TABLE I:

UDF operator equivalences

Commutativity
$\begin{array}{l}{\mathit{group}}_{G,g,{\alpha }_g,{\beta }_g}\left({\mathit{group}}_{G,f,{\alpha }_f,{\beta }_f}(R)\right)\equiv \\ {\mathit{group}}_{G,f,{\alpha }_f,{\beta }_f}\left({\mathit{group}}_{G,g,{\alpha }_g,{\beta }_g}(R)\right)\end{array}$	if αg ∩ βf = ∅˄ αf ∩ βg = ∅
${\mathit{compute}}_{g,{\alpha }_g,{\beta }_g}\left({\mathit{compute}}_{f,{\alpha }_f,{\beta }_f}(R)\right)\equiv {\mathit{compute}}_{f,{\alpha }_f,{\beta }_f}\left({\mathit{compute}}_{g,{\alpha }_g,{\beta }_g}(R)\right)$	if αg ∩ βf = ∅˄ αf ∩ βg = ∅
Compute/Group
${\mathit{compute}}_{g,{\alpha }_g,{\beta }_g}\left({\mathit{group}}_{G,f,{\alpha }_f,{\beta }_f}(R)\right)\equiv {\mathit{group}}_{G\cup {\beta }_q,f,{\alpha }_f,{\beta }_f}\left({\mathit{compute}}_g,{\alpha }_g,{\beta }_g(R)\right)$	if αg ⊆ G, αg ∩ βf = ∅˄ αf ∩ βg = ∅