Advances in Algebra and Combinatorics

The following sections are included:

• Preface

• Contents

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies an identity s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A graph G = (V, E) is called an (x(yz)) ≈ ((x(yy)z) graph if the graph algebra A(G) satisfies the equation (x(yz)) ≈ ((x(yy))z). An identity s ≈ t of terms s and t of any type τ is called a hyperidentity of an algebra A if whenever the operation symbols occurring in s and t are replaced by any term operations of A of the appropriate arity, the resulting identities hold in A.

In this paper, we characterize the class of (x(yz)) ≈ ((x(yy))z) graph algebras, (x(yz)) ≈ ((x(yy))z) class, identities and hyperidentities in the class of (x(yz)) ≈ ((x(yy))z) graph algebras.

The paper contains a survey of some recent results on ring-theoretical properties of quantum affine space and actions on them of finite dimensional pointed Hopf algebras.

In this survey article, we report some new results of Gröbner-Shirshov bases, including new Composition-Diamond lemmas, applications of some known Composition-Diamond lemmas and content of some expository papers.

Using a given ring, we define the Weyl-type non-associative rings in this work. If the ring is simple, then we show that the Weyl-type non-associative ring is simple in this work. All the derivations of a Weyl-type non-associative algebra are found in the papers (see [4, 6]). In this work, we find all the derivations of a Weyl-type algebra which includes the algebras in the papers (see [4, 6]). We show that the associators of the algebras and are zeroes.

Let A^⊔n := {1, …, n} × A be the n-th copower of the set A. An n-ary cooperation is a mapping f : A → A^⊔n. If f is an n-ary cooperation and if g₁, …, g_n are k-ary cooperations on A, we may define a composition and obtain a new k-ary cooperation defined on A. Together with the injection cooperations defined by a ↦ (i, a) for 1 ≤ i ≤ n, all cooperations defined on A form a multibased algebra. This algebra is called the clone of all cooperations defined on A. If we define by a binary operation on the set of all n-ary cooperations, the set of all n-ary cooperations defined on A forms a semigroup with respect to this operation. We determine the order of the elements of this semigroup, characterize all idempotent and regular elements, ask for bands of n-ary cooperations and characterize Green's relations and .

Hypersubstitutions map operation symbols to terms of the corresponding arities. Using hypersubstitutions from each algebra one can derive new algebras. We compare properties of congruence lattices, of subalgebra lattices and of clones of the derived algebras with the corresponding properties of the starting algebras. If both algebras have the same clone of term operations, then they have the most properties in common. This is for instance the case if the hypersubstitution is surjective. We characterize surjective hypersubstitutions and prove that for finite types surjective hypersubstitutions are bijective. This will be applied to i-closed varieties of n-ary type.

Due to properties of the topological tensor product of ultrametric Banach spaces, the algebraic notion of coalgebra has a natural ultrametric counterpart. An important class of ultrametric Banach coalgebras is provided by the spaces of continuous functions from a totally discontinuous compact group G with values in a complete ultrametric valued field K. The coproduct on is induced by the law of multiplication of the group G.

Related to ombral calculus is the characterization of the endomorphisms of a coalgebra. For a wide class of ultrametric Banach colagebras the monoid of the continuous coalgebra endomorphisms is anti-isomorphic to the monoid of continuous and weakly continuous algebra endomorphims of dual algebra. Hence, we recover earlier results obtained on the coalgebra , where ℤ_p, is the additive group of the ring of p-adic integers and K is a complete valued field of the field extension of p-adic numbers ℚ_p. In addition for such ground fields K and more generally for K of residue characteristic p, we consider the coalgebra , where V_q is the infinite compact monothetic subgroup of the group of units of K generated by q which is not a root of unit such that |q^h - 1) < 1, for h an integer ≥ 1. One obtains, as for formal power series, a procedure of substitution on the algebra of p-adic bounded measures M(V_q, K) which gives the continuous and weak*-continuous algebra endomorphisms of M(V_q, K) and which in turn gives the continuous coalgebra endomorphisms of .

Let H, C and D be subgroups of a finite group G. We call the pair (C, D) a θ*-pair for H if it satisfies the following conditions: (i) D = (C∩H)_G, 〈C^g, H〉 = G for every g ∈ G, (ii) KH < G for every K/D ◃ G/D with K/D < C/D. In this paper, we obtain several results on the maximal θ*-pair which implies G to be solvable, supersolvable and nilpotent.

Let L_S denote the language of (right) S-acts over a monoid S and let Σ_S be a set of sentences in L_S which axiomatises S-acts. A general result of model theory says that Σ_S has a model companion, denoted by T_S, precisely when the class of existentially closed S-acts is axiomatisable and in this case, T_S axiomatises . It is known that T_S exists if and only if S is right coherent. Moreover, by a result of Ivanov, T_S has the model-theoretic property of being stable.

