We consider superfluous elements in a bounded lattice with 0 and 1, and introduce various types of graphs associated with these elements. The notions such as superfluous element graph (S(L)), join intersection graph (JI(L)) in a lattice, and in a distributive lattice, superfluous intersection graph (SI(L)) are defined. Dual atoms play an important role to find connections between the lattice-theoretic properties and those of corresponding graph-theoretic properties. Consequently, we derive some important equivalent conditions of graphs involving the cardinality of dual atoms in a lattice. We provide necessary illustrations and investigate properties such as diameter, girth, and cut vertex of these graphs.

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We introduce the concept of essentiality in a lattice [Formula: see text] with respect to an element [Formula: see text]. We define notions such as [Formula: see text]-essential, [Formula: see text]-uniform elements and obtain some of their properties. Examples of lattices are given wherein essentiality can be retained with respect to an arbitrary element (specifically, there are elements in [Formula: see text] which are [Formula: see text]-essential but not essential). We prove Goldie analogue results in terms of [Formula: see text]-uniform elements and [Formula: see text]-∨-independent sets. Furthermore, we define a graph with respect to [Formula: see text]-essential element in a lattice and study its properties.

Let N be a zero-symmetric (right) nearring with identity. We introduce a partial order in the matrix nearring corresponding to the partial order (defined by Pilz in Near-rings: the theory and its applications, North Holland, Amsterdam, 1983) in N. A positive cone in a matrix nearring is defined and a characterization theorem is obtained. For a convex ideal I in N, we prove that the corresponding ideal $${I^*}$$ I ∗ is convex in $$M_n(N)$$ M n ( N ) , and conversely, if I is convex in $$M_n(N)$$ M n ( N ) , then $${I_{*}}$$ I ∗ is convex in N. Consequently, we establish an order-preserving isomorphism between the p.o. quotient matrix nearrings $$M_n(N)/{I^*}$$ M n ( N ) / I ∗ and $$M_n(N')/{(I')^*}$$ M n ( N ′ ) / ( I ′ ) ∗ where I and $$I'$$ I ′ are the convex ideals of p.o. nearrings N and $$N'$$ N ′ , respectively. Finally, we prove some properties of Archimedean ordering in matrix nearrings corresponding to those in nearrings.

Let $G$ be an $N$-group where $N$ is a (right) nearring. We introduce the concept of relative essential ideal (or $N$-subgroup) as a generalization of the concept of essential submodule of a module over a ring or a nearring. We provide suitable examples to distinguish the notions relative essential and essential ideals. We prove the important properties and obtain equivalent conditions for the relative essential ideals (or $N$-subgroups) involving the quotient. Further, we derive results on direct sums, complement ideals of $N$-groups and obtain their properties under homomorphism.