Muhammad Anwar Chaudhry

On some classes of BCH-algebras

The concept of a BCH-algebra is a generalization of the concept of a BCI-algebra. It is shown that weakly commutative BCH-algebras are weakly commutative BCI-algebras. Moreover, the concepts of weakly positive implicative and weakly implicative BCH-algebras are defined and it is shown that every weakly implicative BCH-algebra is a weakly positive implicative BCH-algebra. The weakly positive implicative BCH-algebras are characterized with the help of their self maps. Two open problems are posed. 2000 Mathematics Subject Classification. Primary 06F35, 03G25. 1. Introduction. In 1966, Imai and Iseki introduced two classes of abstract alge- bras, BCK-algebras and BCI-algebras (6, 7). BCI-algebras are a generalization of BCK- algebras. These algebras have been extensively studied since their introduction. In 1983, Hu and Li (4, 5) introduced the notion of a BCH-algebra, which is a generaliza- tion of the notions of BCK- and BCI-algebras. They have studied a few properties of these algebras. Certain other properties have been studied by Chaudhry (2) and Dudek and Thomys (3). It has been shown (3, 4, 5) that there are no proper associative and medial BCH-algebras, that is, associative and medial BCH-algebras are associative and medial BCI-algebras, respectively. The purpose of this paper is to investigate the existence of certain classes of proper BCH-algebras and study their properties. It is shown that proper weakly commutative BCH-algebras do not exist. However, proper weakly positive implicative and proper weakly implicative BCH-algebras exist and every weakly implicative BCH-algebra is a weakly positive implicative BCH-algebra but not conversely. Weakly positive implica- tive BCH-algebras have been characterized in terms of their self maps. The results proved in this paper are general in the sense that corresponding results for BCK- algebras and BCI-algebras become special cases.

Ideals in BCI‐algebras

In this paper we describe the notion of the centre of a BCI‐algebra and show that it is a p‐semisimple subalgebra. Various properties of BCI‐ideals have been studied, and necessary and sufficient conditions for certain ideals to be closed have been investigated.

On branchwise implicative BCI-algebras

We introduce a new class of BCI-algebras, namely the class of branchwise implicative BCI-algebras. This class contains the class of implicative BCK-algebras, the class of weakly implicative BCI-algebras (Chaudhry, 1990), and the class of medial BCI-algebras. We investigate necessary and sufficient conditions for two types of BCI-algebras to be branchwise implicative BCI-algebras.

A NOTE ON A PAIR OF DERIVATIONS OF SEMIPRIME RINGS

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with center Z(R) ,a nd letf, g be derivations of R such that f( x)x+ xg(x) ∈ Z(R) for all x ∈ R, then f and g are central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation f of a semiprime ring R is skew-commuting.

Some categorical aspects of BCH-algebras

We show that the category BCH of BCH-algebras and BCH-homomorphisms is complete. We also show that it has coequalizers, kernel pairs, and an image factorization system. It is also proved that onto homomorphisms and coequalizers, and monomorphisms and one-to-one homomorphisms coincide, respectively, in BCH. It is shown that MBCI is a coreflexive subcategory of BCH. Regular homomorphisms have been defined and their properties are studied. An open problem has been posed.

On the metric dimension of generalized Petersen graphs

Abstract A family G of connected graphs is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of the generalized Petersen graphs P(2m, m − 1) and give a partial answer to an open problem raised in [13]: Is the generalized Petersen graphs P(s, t), for s ≥ 7 and 3 ≤ t ≤ , a family of graphs with constant metric dimension? We prove that the generalized Petersen graphs P(2m, m − 1) have metric dimension equal to 3 for all odd m ≥ 3, and equal to 4 for all even m ≥ 4.

Dependent Elements of Derivations on Semiprime Rings

We characterize dependent elements of a commuting derivation on a semiprime ring and investigate a decomposition of using dependent elements of . We show that there exist ideals and of such that is an essential ideal of , , on , , and acts freely on .

Centralizing mappings and derivations on semiprime rings

In this paper we study some properties of centralizing mappings on semiprime rings. The main purpose is to prove the result: Let R be a semiprime ring and / an endomorphism of R, g an epimorphism of R such that the mapping x —> [/(x),g(x)] is central. Then [/(x),g(a;)] = 0 holds for all x € R. We also establish some results about (a, /?)-derivations.

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S 1, S 2, …, S k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n−1 and n−2. Some realizable results are also given in this paper.

On generalized (α, β)-derivations on lattices

Abstract In this paper we introduce the notion of a generalized (α, β, d)-derivation on a lattice, with associated two sided α-derivation d, and investigate some related results. Among some other results we prove that “Let be a modular lattice and G a generalized (α, β, d)-derivation, with associated two sided α-derivation d, on L. If a is an increasing endomorphism on L satisfying α(x) ≤ β(x) for x ∈ L, then the following conditions are equivalent: (a) G is isotone generalized (α, β, d)-derivation on L, ”