Changes

While operating in this context, it is necessary to distinguish ''domains'' in the broad sense from ''domains of definition'' in the narrow sense.  For <math>k\!</math>-place relations it is convenient to use the terms ''domain'' and ''quorum'' as references to the wider and narrower sets, respectively.

While operating in this context, it is necessary to distinguish ''domains'' in the broad sense from ''domains of definition'' in the narrow sense.  For <math>k\!</math>-place relations it is convenient to use the terms ''domain'' and ''quorum'' as references to the wider and narrower sets, respectively.

For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> I maintain the following usages.

For a <math>k\!</math>-place relation <math>L \subseteq X_1 \times \ldots \times X_k,\!</math> we have the following usages.

# The notation <math>{}^{\backprime\backprime} \operatorname{Dom}_j (L) {}^{\prime\prime}\!</math> denotes the set <math>X_j,\!</math> called the ''domain of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> domain of <math>L.\!</math>''.

# The notation <math>{}^{\backprime\backprime} \operatorname{Dom}_j (L) {}^{\prime\prime}\!</math> denotes the set <math>X_j,\!</math> called the ''domain of <math>L\!</math> at <math>j\!</math>'' or the ''<math>j^\text{th}\!</math> domain of <math>L.\!</math>''.

{| align="center" cellspacing="8" width="90%"

{| align="center" cellspacing="8" width="90%"

| <math>\operatorname{Quo}_j (L)\!</math>

|

| =

<math>\begin{array}{lll}

| the largest <math>Q \subseteq X_j\!</math> such that <math>L_{Q \,\text{at}\, j}\!</math> is <math>(> 1)\text{-regular at}~ j,\!</math>

\operatorname{Quo}_j (L)

& = &

| &nbsp;

\text{the largest}~ Q \subseteq X_j ~\text{such that}~ ~L_{Q \,\text{at}\, j}~ ~\text{is}~ (> 1)\text{-regular at}~ j,

| =

\\[6pt]

| the largest <math>Q \subseteq X_j\!</math> such that <math>|L_{Q \,\text{at}\, j}| > 1\!</math> for all <math>x \in Q \subseteq X_j.\!</math>

& = &

\text{the largest}~ Q \subseteq X_j ~\text{such that}~ |L_{Q \,\text{at}\, j}| > 1 ~\text{for all}~ x \in Q \subseteq X_j.

\end{array}</math>

|}

|}