Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra
We introduce the \emph{tangloid algebra} $\mathcal{T}_n$, a diagrammatic algebra arising from the extended tangloid category $\Ti$, and its subalgebra, the \emph{braidoid algebra} $\mathcal{B}_n$. The construction begins with the unoriented tangloid category $UTC$, inspired by Turaev's theory of knotoids, which is obtained from the unoriented welded tangleoid category $UWTC$ by removing the welded relation while retaining the forbidden moves and imposing some additional relations. Within this framework, the braidoid category arises naturally as a subcategory of $UTC$. We then introduce the extended tangloid category $\Ti$ by adjoining a placeholder morphism $\emptyset$, which allows the generators to be extended to endomorphisms of a fixed $n$-box. The interaction of the placeholder morphism with the crossing morphisms gives rise to diagonal morphisms, $/$ and $\backslash$, which are used in the defining relations of $\Ti$. The corresponding extended braidoid category $\mathbf{Brd}_{\Ti}$ is realised as a subcategory of $\Ti$. For each $n\geq 0$, we then define the tangloid algebra $\mathcal{T}_n$ by linearising the endomorphism algebra $\operatorname{End}_{\Ti}(n)$, as a unital $\mathbb{C}$--algebra presented by diagrammatic generators and relations. We also define the braidoid subalgebra $\mathcal{B}_n$, which provides an algebraic framework for the theory of braidoids. We further introduce the reduced tangloid algebra $δ\mathcal{T}_n$ by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic \textit{bilinear pairing} on $δ\mathcal{T}_n$ using diagrammatic reflection, composition, and closure. Our categorical and algebraic setting provides a theoretical framework for studying diagrammatic structures with intrinsic endpoints, namely knotoids, linkoids, and braidoids, and related diagram algebras.