a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid)
2. a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra.

Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.     

Commutative basic algebras and non-associative fuzzy logics

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. ?ukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L _CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers and includes the class of MV-algebras. We show that the logic L _CBA is very close to the ?ukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization.     

Multiplicative Semigroup of a Residue Class Ring

In this paper, we consider the structure of the multiplicative semigroup of a residue class ring R/I of a commutative ring R with identity modulo its nonzero ideal I. For the general case, we investigate the H-classes, maximal subgroups and the structure of Reg(R/I) which is the set of regular elements of R/I. If R is any integral domain and if I is a product of powers of invertible maximal ideals, we show that R/I is an epigroup, every H^*-class of R/I is a nil-extension of a group (:unipotent epigroup) and that R/I is a complete lattice of unipotent epigroups.     

On pseudo-amenability of commutative semigroup algebras and their second duals

In this paper, we first characterize pseudo-amenability of semigroup algebras \(\ell ^1(S),\) for a certain class of commutative semigroups S,  the so-called archimedean semigroups. We show that for an archimedean semigroup S,  pseudo-amenability, amenability and approximate amenability of \(\ell ^1(S)\) are equivalent. Then for a commutative semigroup S,  we show that pseudo-amenability of \(\ell ^{1}(S)\) implies that S is a Clifford semigroup. Finally, we give some results on pseudo-amenability and approximate amenability of the second dual of a certain class of commutative semigroup algebras \(\ell ^1(S)\).     

Weakly central-vertex complete graphs with applications to commutative rings

The zero-divisor graph of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. In this paper, a decomposition theorem is provided to describe weakly central-vertex complete graphs of radius 1. This characterization is then applied to the class of zero-divisor graphs of commutative rings. For finite commutative rings whose zero-divisor graphs are not isomorphic to that of Z₄[X]/(X²), it is shown that weak central-vertex completeness is equivalent to the annihilator condition. Furthermore, a schema for describing zero-divisor graphs of radius 1 is provided.     

Ordered semigroups which are both right commutative and right cancellative

In this paper we prove that each right commutative, right cancellative ordered semigroup (S,.,??) can be embedded into a right cancellative ordered semigroup (T,??,?) such that (T,??) is left simple and right commutative. As a consequence, an ordered semigroup S which is both right commutative and right cancellative is embedded into an ordered semigroup T which is union of pairwise disjoint abelian groups, indexed by a left zero subsemigroup of?T.     

On the Fully Commutative Elements of Coxeter Groups

Let W be a Coxeter group. We define an element w ε W to be fully commutative if any reduced expression for w can be obtained from any other by means of braid relations that only involve commuting generators. We give several combinatorial characterizations of this property, classify the Coxeter groups with finitely many fully commutative elements, and classify the parabolic quotients whose members are all fully commutative. As applications of the latter, we classify all parabolic quotients with the property that (1) the Bruhat ordering is a lattice, (2) the Bruhat ordering is a distributive lattice, (3) the weak ordering is a distributive lattice, and (4) the weak ordering and Bruhat ordering coincide. Partially supported by NSF Grants DMS-9057192 and DMS-9401575.     

On the density of images of the power maps in Lie groups

Let G be a connected Lie group. In this paper, we study the density of the images of individual power maps \(P_k:G\rightarrow G:g\mapsto g^k\). We give criteria for the density of \(P_k(G)\) in terms of regular elements, as well as Cartan subgroups. In fact, we prove that if \(\mathrm{Reg}(G)\) is the set of regular elements of G, then \(P_k(G)\cap \mathrm{Reg}(G)\) is closed in \(\mathrm{Reg}(G)\). On the other hand, the weak exponentiality of G turns out to be equivalent to the density of all the power maps \(P_k\). In linear Lie groups, weak exponentiality reduces to the density of \(P_2(G)\). We also prove that the density of the image of \(P_k\) for G implies the same for any connected full rank subgroup.     

Graphs admitting transitive commutative group actions

Let X be a compact metric space, and Homeo(X) be the group consisting of all homeomorphisms from X to X. A subgroup H of Homeo(X) is said to be transitive if there exists a point x∈X such that {k(x):k∈H} is dense in X. In this paper we show that, if X=G is a connected graph, then the following five conditions are equivalent: (1) Homeo(G) has a transitive commutative subgroup; (2) G admits a transitive Z²-action; (3) G admits an edge-transitive commutative group action; (4) G admits an edge-transitive Z²-action; (5) G is a circle, or a k-fold loop with k?2, or a k-fold polygon with k?2, or a k-fold complete bigraph with k?1. As a corollary of this result, we show that a finite connected simple graph whose automorphism group contains an edge-transitive commutative subgroup is either a cycle or a complete bigraph.     

On the fractions of semi-Mackey and Tambara functors

For a finite group G, a semi-Mackey (resp. Tambara) functor is regarded as a G-bivariant analog of a commutative monoid (resp. ring). As such, some naive algebraic constructions are generalized to this G-bivariant setting. In this article, as a G-bivariant analog of the fraction of a ring, we consider fraction of a Tambara (and a semi-Mackey) functor, by a multiplicative semi-Mackey subfunctor.     

Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

On non-symmetric commutative association schemes with exactly one pair of non-symmetric relations

For a given non-symmetric commutative association scheme, by fusing all the non-symmetric relations pairwise with their symmetric counterparts, we can obtain a new symmetric association scheme. In this paper, we introduce a set of feasibility and realizability conditions for a class e symmetric association scheme to be split into a class e+1 non-symmetric commutative association scheme. By applying the feasibility and realizability conditions, we obtain a classification into six categories of the class 4 non-symmetric fission schemes of group-divisible 3-schemes. Complete solutions for three of the six categories and partial results for the remaining cases are presented.     

Intuitionistic fuzzy commutative hyper K-ideals

In this paper first we introduce the notion of intuitionistic fuzzy (weak)hyperK-ideals and also commutative hyperK-ideals of types 1–4 and by some examples we show that there are different notions. Then we obtain some relationships between these notions. Finally by given the notion of the product of two intuitionistic fuzzy commutative hyperK-ideals we give a Decomposition Theorem.