Changes

\star ---> \,\text{at}\,
Line 3,898: Line 3,898:     
{| 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>
 
|}
 
|}
   −
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:
Line 3,908: Line 3,908:  
|
 
|
 
<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 \}
Line 3,915: Line 3,915:  
\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 \}
Line 3,939: Line 3,939:  
|}
 
|}
   −
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%"
Line 3,954: Line 3,954:  
|}
 
|}
   −
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%"
Line 3,973: Line 3,973:  
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:
Line 3,982: Line 3,982:  
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>
 
|}
 
|}
12,080

edits