Asian-European Journal of Mathematics

Let X be any set and P(X) be the partial transformation semigroup on X. It is well-known that P(X) is regular. To generalize this, let X and Y be any sets and P(X, Y) be the set of all partial transformations from X to Y. For θ ∈ P(Y, X), let (P(X, Y), θ) be a semigroup (P(X, Y), *) where α * β = αθβ for all α, β ∈ P(X, Y). In this paper, we characterize the semigroup (P(X, Y), θ) to be regular, regular elements of the semigroup (P(X, Y), θ), -classes, -classes, -classes and -classes of the semigroup (P(X, Y), θ).

Many kinds of errors in coding theory have been dealt with for which codes have been constructed to combat such errors. Though there is a long history concerning the growth of the subject and many of the codes developed have found applications in numerous areas of practical interest, one of the areas of practical importance in which a parallel growth of the subject took place is that of burst error detecting and correcting codes. The nature of burst errors differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of data transmission. In very busy communication channels, errors repeat themselves more frequently. In view of this, it is desirable to consider repeated burst errors. The paper presents lower and upper bounds on the number of parity-check digits required for a linear code correcting errors in the form of repeated bursts. An upper bound for a code that detects m-repeated bursts has also been derived. Illustrations of several codes that correct 2-repeated bursts of different lengths have also been given.

Many-sorted algebras are used in Computer Science for abstract data type specifications. It is widely believed that many-sorted algebras are the appropriate mathematical tools to explain what abstract data types are ([1]). In this paper we want to extend the concept of a hypersubstitution from the one-sorted to the many-sorted case. Hypersubstitutions are used for one-sorted algebras to define hyperidentities and M-solid varieties ([2]). We will prove that extensions of hypersubstitutions for many-sorted algebras are endomorphisms of the many-sorted clone of a given type. As in the one-sorted case we define a binary operation for hypersubstitutions and prove that with respect to this operation all many-sorted hypersubstitutions form a monoid.

Groups with finitely many maximal normalizers of non-abelian subgroups are studied in this paper. It is proved in particular that (generalized) soluble groups with such property have finite commutator subgroup. This result is an extension of a well-known theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

It is well-known that a ring R is right PF if and only if it is semiperfect and right self-injective with essential right socle. In this note, it is shown that a ring R is right PF if and only if u.dim(R_R) < ∞ and every colocal, injective right R-module is projective. Consequently, a semiperfect ring R is right PF if and only if the two classes of colocal injective right R-modules and local projective right modules coincide.

A semigroup S is called a ∇-semigroup if the set of all its full subsemigroups forms a chain under set inclusion. In this paper, we investigate some properties of such kind of semigroups and establish several characterization theorems of type-A ∇-semigroups. Our theorems generalize the known results of P. R. Jones obtained in 1981 on inverse semigroups whose full inverse subsemigroups form a chain.

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let H_sG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ H_sG.

Our main result here is the following theorem. Let be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that . Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then .

In this article the author proves local Lipschitz properties of two selections for the space of convex hulls of n points and compacts of the complete convex Busemann metric space.

We give a complete description of the lattice of varieties of rings with involution satisfying x⁷ ≈ x by identity bases. There are 90 such varieties. If we substitute in each ring of such a variety the operations by term operations of the same arity we obtain a so-called class of derived rings. We discuss the case when the class of derived rings belongs to the original variety. In particular, we describe the class of derived rings for the variety of rings generated by the two-element Galois-field.

Let 𝕂 be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. Similarly to the Hayman problem, here we study meromorphic functions in 𝕂 or in an open disk that are of the form f′ fⁿ(f − a)^k − α with α a small function, in order to find sufficient conditions on n, k assuring that they have infinitely many zeros. We first define and characterize a special value for a meromorphic function and check that, if it exists, it is unique. So, such values generalize Picard exceptional values.

We prove that a hyper(abelian of finite rank) group, whose images are all hypo (abelian of finite rank), is hypo(central and rank-1 divisible) by hypo(finite abelian).

In the paper some trigonometric nature identities for finite sums of inverses of central binomial numbers and Catalan numbers are given.