If \(E\) is an ordered vector space, its centre \(Z (E)\) is the space of all linear operators from \(E\) into itself such that there exists \(\alpha \in \mathbb{R}\) with \(- \alpha I_E \leq T \leq \alpha I_E\), where \(I_E\) is the identity of \(E\). \(L^r (E,F)\) denotes the space of regular operators from \(E\) into \(F\). If \(\varphi = \sum_{k=1}^n U_k \otimes V_k\) is an element of the algebraic tensor product \(Z (E) \odot Z (F)\), then an operator \(\Phi\) on \(L^r (E,F)\) may be defined by \(\Phi (T) = \sum_{k=1}^m V_k \circ T \circ U_K\). In Theorem 2.1, it is proved that if \(E\) and \(F\) are uniformly complete vector lattices with order dual of \(E\) separating the points of \(E\), then the operator \(\Phi \in Z (L^r (E,F))\) for each \(\varphi^{.} \in Z (E) \odot Z (F)\). This correspondence is norm preserving and extends to an isometry of \(Z (E) \otimes_\lambda Z (F)\) into \(Z (L^r (E,F))\), which is also an algebra and order isomorphism. These considerations give rise to a characterisation of pre-regular operators. In the third part of the paper, the author studies the density of the embedding \(Z (E) \otimes_\lambda Z (F)\) into \(Z (L^r (E,F))\), as it is rarely onto. To this end, a topology \(\tau\) is introduced for \(Z (L^r (E,F))\) whose subbasic open sets are \(\{W : \||W (T) - V (T) |\| 0\). In Theorem 3.2, it is proved that if \(E\) is a Banach lattice which is \(\sigma\)-Dedekind complete or has a topological order unit and \(F\) is a Banach lattice with order continuous norm, then \(\{\Phi : \varphi \in Z (E) \otimes Z (F)\}\) is \(\tau\)-dense in \(Z (L^r (E,F))\). Let \(W^r (E,F)\) be the linear span of the positive weakly compact operators from \(E\) into \(F\), equipped with the norm \(\|T \|_w = \inf \{ \|S \|: \pm T \leq S \in W^r (E,F)\}\). For \(\varphi\) in \(Z (E^{\ast \ast}) \otimes Z (F)\), the corresponding operator \(\Phi (T)\) on \(Z (W^r (E,F))\) is defined as \(\Phi (T) = (\sum_{i=1}^m V_k \circ T^{\ast \ast} \circ U_k) |_E\) whenever \(\varphi = \displaystyle \sum_{k=1}^m U_k \otimes V_k\) is in \(Z (E^{\ast \ast}) \otimes Z (F)\). If \(E^\ast\) has order continuous norm, \(\Phi (T) \in Z (W^r (E,F))\). Furthermore, the norm preserving correspondence \(\varphi \rightarrow \Phi\) extends to an isometry of \(Z (E^{\ast \ast}) \otimes_\lambda Z (F)\) into \(Z (W^r (E,F))\). In Theorem 3.6, it is proved that when both \(E^\ast\) and \(F\) have order continuous norms, then \(\{\Phi : \varphi \in Z (E^{\ast \ast}) \otimes Z \subset F)\}\) is \(\tau\)-dense in \(Z (W^r (E,F))\). In Theorems 3.7 and 3.8, the author obtains stronger density results for norm compact operators. In the final section of the paper, the results are applied to obtain a domination theorem for \(r\)-compact operators.