ua citation

app_notation-background.tex

@@ -66,7 +66,7 @@ In $\db_{\semNX}$ we assign each tuple $\tup$ the polynomial:
   \]
 The probability distribution $\pd'$ assigns all world vectors zero probability except for $\abs{\idb}$ world vectors (representing the possible worlds) $\vct{w}_i$. All elements of $\vct{w}_i$ are zero except for the position corresponding to variables $X_{i}$ which is set to $1$. Unfolding definitions it is trivial to show that $\rmod(\pxdb) = \pdb$. Thus, \abbrNXPDB\xplural are a complete representation system.
 
-Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}$ used here, uniquely extends to the semiring homomorphism alluded to above, $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}: \semNX \to \semN$. For a polynomial $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}$ substitutes variables based on $\vct{w}$ and then evaluates the resulting expression in $\semN$. For instance, consider the polynomial $\pxdb\inparen{\tup} = \poly = X + Y$ and assignment $\vct{w} := X = 0, Y=1$. We get $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup} = 0 + 1 = 1$. % It is trivial to show that an assignment  is a semiring homomorphism.
+Since $\semNX$ is the free object in the variety of semirings, Birkhoff's HSP theorem~\cite{graetzer-08-un} implies that any assignment $\vct{X} \to \semN$, which includes as a special case the assignments $\assign_{\vct{w}}$ used here, uniquely extends to the semiring homomorphism alluded to above, $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}: \semNX \to \semN$. For a polynomial $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup}$ substitutes variables based on $\vct{w}$ and then evaluates the resulting expression in $\semN$. For instance, consider the polynomial $\pxdb\inparen{\tup} = \poly = X + Y$ and assignment $\vct{w} := X = 0, Y=1$. We get $\assign_\vct{w}\inparen{\pxdb}\inparen{\tup} = 0 + 1 = 1$. % It is trivial to show that an assignment  is a semiring homomorphism.
 Closure under $\raPlus$ queries follows from this and from \cite{DBLP:conf/pods/GreenKT07}'s Proposition 3.5, which states that semiring homomorphisms commute with queries over $\semK$-relations.