The notion of hemicomplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemicomplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemicomplemented lattice to become a D-Stone lattice. Hemicom-complemented lattices and D-Stone lattices are characterized in topological terms.

The notion of hemicomplemented lattices is introduced and some of the properties of these algebras are studied. Some characterization theorems of hemicomplemented lattices are derived with the help of minimal prime D-filters, ideals, and congruences. The notion of D-Stone lattices is introduced and then derived a set of equivalent conditions for a hemicomplemented lattice to become a D-Stone lattice. Hemicom-complemented lattices and D-Stone lattices are characterized in topological terms.

Let M be a unitary left R-module, where R is a (not necessarily commutative) ring with identity. The small intersection graph of nontrivial submodules of M, denoted by 􀀀, (M), is an undirected simple graph whose vertices are in one-to-one correspondence with all nontrivial submodules ofM and two distinct vertices are adjacent if and only if the intersection of corresponding submodules is a small submodule of M. In this paper, we investigate the fundamental properties of these graphs to relate the combinatorial properties of ℾ, (M) to the algebraic properties of the module M. We determine the diameter and the girth of ℾ, (M). We obtain some results for connectivity and planarity of these graphs. Moreover, we study orthogonal vertex, domination number and the conditions under which the graph ℾ, (M) is complemented.

In this paper, we introduced the concepts of normlet and normal ideal in a pseudo-complemented almost distributive lattice and studied its properties. We have char-acterized normal ideals and established equivalent conditions for every ideal to become a normal ideal. Also, derived equivalent conditions for the set of all prime normal ideals of a pseudo-complemented ADL to become a Hausdorff space.

Let L be a  nite-dimensional Lie algebra. We say a subalgebra H of L is per-mutably complemented in L if there is a subalgebra K of L such that L = H + K and H \K = 0. Also, if every subalgebra of L is permutably complemented in L, then L is called completely factorisable. In this article, we consider the in uence of these concepts on the structure of a Lie algebra, in particular, we obtain some characterizations for supersolvability of a  nite-dimensional Lie algebra in terms of permutably complemented subalgebras.

We prove that the following families of (in, nite) groups have complemented subgroup lattice: alternating groups, , nitary symmetric groups, Suzuki groups over an in, nite locally , nite , eld of characteristic 2, Ree groups over an in, nite locally , nite , eld of characteristic 3. We also show that if the Sylow primary subgroups of a locally , nite simple group G have complemented subgroup lattice, then this is also the case for G.

P. Hall proved that a , nite group G is supersoluble with elementary abelian Sylow subgroups if and only if every subgroup of G is complemented in G. He called such groups complemented. A. Ballester-Bolinches and X. Guo established the structure of minimal non-complemented groups. We give the classi, cation of , nite non-soluble groups all of whose second maximal subgroups are complemented groups. We also prove that every , nite group with less than 21 non-complemented non-minimal f 2,3,5 g-subgroups is soluble.

The independence graph Ind(G) of a graph G is the graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence graph of Cartesian product of d copies of complete graphs Kq, which is known as the Hamming graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.