In this work, we construct a homotopy theory for a class of intersection graphs arising from topological monoids. We introduce the M-intersection graph of a ${\tau }_e$-monoid, where the vertices correspond to proper ${\tau }_e$-submonoids and adjacency is defined by trivial intersection. Several structural properties of the graph, including total disconnectedness, bipartiteness and planarity, are investigated and shown to be closely related to the algebraic structure and decomposition of finite ${\tau }_e$-monoids. Based on this framework, we develop a graphical homotopy theory by introducing graphical ${\tau }_e$-monoids, graphical homomorphisms, and graphical homotopies. We study graphical homotopy equivalence, graphical contractibility, and path monoids, and examine retraction properties through graphical retracts, D-graphical retracts and graphical homotopy extension properties. Furthermore, we present an example of graphical comprehensive monoids and construct a $\theta$-homogeneous topology on the set of graphical path homotopy classes. We show that this topology is compatible with the induced monoid operation, yielding a well-behaved functorial topological monoid structure.

A near-field is an algebraic structure akin to a division ring where distributivity is relaxed on one side. The smallest such object is the Dickson near-field of order nine. We lay the foundations of linear codes over that near-field. This includes a systematic form for their generator matrices based on a one-sided Gauss pivot algorithm. Dual codes are defined and a formula is given for the parity check matrix. LCD codes are characterized by a direct sum property. Self-orthogonal codes are classified in short lengths.

This paper introduces a graphical framework for smooth Lie group actions based on tangent orbit interactions. In contrast with classical intersection graphs, where vertices usually represent algebraic subobjects and edges record set-theoretic intersections, the present construction uses non-trivial orbits as vertices and creates edges from common nonzero tangent directions inside the fixed ambient embedding. Starting from infinitesimal tangent spaces generated by the action, we construct Lie orbit tangent graphs and analyze their adjacency structure, connectedness, completeness, degrees and diameter estimates. To describe local and global interactions, tangent fibers, local tangent orbit cliques, tangent orbit hypergraphs and incidence structures are introduced. We further develop separating tangent paths and use them to construct neighborhood systems and tangent-separating topologies. The framework gives a unified way to encode orbit-level tangent interactions and may be useful in geometric analysis, symmetry-based dynamical systems, differential topology and mathematical physics, where orbits and infinitesimal directions describe invariant motions, constraints or symmetry-reduced configurations. Several examples are included to illustrate how Lie group actions, graph structures, hypergraphs and tangent geometry interact within the proposed scheme.

In this paper, we study the homotopy theory of single intersection graphs arising from acting spaces over topological semigroups. An acting space $(\mathcal{S},\mathcal{B})$ is defined as a topological space $\mathcal{B}$ equipped with a continuous action of a topological semigroup $\mathcal{S}$, generalizing the notion of algebraic actions in a topological setting. To connect this structure with graph theory, we associate to each acting space a single intersection graph ${\mathcal{G}}_{\mathcal{S}\mathcal{B}}$, whose vertices are proper $\mathcal{S}\mathcal{B}$-subacting spaces, and two vertices are adjacent if their intersection is a singleton set. This graph construction encodes both algebraic and topological interactions between subacting spaces and provides a framework to study connectivity and homotopical properties via combinatorial methods. We then work within a categorical framework, where objects are graphical acting semigroups and morphisms are $\mathcal{S}$-acting maps, allowing us to systematically study structural properties and their invariance under morphisms. In this setting, we introduce the notion of acting fibrations and formulate the corresponding lifting problem. Our main result establishes that an $\mathcal{S}$-acting map is an acting fibration if and only if it admits an $\mathcal{A}$-lifting function, providing a characterization analogous to classical fibration theory. Furthermore, we introduce $\mathcal{A}$-regular lifting functions and analyze their role in preserving homotopical structures, including a natural homotopy extension property.

In this article, we study the homotopy aspects of intersection graphs of topological semigroups. We begin by defining the top intersection graph $T{G}_{\mathbb{X}}$ and investigating how the algebraic and topological properties of a topological semigroup are reflected in the global structure of this graph. In particular, we characterize when $T{G}_{\mathbb{X}}$ is totally disconnected, bipartite, or planar in terms of the order and factorization of the underlying semigroup. We then introduce the notions of $HTG$-semigroups, graphical homomorphisms, and graphical homotopy relations, thereby developing a graphical homotopy framework. Within this setting, we study $Gr$-homotopy equivalences, $Gr$-contractible spaces, and retraction phenomena, including $DGr$-retracts and homotopy extension properties. Finally, we introduce graphical total semigroups and equip the set of $Gr$-path homotopy classes $\left[{\mathbb{X}}_{pe}\right]$ with a natural $\Delta$-uniform topology. We show that this topology is compatible with the induced semigroup operation, yielding a topological semigroup structure. Overall, this work provides a unified algebraic, topological, and graph-theoretic perspective, and opens the door to further applications of homotopy theory in the study of intersection graphs of topological semigroups.