In the study of stable first order theories, superstable and totally transcendental theories are of particular interest. These concepts depend upon the notion of type: we describe types over T_S algebraically, thus reducing our examination of T_S to consideration of the lattice of right congruences of S. We indicate how to use our result to confirm that T_S is stable and to prove another result of Ivanov, namely that T_S is superstable if and only if S satisfies the maximal condition for right ideals. The situation for total transcendence is more complicated but again we can use our description of types to ascertain for which right coherent monoids S we have that T_S is totally transcendental and is such that the U-rank of any type coincides with its Morley rank.

The Greeks understood, around 350 B.C., how to construct, with Euclidean tools, regular N-gons for N = 2^cN₀, where N₀ = 1, 3, 5 or 15. Two thousand years later, Gauss proved that a Euclidean construction of a regular N-gon is possible if and only if N = 2^c × (product of distinct Fermat primes).

Here we are content to constuct arbitrarily close approximations to regular polygons. Our constructions lead to some interesting number theory involving the Euler totient function. For a given odd number b, and a given odd number , we construct a numerical array, called a symbol and describe an explicit procedure based on the symbol for constructing a regular star -gon.

We can combine the symbols for a given b to produce a complete symbol where each constituent symbol is called a coach. We present two theorems, the Quasi-Order Theorem and the Coach Theorem, which show how the numbers appearing in the symbol (and hence the steps in our constructions) are governed by the values of Φ(b) and the quasi-order of b mod 2. We then generalize the results to any arbitrary base.

This is a survey to apply theory from noncommutative graded algebras to questions about the holonomy algebra and the Orlik-Solomon algebra of a hyperplane arrangement. We first recall the main properties of Koszul algebras and hyperplane arrangements. Then, we focus our interest on the class of hypersolvable arrangements which includes both the fiber-type and the generic arrangements. For these hypersolvable arrangements, the holonomy algebra is Koszul and koszulness of the Orlik-Solomon algebra characterizes the subclass of fiber-type's.

In this paper, we discuss some problems in PI-theory. Some developments of this theory are presented.

We observe an application of conformal algebra theory to the following natural problem: describe those subalgebras of the algebra of differential operators with polynomial coefficients (Weyl algebra) that act irreducibly on the space of polynomials.

We investigate the influences of lengths of conjugacy classes of finite groups on the structure of finite groups. Some sufficient conditions for a finite group to be p-nilpotent and supersolvable are obtained. Some known results are generalized.

Finite index subgroups of the modular group are of great arithmetic importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for working with these groups. Given such a group Γ, a Farey symbol for Γ is a certain finite sequence of rational numbers (representing vertices of a fundamental domain of Γ) together with pairing information for the edges between the vertices. They are a compact way of encoding the information about the group and they provide a simple way to do calculations with the group. For example: calculating an independent set of generators and decomposing group elements into a word in these generators, finding coset representatives, elliptic points, and genus of the group, testing if the group is congruence, etc. We will discuss Farey Symbols and explicit algorithms for working with them.

In this paper, Gröbner—Shirshov bases and normal forms of elements for the Coxeter groups E₆ and E₇ are found. These results support the conjecture in [4] about the general form of Gröbner—Shirshov bases for all Coxeter groups.

Let F be a field and K a subfield of F. We classify the overgroups in GL(n, F) of an Sp(n, K, f_K), SU(n, K, f_K) or Ω(n, K, Q_K) provided that the index [F : K] is not much bigger than the Witt index ν(f_k) or ν(Q_K).

A symbolic calculus named as the R-calculus is built to revise the defects of axiomatic systems mechanically when some counterexamples are found. The R-calculus consists of the rules of logical connective symbols and logical quantifier symbols of first order languages. The concept of reachability, soundness and completeness of the R-calculus are introduced. The basic theorem of software testing based on the R-calculus is also introduced to help mechanizing model checking.

In a previous paper we have begun to study the Restricted Burnside Problem for some classes of loops reducing it to the positive solution for the class of groups. Here we continue this work, giving a positive answer for those Bruck loops which can be obtained by Glauberman's construction from a Moufang A–loops.

In this survey article, we will briefly introduce the recent development of rpp semigroups, its generalizations and some of its special subclasses. Some methods of constructions for such semigroups are introduced. Several structure theorems of such semigroups are described.

The following sections are included:

• Introduction

• Modular Functor

• Conformal Field Theory
  • Non-Abelian Conformal Field Theory
  • Abelian Conformal Field Theory

• Modular Functor
  • Teichmüller space
  • Construction of Modular Functor

• References

In this paper, we study a subclass of the class of MD-algebras, i.e., the class of solvable real Lie algebras such that the K-orbits of its corresponding connected and simply connected Lie groups are either orbits of dimension zero or orbits with maximal dimensions. Our main result is to classify, up to isomorphism, all the 5-dimensional MD-algebras having commutative derived ideals.