A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_n=\{0,1,\cdots ,n-1\}$ as the set of vertices and $E=\{(a,b):a^k\equiv b(modn)\}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed.

Let $G$ be a finite group, and let $N\left(G\right)$ be the set of conjugacy class sizes of $G$. By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N\left(G\right)=N\left(L\right)$, then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation). In...

An endomorphism f of an Abelian group A is said to be inessentia! (in the category of Abelian groups) if it can be extended to an endomorphism of any Abelian group which contains A as a subgroup. In this paper we show that f is as above if and only if (f - v idA)(A) is contained in the rnaximal divisible subgroup of A for some v belonging to Z.

In survival studies and life testing, the data are generally truncated. Recently, authors have studied a weighted version of Kerridge inaccuracy measure for truncated distributions. In the present paper we consider weighted residual and weighted past inaccuracy measure and study various aspects of their bounds. Characterizations of several important continuous distributions are provided based on weighted residual (past) inaccuracy measure.

We investigate the structure and properties of $TL$-sub-semihypergroups, where $T$ is an arbitrary triangular norm on a given complete lattice $L$. We study its structure under the direct product and with respect to the fundamental relation. In particular, we consider $L=[0,1]$ and $T=min$, and investigate the connection between $TL$-sub-semihypergroups and the probability space.

We characterize totally ordered sets within the class of all ordered sets containing at least three-element chains using a simple relationship between their isotone transformations and the so called 2-, 3-, 4-endomorphisms which are introduced in the paper. Another characterization of totally ordered sets within the class of ordered sets of a locally finite height with at least four-element chains in terms of the regular semigroup theory is also given.

In this paper, we give some theorems which characterize the intraregular semigroups in terms of intuitionistic fuzzy left, right, and biideals.

Let $G$ be a finite group. Let $X_1\left(G\right)$ be the first column of the ordinary character table of $G$. We will show that if $X_1\left(G\right)=X_1\left({\mathrm{PGU}}_3\left(q^2\right)\right)$, then $G\cong {\mathrm{PGU}}_3\left(q^2\right)$. As a consequence, we show that the projective general unitary groups ${\mathrm{PGU}}_3\left(q^2\right)$ are uniquely determined by the structure of their complex group algebras.

An inverse semigroup $S$ is pure if $e=e^2$, $a\in S$, $e<a$ implies $a^2=a$; it is cryptic if Green’s relation $\mathscr{H}$ on $S$ is a congruence; it is a Clifford semigroup if it is a semillatice of groups. We characterize the pure ones by the absence of certain subsemigroups and a homomorphism from a concrete semigroup, and determine minimal nonpure varieties. Next we characterize the cryptic ones in terms of their group elements and also by a homomorphism of a semigroup constructed in the paper. We also characterize groups and...