We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring $I=⟨a,b|2a=2b=0,a^2=b,\mathit{ab}=0⟩$ and the noncommutative ring $E=⟨a,b|2a=2b=0,a^2=a,b^2=b,\mathit{ab}=a,\mathit{ba}=b⟩.$ We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, $E_p=⟨a,b|pa=pb=0,a^2=a,b^2=b,ab=a,ba=b⟩,$ where $a\ne b$ and p is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over $E_p$, where $p=3,5,$ and 7, in short lengths.

Some basic theory on the duality of codes over two non-unital rings of order $ 6 $, namely $ H_{23} $ and $ H_{32} $ is presented. For a code $ {{\mathcal C}} $ over these rings, there is an associated binary code $ {{\mathcal C}}_a $ and a ternary code $ {{\mathcal C}}_b $. Self-orthogonal, self-dual, and quasi self-dual (QSD) codes over these rings are characterized using the associated codes $ {{\mathcal C}}_a $ and $ {{\mathcal C}}_b $, and a classification of self-orthogonal codes for short lengths is given. In addition, a building-up construction for self-orthogonal codes is presented, and cyclic and linear complementary dual (LCD) codes over the said rings are introduced.

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.

In this study, we introduce new versions of fuzzy fractional integral operators for both one- and two-variable cases. Using these operators, several Hermite–Hadamard-type ($H$-type) inclusions are established for fuzzy-number-valued convex functions ($F$·$N\cdot V$-functions) and $F$·$N\cdot V$-coordinated convex functions. These results are obtained by employing $F$·$N\cdot V$-weighted functions within the framework of the newly defined Hadamard and generalized fractional integrals in one- and two-dimensional settings. The use of generalized fractional integral operators provides a unified approach that encompasses a wide class of classical and modern fractional integrals, including the fuzzy Riemann–Liouville and Hadamard types. This unified setting enables the derivation of more comprehensive and flexible inequality results in the fuzzy-number context. The inclusions obtained in this work significantly extend and generalize several known $H\cdot H$-type inequalities previously established for real-valued and interval-valued functions ($I\cdot V$-functions). Furthermore, the proposed results yield a variety of meaningful special cases by specifying suitable kernel functions and parameters of the generalized fractional integrals. In particular, we derive new weighted $H\cdot H$-type inclusions involving logarithmic functions in the fuzzy-number framework. These findings underscore the effectiveness of generalized fractional integrals in capturing nonlocal behavior and uncertainty, and they provide new tools for further investigations in fuzzy analysis, fractional calculus, and generalized convexity.