Narrow Results

In this paper, we investigate the Fischer group $F_{i_{22}}$. This group is generated by a conjugacy class of involutions, any non-commuting pair of which has product of order 3. Such involutions are called transpositions and their conjugacy class is denoted by D. Subgroups generated by elements of D are called D-groups as they have been called by Enright [G. M. Enright, The structure and subgroups of the Fischer groups $F_{22}$ and $F_{23}$, Ph.D. thesis, University of Cambridge (1976)], Fischer embedded or 3-transposition groups. Here, we obtain the following main results:

• •The rank of the groups $F=3\cdot F_{i_{22}}$, the Weyl group W of type $E_6$ over fields of characteristic 2, the group M of shape $3^6⋊U_4(2)\cdot 2$ and the group $H_1=3{\mathrm{\Omega }}_7(3)$ are computed. If G is a 3-transposition group (Fischer embedded) for a class $D$ of transpositions, then $\text{rank}(G)$ is defined as $\text{max}\{|X\left|\right|X\subseteq D$, all elements in X commute$\}$.
• •We prove that the subgroups ${\mathrm{\Omega }}_8^{+}(2)\cdot S_3,3^5\cdot S_6,3\cdot U_4(3)\cdot 2$ and $S{U}_6(2)$ are Fischer embedded and we give an explicit construction for them. It is remarkable to mention that Enright studied certain D-groups under a very strong condition that is $O_2(G)\le Z(G)\ge O_3(G)$ and $G^{\prime }=G^{″}$ where G is the group generated by a proper subset E of D, $G^{\prime }$ is the derived subgroup of G and E is a single conjugacy class in G. Our study will be carried out without such conditions.

In this paper, we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ of all weak endomorphisms of a directed path with $n$ vertices, which is a submonoid of the widely studied monoid ${{𝒪}}_n$ of all order-preserving transformations of an $n$-chain. Also, we describe the regular elements of $\mathrm{\text{wEnd}}{\overrightarrow{P}}_n$ and calculate its size and number of idempotents.

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner–Preston Theorem. Let $n$ be a positive natural number and ${\mathrm{\text{PFI}}}_n$ be the semigroup of all fence-preserving partial one-to-one maps of $\{1,\dots ,n\}$ into itself with respect to composition of maps and the fence $1\prec 2\succ 3\prec \cdots n$. There is considered the inverse semigroup ${\mathrm{\text{IOF}}}_n^{\mathrm{\text{par}}}$ of all $\alpha \in {\mathrm{\text{PFI}}}_n$ such that $\alpha$ is regular in ${\mathrm{\text{PFI}}}_n$, order-preserving with respect to the order $1<2<\cdots <n$ and parity-preserving. According to the main result of the paper, it is $3n-6$ the least of the cardinalities of the generating sets of ${\mathrm{\text{IOF}}}_n^{\mathrm{\text{par}}}$ for $4\le n$. There is determined a concrete representation of a generating set of minimal size.

In this paper, we consider the monoid ${{𝒟}{𝒫}{𝒲}}_n$ of all partial isometries of a wheel graph $W_n$ with $n+1$ vertices. Our main objective is to determine the rank of ${{𝒟}{𝒫}{𝒲}}_n$. In the process, we also compute the ranks of three notable subsemigroups of ${{𝒟}{𝒫}{𝒲}}_n$. We also describe Green’s relations of ${{𝒟}{𝒫}{𝒲}}_n$ and of its three considered subsemigroups.

In this paper, we study the cyclic inverse monoid ${{𝒞}{ℐ}}_n$ on a set ${\mathrm{\Omega }}_n$ with $n$ elements, i.e. the inverse submonoid of the symmetric inverse monoid on ${\mathrm{\Omega }}_n$ consisting of all restrictions of the elements of a cyclic subgroup of order $n$ acting cyclically on ${\mathrm{\Omega }}_n$. We show that ${{𝒞}{ℐ}}_n$ has rank $2$ (for $n\ge 2$) and $n{2}^n-n+1$ elements. Moreover, we give presentations of ${{𝒞}{ℐ}}_n$ on $n+1$ generators and $\frac{1}{2}(n^2+3n+4)$ relations and on $2$ generators and $\frac{1}{2}(n^2-n+6)$ relations. We also consider the remarkable inverse submonoid ${{𝒪}{𝒞}{ℐ}}_n$ of ${{𝒞}{ℐ}}_n$ constituted by all its order-preserving transformations. We show that ${{𝒪}{𝒞}{ℐ}}_n$ has rank $n$ and $3\cdot 2^n-2n-1$ elements. Furthermore, we exhibit presentations of ${{𝒪}{𝒞}{ℐ}}_n$ on $n+2$ generators and $\frac{1}{2}(n^2+3n+8)$ relations and on $n$ generators and $\frac{1}{2}(n^2+3n)$ relations.

Let $I_n$ be the symmetric inverse semigroup on the finite chain $X_n=\{1,\dots ,n\}$ and let $I_{n,r}=\{\alpha \in I_n:|\mathrm{\text{im}}(\alpha )|\le r\}$ for $1\le r\le n-1$. A quasi-idempotent element is an element $\alpha \in I_n$ with the property that $\alpha \ne {\alpha }^2={\alpha }^4$. In this paper, we obtain a useful method by listing the subsets of $X_n$ to build a (minimal) quasi-idempotent generating set of $I_{n,r}$ both as a semigroup and also as an inverse semigroup for $n\ge 2$ and $1\le r\le n-1$.

A semigraph, defined as a generalization of graph by Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs.

In this paper, we define a representing matrix of a semigraph $G$ and call it binomial incidence matrix of the semigraph $G$. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on $n$ vertices can be obtained from the incidence matrix of the complete graph $K_n$.

Let R be a commutative ring with unity, and let $\mathrm{\Gamma }(R)$ denote the comaximal graph of R. The comaximal graph $\mathrm{\Gamma }(R)$ has vertex set as R, and any two distinct vertices x, y of $\mathrm{\Gamma }(R)$ are adjacent if $Rx+Ry=R$. Let ${\mathrm{\Gamma }}_2(R)$ denote the induced subgraph of $\mathrm{\Gamma }(R)$ on the set of all nonzero non-unit elements of R, and any two distinct vertices x, y of ${\mathrm{\Gamma }}_2(R)$ are adjacent if $Rx+Ry=R$. In this paper, we study the graphical structure as well the adjacency spectrum of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$, where $n\ge 4$ is a non-prime positive integer, and ${\mathbb{Z}}_n$ is the ring of integers modulo n. We show that for a given non-prime positive integer n with D number of positive proper divisors, the eigenvalues of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ are $0$ with multiplicity $n-\phi (n)-D-1$, and remaining eigenvalues are contained in the spectrum of a symmetric $D\times D$ matrix. We further calculate the rank and nullity of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$. We also determine all the eigenvalues of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ whenever ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$ is a bipartite graph. Finally, apart from determining certain structural properties of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$, we conclude the paper by determining the metric dimension of ${\mathrm{\Gamma }}_2({\mathbb{Z}}_n)$.

The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG $\mathrm{\Gamma }$ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG $\mathrm{\Gamma }(v,k,t,\lambda ,\mu )$ with $\mu =1$, including some mixed Moore graphs and corresponding codes arising from them, in this paper.