This section of the article, dedicated to Professor E. Witt on his 60th birthday, introduces the concept of linear representations of a structure $\overline{K}$, which is obtained by forgetting the composition law of morphisms in a given category. A linear representation of $\overline{K}$ is defined by a map $V$ that associates with each morphism $\varphi: a \rightarrow e$ a linear vector space map $V(\varphi): V(a) \rightarrow V(e)$. The focus is on classifying $\overline{K}$ structures that have only finitely many isomorphism classes of indecomposable linear representations, a classification related to an old paper by Yoshii. The introduction defines a 4-tuple $\overline{K} = (\overline{F(K)}, \overline{P(K)}, \overline{n_K}, \overline{s_K})$ consisting of a set of points $\overline{P(K)}$, a set of arrows $\overline{F(K)}$, and two mappings $\overline{n_K}$ and $\overline{s_K}$ that assign to each arrow its node and endpoint. These 4-tuples are referred to as "kästen" (cups) rather than graphs to avoid confusion with other related concepts. The set of all arrows with a common node $a$ and endpoint $e$ is denoted by $\overline{K}[a, e]$ or $[a, e]$, and arrows are written as $\varphi: a \rightarrow e$. The article then discusses the structure of these käten and introduces the concept of linear representations of a k-ordered graph. A (k-linear) representation $V$ of a k-ordered graph $K$ consists of two mappings: one assigning to each point $p \in P(K)$ a k-vector space $V(p)$ and another assigning to each arrow $\varphi: a \rightarrow e$ a k-linear map $V(\varphi): V(a) \rightarrow V(e)$. Morphisms between representations are defined by commuting diagrams, and the category of k-linear representations of $K$ is abelian. The dimension of a representation is given by the product of the dimensions of its vector spaces, and a representation is called indecomposable if its dimension is non-zero and if $V \cong V' \oplus V''$ implies $[V' : k] \cdot [V'' : k] = 0$.This section of the article, dedicated to Professor E. Witt on his 60th birthday, introduces the concept of linear representations of a structure $\overline{K}$, which is obtained by forgetting the composition law of morphisms in a given category. A linear representation of $\overline{K}$ is defined by a map $V$ that associates with each morphism $\varphi: a \rightarrow e$ a linear vector space map $V(\varphi): V(a) \rightarrow V(e)$. The focus is on classifying $\overline{K}$ structures that have only finitely many isomorphism classes of indecomposable linear representations, a classification related to an old paper by Yoshii. The introduction defines a 4-tuple $\overline{K} = (\overline{F(K)}, \overline{P(K)}, \overline{n_K}, \overline{s_K})$ consisting of a set of points $\overline{P(K)}$, a set of arrows $\overline{F(K)}$, and two mappings $\overline{n_K}$ and $\overline{s_K}$ that assign to each arrow its node and endpoint. These 4-tuples are referred to as "kästen" (cups) rather than graphs to avoid confusion with other related concepts. The set of all arrows with a common node $a$ and endpoint $e$ is denoted by $\overline{K}[a, e]$ or $[a, e]$, and arrows are written as $\varphi: a \rightarrow e$. The article then discusses the structure of these käten and introduces the concept of linear representations of a k-ordered graph. A (k-linear) representation $V$ of a k-ordered graph $K$ consists of two mappings: one assigning to each point $p \in P(K)$ a k-vector space $V(p)$ and another assigning to each arrow $\varphi: a \rightarrow e$ a k-linear map $V(\varphi): V(a) \rightarrow V(e)$. Morphisms between representations are defined by commuting diagrams, and the category of k-linear representations of $K$ is abelian. The dimension of a representation is given by the product of the dimensions of its vector spaces, and a representation is called indecomposable if its dimension is non-zero and if $V \cong V' \oplus V''$ implies $[V' : k] \cdot [V'' : k] = 0$.