In this note, we shall prove that the transcendency degree of a finitely generated field extension is equal to a certain integer associated with a restriction mapping of spaces of derivations.

In this paper, using the idea of upper and lower faithful d.g. near-rings, introduced in (6), we show that the category of all faithful d.g. near-rings is a reflective as well as coreflective subcategory of the category of all d.g. near-rings. We also prove that both and are complete and cocomplete categories.

It is known that while the Shannon and the Rényi entropies are additive, the measure entropy of degree β proposed by Havrda and Charvat (7) is non-additive. Ever since Chaundy and McLeod (4) considered the following functional equation

which arose in statistical thermodynamics, (1.1) has been extensively studied (1, 5, 6, 8). From the algebraic properties of symmetry, expansibility and branching of the entropy (viz. Shannon entropy Hn, etc.) one obtains the sum representation

which with the property of additivity yields the functional equation (1.1), (9, 10).

A class of finite semigroups is called a genus if it is closed under homomorphic images, subsemigroups and finite direct products. During a talk at the Symposium on Semigroups held at the University of St Andrews, in 1976, M. P. Schützenberger posed the problem of characterising the smallest genus which contains finite groups and finite semigroups, all of whose subgroups are trivial.

A permutation group G operating on a set M is called a ZT-group (Zassenhaus transitive group) if G has the properties (i) and (ii):

(i) G operates 2-transitively on M;

(ii) Ga,b≠{id} and Ga,b,c = {id} for distinct elements a,b,c ∈ M.

Here Ga,b = {α ∈ G | α(a) = a and α(b) = b} denotes the stabilizer of {a, b}, and Ga,b,c the stabilizer of {a, b, c}, respectively.

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series

.

Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} and

The summation is over all matrices

with different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

In somes categories, there are structures that look very much like groups, and they usually are. These structures are called group-objects and were first studied by Eckmann and Hilton (1). If our category has an object T such that hom(X, T)= {tx}, a singleton, for each object X ∈ Ob , T is called a terminal object. Our category must have products; i.e. for A1,…, An ∈;. Ob , there is an object A1 × … × An ∈ Ob and morphisms pi: A1 × … × An → Ai so that if fi: X → Ai, i = 1, 2, …, n, are morphisms of , then there is a unique morphism [f1, …, fn]: X → A1 × … × An such that for i = 1, 2, …, n.

With differential equations in the neighbourhood of an irregular singular point, it sometimes happens that formal solutions may converge. For example, this occurs for Bessel's equation at∞ when the parameter is half of an odd integer. In addition, there are some classical theorems of Perron and Lettenmeyer which give sufficient conditions for the existence of linearly independent analytic solutions at (generally) an irregular singular point. Using the principle of reduction of order, such a solution may be used to transform the differential equation into one whose coefficient matrix is triangularly blocked with an (n – 1) and 1-block on the diagonal. The solutions of the given differential equation can thus be obtained by solving a lower dimensional differential equation plus quadrature.

A triple (or monad) in a category K is a triple = (T, μ, η) where, T: K → K is a functor and μ: TT →, T, η: 1k → T are natural transformations for which (1.1) and (1.2) commute:

In these diagrams the component of a natural transformation α at an object x is denoted xα. Thus for example (kη)T is the value of the functor T applied to the component of η at k, whereas (kT)η is the component of η at the object kT. I write functions and functors on the right and composition from left to right.

A strongly regular ring R is one in which for all x ∈ R, there is an a ∈ R with x = x2a. Equivalently, for all x there is an a with x = ax2. Such a ring is regular, duo, biregular, and a left and right V-ring. Moreover since R is reduced, all nilpotent elements are central (vacuously) and so all idempotent elements are central.

A ring R is said to satisfy the right Ore condition with respect to a subset C of R if, given a ∈ R and e ∈ C, there exist b ∈ R and D ∈ C such that ad = cb. It is well known that R has a classical right quotient ring if and only if R satisfies the right Ore condition with respect to C when C is the set of regular elements of R (a regular elemept of R being an element of R which is not a zero-divisor). It is also well known that not every ring has a classical right quotient ring. If we make the non-trivial assumption that R has a classical right quotient ring, it is natural to ask whether this property also holds in certain rings related to R such as the ring Mn(R) of all n by n matrices over R. Some answers to this question are known when extra assumptions are made. For example, it was shown by L. W. Small in (5) that if R has a classical right quotient ring Q which is right Artinian then Mn(Q) is the right quotient ring of Mn(R) and eQe is the right quotient ring of eRe where e is an idempotent element of R. Also it was shown by C. R. Hajarnavis in (3) that if R is a Noetherian ring all of whose ideals satisfy the Artin-Rees property then R has a quotient ring Q and Mn(Q) is the quotient ring of Mn(R).

In this paper radicals in the sense of Kuroš and Amitsur (KA-radicals) for Ω-groups will be studied. For the sake of simplicity these radicals will be considered on varieties, the results remaining valid for more general classes.

The class of non-metacyclic finite soluble groups known to have 2-generator 2-relation presentations is small. Classes of such groups are given in (3), (4), (8) and (9). Some subclasses of the groups discussed in (1) and (2) also provide examples, while a class of finite nilpotent 2-generator 2-relation groups is given by Macdonald in (7).

The purpose of this note is to extend the results of Reilly and Scheiblich (6) (see also Scheiblich (7) and Hall (2)) on the θ-class decomposition of the congruence lattice of a regular semigroup and, at the same time, to provide an alternative proof of these results.

Some techniques developed by Eilenberg (1) in the theory of automata are used to investigate certain types of near-ring.

Throughout this paper any near-ring N will be left distributive, zero symmetric and have an identity. Furthermore, all N-groups will be unitary. All groups considered will be written additively. This does not imply commutativity.

Consider a formal series with partial sums and the corresponding power series . Throughout we will assume that f is analytic for |z| <1, i.e. that A classical theorem of Fatou-Riesz (see (1, 4)) states that if and

then is convergent to 0.

Throughout this paper all near-rings will be zero-symmetric and left distributive. A near-ring with minimal condition on right N-subgroups will be called an near-ring. It is well known (see (1), 3.40, p. 90)) that a nil right N-subgroup of an near-ring is nilpotent. However, in a deeper study of near-rings a stronger result than this is sometimes required (2, p. 77).

Many basic definitions and results in the theory of near-rings can be found in G. Pilz (4). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S, which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by (R, S). Then (R, +) = Gp < S; > where is a set of defining relations in S. Let (T, U) be a d.g. near-ring. Then a d.g. homomorphism θ from (R, S) to (T, U) is a near-ring homomorphism from R to T which satisfies Sθ ⊆ U. If (G, +) is a group, let T0(G) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G. Then (E(G), End G) is a d.g. near-ring. A d.g. near-ring (R, S) is faithful if there exists a d.g. monomorphism θ:(R, S) → (E(G), End G) for some group G.