Quoting an example from C.L Liu's Discrete Mathematics: Let R be a binary relation on A. The transitive extension of R (let's denote it as $R_1$) is a binary relation on A such that $R_1$ contains R. Doesn't that make $R_1$ the transitive closure to R or is a transitive extension different from a closure?
Also, in the following two matrices, how is the second one the transitive extension to the first one?
$$
\begin{matrix}
\verb|bmatrix| & \begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix} \\[15pt]
\end{matrix}
$$
$$
\begin{matrix}
\verb|bmatrix| & \begin{bmatrix}
0 & 1 & 1 & 0 \\
0 & 1 & 1 & 1 \\
0 & 1 & 1 & 1 \\
0 & 0 & 0 & 0
\end{bmatrix} \\[15pt]
\end{matrix}
$$
Best Answer
Yes, transitive extension and transitive closure are different notions. If you start from a relation $R$ on some set, then the transitive extension $R_1$ of $R$ is obtained by adding to $R$ a pair $(a,c)$ whenever $(a,b)$ and $(b,c)$ are in $R$ for some $b$. Note that the transitive extension is in general not-transitive, while transitive closures are. For example, consider the two matrices of your question, and let $(i,j)$ denote the place corresponding to row $i$ and column $j$. Let $R$ be the relation represented by the first matrix, $R_1$ the relation represented by the second one. Then we say that $i$ is related to $j$ under $R$ if there is $1$ in place $(i,j)$ of the matrix representing $R$, we say they are not related if there is $0$. The same for $R_1$. In the first matrix, you have $1$ at place $(1,2)$, and $1$ in place $(2,3)$, meaning that $1$ is related to $2$ and $2$ to $3$. In the matrix, you have $0$ at place $(1,3)$, meaning that $R$ is not transitive, but you have $1$ in matrix $R_1$, which represents the transitive extension of $R$. Still $R_1$ is not transitive, because for example, using the same notation, $(1,2)$ and $(2, 4)$ are in $R_1$, but $(1,4)$ is not.
To obtain a transitive relation starting from a general relation $R$, you have to build its transitive closure, which is the least relation containing $R$ which is transitive, or equivalently, the set intersection of all transitive relations containing $R$.
You can relate the two notions by observing that, if you denote by $R_1$ the transitive extension of $R$, by $R_2$ the transitive extension of $R_1,\ldots,$ by $R_{i+1}$ the transitive extension of $R_i\ldots$, then the transitive closure of $R$ is the set union of all $R_i$'s.