Let $G$ be a compact group and ${\cal C}(G)$ be the $C^{*}$ -algebra of continuous complex-valued functions on $G$ . The paper constructs an imbedding of the Fourier algebra ${\rm A}(G)$ of $G$ into the algebra ${\rm V}(G) = {\cal C}(G)\otimes^h {\cal C}(G)$ (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in $G \times G$ are sets of operator synthesis with respect to the Haar measure, using the definition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders defined by closed unital subsemigroups of $G$ .