Narrow Results

A theory of representation of semiautomata by canonical words and equivalences was developed in [7]. That work was motivated by trace-assertion specifications of software modules, but its focus was entirely on the underlying mathematical model. In the present paper we extend that theory to automata with Moore and Mealy outputs, and show how to apply the extended theory to the specification of modules. In particular, we present a unified view of the trace-assertion methodology, as guided by our theory. We illustrate this approach, and some specific issues, using several nontrivial examples. We include a discussion of finite versus infinite modules, methods of error handling, some awkward features of the trace-assertion method, and a comparison to specifications by automata. While specifications by trace assertions and automata are equivalent in power, there are cases where one approach appears to be more natural than the other. We conclude that, for certain types of system modules, formal specification by automata, as opposed to informal state machines, is not only possible, but practical.

This paper proposes an action generation model which consists of many motor primitive modules. The motor primitive modules output motor commands based on sensory information. Complicated behavior is generated by sequentially switching the modules. The model also has a prediction unit. This unit predicts which module will be used for current action generation. We have confirmed the effectiveness of the model by applying it to a robot navigation task simulation, and have investigated the influence of the prediction on the action generation.

Various educational Arduino modules with its simplicity have been developed since Arduino's release into the market. In this study, the helium gas airship was employed to make an Arduino module by applying Arduino Mini, Bluetooth and Android applications.

For a variety of comtrans algebras over a commutative ring, representations of algebras in the variety are identified as modules over an enveloping algebra. In particular, a new, simpler approach to representations of Lie triple systems is provided.

Given a commutative ring with identity $A$ and an $A$-module $M$, a subset $H$ of $M$ is a cyclic covering of $M$, if this module is the union of the cyclic submodules $[h]=\{ah:a\in A\}$, where $h\in H$. Such covering is said to be irredundant, if no proper subset of $H$ is a cyclic covering of $M$. In this work, an irredundant cyclic covering of $A^n$ is constructed for every Artinian commutative ring $A$. As a consequence, a cyclic covering of minimal cardinality of $A^n$ is obtained for every finite commutative ring $A$, extending previous results in the literature.

In this paper, we show that unknotting number of the connected sum of n identical knots k is at least n when k has nontrivial Alexander polynomial.

A finitely generated ℤ[t, t^-1]-module without ℤ-torsion and having nonzero order Δ(M) of degree d is determined by a pair of sub-lattices of ℤ^d. Their indices are the absolute values of the leading and trailing coefficients of Δ(M). This description has applications in knot theory.

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo $4$T-relations admit a Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define a multiplication is not known yet. In this paper, we first define a new module ${{ℳ}}_2$ which is generated by chord diagrams on two circles and factored by $4$T-relations. Then we construct a “parity” map from the module of framed chord diagrams into ${{ℳ}}_2$ and a weight system on ${{ℳ}}_2$. Using the map and weight system we show that a connected sum for framed chord diagrams is not a well-defined operation. In the end of the paper we touch linear diagrams, the circle replaced by a directed line.

Inspired by Borcherds' work on "G-vertex algebras," we formulate and study an axiomatic counterpart of Borcherds' notion of G-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by G₁. Specifically, we formulate a notion of axiomatic G₁-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic G₁-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic G₁-vertex algebras from a set of compatible G₁-vertex operators.

Two observations in support of the thesis that trusses are inherent in ring theory are made. First, it is shown that every equivalence class of a congruence relation on a ring or, equivalently, any element of the quotient of a ring $R$ by an ideal $I$ is a paragon in the truss $\mathrm{\text{T}}(R)$ associated to $R$. Second, an extension of a truss by a one-sided module is described. Even if the extended truss is associated to a ring, the resulting object is a truss, never a ring, unless the module is trivial. On the other hand, if the extended truss is associated to a brace, the resulting truss is also associated to a brace, irrespective of the module used.

Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle has been defined and studied in earlier paper. Here, by using cohomological methods, we establish that an algebra bundle is a semidirect product of its radical bundle and a semisimple subalgebra bundle. Further we define multiplication algebra bundle of an algebra bundle and representation of an algebra bundle. We study special representations of an algebra bundle using Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle. We observe that if a representation of an algebra bundle is special then its obstruction is zero. Further we show that a subgroup H of H²(ξ, N) is faithfully represented as a transitive group of translations operating on the set of those equivalence classes of algebra bundle extensions of ξ which determine a given representation [φ, K].

We consider those injective modules that determine every left exact preradical and we call them main injective modules. We construct a main injective module for every ring and we prove some of its properties. In particular we give a characterization, in terms of main injective modules, of rings with a dimension defined by a filtration in the lattice of left exact preradicals. We define also the concept of basic preradical and prove some of its properties. In particular we prove that the class of all basic preradicals is a set, giving a bijective correspondence with the set of all left exact preradicals.

Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom(M, R). We observe that over a commutative ring R, Γ(M) is connected and diam(Γ(M)) ≤ 3. Moreover, if Γ(M) contains a cycle, then gr(Γ(M)) ≤ 4. Furthermore if ∣Γ(M)∣ ≥ 1, then Γ(M) is finite if and only if M is finite. Also if Γ(M) = ∅, then any nonzero f ∈ Hom(M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

We study direct sum decompositions of modules satisfying the descending chain condition on direct summands. We call modules satisfying this condition Krull–Schmidt artinian. We prove that all direct sum decompositions of Krull–Schmidt artinian modules refine into finite indecomposable direct sum decompositions and we prove that this condition is strictly stronger than the condition of a module admitting finite indecomposable direct sum decompositions. We also study the problem of existence and uniqueness of direct sum decompositions of Krull–Schmidt artinian modules in terms of given classes of modules. We present also brief studies of direct sum decompositions of modules with deviation on direct summands and of modules with finite Krull–Schmidt length.

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.

Let R be a ring with identity and M be a unitary left R-module. The complement of the intersection graph of submodules of M, denoted by Γ(M), is defined to be a graph whose vertices are in one-to-one correspondence with all nontrivial submodules of M and two distinct vertices are adjacent if and only if the corresponding submodules of M have zero intersection. In this paper, we consider the complement of the intersection graph of submodules of a module. We prove that, if Γ(M) is connected and Δ(Γ(M)) < ∞, then M is semisimple, where Δ(Γ(M)) is the maximum degree of Γ(M). We show that, if Γ(M) is a forest, then each component of Γ(M) is a star graph. Moreover, it is proved that, if Γ(M) is a tree, then Γ(M) is isomorphic to a complete graph of order at most two.

The zero-divisor graphs of modules introduced and studied in [S. Safaeeyan, M. Baziar and E. Momtahan, A generalization of the zero-divisor graph for modules, J. Korean Math. Soc. 51(1) (2014) 87–98]. Basic results for zero-divisor graphs of $\mathbb{Z}$-modules were obtained in [M. Baziar, E. Momtahan and S. Safaeeyan, Zero-divisor graph of abelian groups, J. Algebra Appl.13(6) (2014) 13]. In this paper, zero-divisor graphs in the title are studied. Here, among other things, we generalize results stated in [M. Baziar, E. Momtahan and S. Safaeeyan, Zero-divisor graph of abelian groups, J. Algebra Appl.13(6) (2014) 13]. Some results for modules over non-integral domains are also obtained.

We introduce and study (strongly) $\pi$-Rickart objects and their duals in abelian categories, which generalize (strongly) self-Rickart objects and their duals. We establish general properties of such objects, we analyze their behavior with respect to coproducts, and we study classes all of whose objects are (strongly) $\pi$-Rickart. We derive consequences for module and comodule categories.

We adapt and generalize results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation with cocycles.

We investigate relative CS-Baer objects in abelian categories in relationship with other relevant classes of objects such as relative Baer objects, extending objects, objects having certain summand intersection properties and relative CS-Rickart objects. Dual results are automatically obtained by applying the duality principle in abelian categories. We also study direct sums of relative CS-Baer objects, and we determine the complete structure of dual self-CS-Baer modules over Dedekind domains. Further applications are given to module categories.