Changes

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

| <math>L_{x \star j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.</math>

| <math>L_{x \,\text{at}\, j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = x \}.</math>

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In the case of a 2-adic relation <math>L \subseteq X_1 \times X_2 = X \times Y,</math> we can reap the benefits of a radical simplification in the definitions of the local flags.  Also in this case, we tend to refer to <math>L_{u \star 1}</math> as <math>L_{u \star X}</math> and <math>L_{v \star 2}</math> as <math>L_{v \star Y}.</math>

In the case of a 2-adic relation <math>L \subseteq X_1 \times X_2 = X \times Y,</math> we can reap the benefits of a radical simplification in the definitions of the local flags.  Also in this case, we tend to refer to <math>L_{u \,\text{at}\, 1}</math> as <math>L_{u \,\text{at}\, X}</math> and <math>L_{v \,\text{at}\, 2}</math> as <math>L_{v \,\text{at}\, Y}.</math>

In the light of these considerations, the local flags of a 2-adic relation <math>L \subseteq X \times Y</math> may be formulated as follows:

In the light of these considerations, the local flags of a 2-adic relation <math>L \subseteq X \times Y</math> may be formulated as follows:

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<math>\begin{array}{lll}

<math>\begin{array}{lll}

L_{u \star X}

L_{u \,\text{at}\, X}

& = &

& = &

\{ (x, y) \in L : x = u \}

\{ (x, y) \in L : x = u \}

\text{the set of all ordered pairs in}~ L ~\text{incident with}~ u \in X.

\text{the set of all ordered pairs in}~ L ~\text{incident with}~ u \in X.

\\[9pt]

\\[9pt]

L_{v \star Y}

L_{v \,\text{at}\, Y}

& = &

& = &

\{ (x, y) \in L : y = v \}

\{ (x, y) \in L : y = v \}

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The local flag <math>E_{3 \star X}</math> is displayed here:

The local flag <math>E_{3 \,\text{at}\, X}</math> is displayed here:

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

|}

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The local flag <math>E_{2 \star Y}</math> is displayed here:

The local flag <math>E_{2 \,\text{at}\, Y}</math> is displayed here:

{| align="center" cellspacing="6" width="90%"

{| align="center" cellspacing="6" width="90%"

Next let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.

Next let's re-examine the ''numerical incidence properties'' of relations, concentrating on the definitions of the assorted regularity conditions.

For instance, <math>L\!</math> is said to be <math>^{\backprime\backprime} c\text{-regular at}~ j \, ^{\prime\prime}</math> if and only if the cardinality of the local flag <math>L_{x \star j}</math> is equal to <math>c\!</math> for all <math>x \in X_j,</math> coded in symbols, if and only if <math>|L_{x \star j}| = c</math> for all <math>x \in X_j.</math>

For instance, <math>L\!</math> is said to be <math>^{\backprime\backprime} c\text{-regular at}~ j \, ^{\prime\prime}</math> if and only if the cardinality of the local flag <math>L_{x \,\text{at}\, j}</math> is equal to <math>c\!</math> for all <math>x \in X_j,</math> coded in symbols, if and only if <math>|L_{x \,\text{at}\, j}| = c</math> for all <math>x \in X_j.</math>

In a similar fashion, one can define the numerical incidence properties <math>^{\backprime\backprime}(< c)\text{-regular at}~ j \, ^{\prime\prime},</math> <math>^{\backprime\backprime}(> c)\text{-regular at}~ j \, ^{\prime\prime},</math> and so on.  For ease of reference, I record a few of these definitions here:

In a similar fashion, one can define the numerical incidence properties <math>^{\backprime\backprime}(< c)\text{-regular at}~ j \, ^{\prime\prime},</math> <math>^{\backprime\backprime}(> c)\text{-regular at}~ j \, ^{\prime\prime},</math> and so on.  For ease of reference, I record a few of these definitions here:

L ~\text{is}~ c\text{-regular at}~ j

L ~\text{is}~ c\text{-regular at}~ j

& \iff &

& \iff &

|L_{x \star j}| = c ~\text{for all}~ x \in X_j.

|L_{x \,\text{at}\, j}| = c ~\text{for all}~ x \in X_j.

\\[6pt]

\\[6pt]

L ~\text{is}~ (< c)\text{-regular at}~ j

L ~\text{is}~ (< c)\text{-regular at}~ j

& \iff &

& \iff &

|L_{x \star j}| < c ~\text{for all}~ x \in X_j.

|L_{x \,\text{at}\, j}| < c ~\text{for all}~ x \in X_j.

\\[6pt]

\\[6pt]

L ~\text{is}~ (> c)\text{-regular at}~ j

L ~\text{is}~ (> c)\text{-regular at}~ j

& \iff &

& \iff &

|L_{x \star j}| > c ~\text{for all}~ x \in X_j.

|L_{x \,\text{at}\, j}| > c ~\text{for all}~ x \in X_j.

\\[6pt]

\\[6pt]

L ~\text{is}~ (\le c)\text{-regular at}~ j

L ~\text{is}~ (\le c)\text{-regular at}~ j

& \iff &

& \iff &

|L_{x \star j}| \le c ~\text{for all}~ x \in X_j.

|L_{x \,\text{at}\, j}| \le c ~\text{for all}~ x \in X_j.

\\[6pt]

\\[6pt]

L ~\text{is}~ (\ge c)\text{-regular at}~ j

L ~\text{is}~ (\ge c)\text{-regular at}~ j

& \iff &

& \iff &

|L_{x \star j}| \ge c ~\text{for all}~ x \in X_j.

|L_{x \,\text{at}\, j}| \ge c ~\text{for all}~ x \in X_j.

\end{array}</math>

\end{array}</math>

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