2. Dashed lines, dashes and spaces: some definitions and simple properties

Let’s resume now from a formal point of view what we treated conceptually in the post From prime numbers to dashed lines. Let’s start from the definition of dashed line.

Dashed line

Let $k$ be a positive integer and $I$ a set of $k$ natural numbers (usually $\{1, \ldots, k\}$). A dashed line of order $k$ is a function $T: I \times \mathbb{N} \rightarrow \mathbb{N}$ such that

$$m < n \Rightarrow T(i, m) < T(i, n),$$

for all $i, m, n$.
The elements of the set $I \times \mathbb{N}$ are called dashed. They are distinguished into rows and columns:

• The set of dashes having as first element a certain $i$, $\{(i, 0), (i, 1), (i, 2), \dots\}$, is called row $i$ of the dashed line.
• The set of dashes for which the dashed line assumes a certain value $j \in \mathbb{N}^{\star}$, $\{(i, n) \in I \times \mathbb{N} \mid T(i, n) = j\}$, is called column $j$ of the dashed line.

The elements of the set $I$ are called indexes.
The dashes of each row are ordered with respect to their second element: $(i, 0) < (i, 1) < (i, 2) < \dots$. The dash $(i, 1)$ is called the first dash of row $i$ (even if the first in the just defined order is $(i, 0)$, but this dash is often disregarded); in general the dash $(i, n)$ is called $n$-th dash, or dash number $n$, of row $i$. The number $T(i, n) \in \mathbb{N}$ is called the value of the dash $(i, n)$.
We agree on denoting dashed lines with the capital letters $T, U, \dots$ and dashed with the lower case letters $t, u, \dots$.

The correspondence between this definition and the tabular representation we have previously shown, is rather immediate: the columns of the table are the possible values assumed by the function $T$, and every dash $(i, n)$ is represented in the table with a dash sign (“-“) on the intersection between row $i$, numbering rows with increasing indexes, from top to down, and the column numbered with the dash value. For example, Table 10 of the previous post contains the following dashes, assuming $I = \{1, 2, 3, 4\}$ as indexes set:

Table 11: graphical representation of the dashed line illustrated in Table 10 of the previous post, highlighting the dashes from a mathematical point of view
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
$(1, 0)$ $(1, 1)$ $(1, 2)$ $(1, 3)$ $(1, 4)$ $(1, 5)$ $(1, 6)$ $(1, 7)$ $(1, 8)$ $(1, 9)$
$(2, 0)$ $(2, 1)$ $(2, 2)$ $(2, 3)$ $(2, 4)$ $(2, 5)$
$(3, 0)$ $(3, 1)$ $(3, 2)$ $(3, 3)$ $(3, 4)$
$(4, 0)$ $(4, 1)$ $(4, 2)$ $(4, 3)$

You can note for example that, by Definition T.1, row 1 is the following set of dashes; $\{(1, 0), (1, 1), (1, 2), (1, 3), \ldots\}$; column 6 is the set $\{(1, 2), (2, 1)\}$, while column 5 is the empty set.

The dashed line represented in the table is the following function $T: \{1, 2, 3, 4\} \times \mathbb{N} \rightarrow \mathbb{N}$:

$$T(i, n) := \begin{cases} 0 & \text{if n = 0}\\\left(i + 1\right)\left(n + 1\right) & \text{otherwise} \end{cases}$$

In order to verify this mathematical definition to coincide with the representation of Table 11 indeed, we have to check that:

• The dash $(i, n)$ is in row $i$, whatever $n$ is
• The dash $(i, n)$ is in column $T(i, n)$, whatever $i$ and $n$ are

The first property is immediately verified: in Table 11, all the dashes in the first row are of the kind $(1, n)$, those ones of the second row are of the kind $(2, n)$, etcetera. Concerning the second point instead:

• All the dashes of the kind $(i, 0)$ are placed in column 0
• The dash $(i, n) = (1, 1)$ is placed in column 4 and indeed $(i + 1)(n + 1) = 2 \cdot 2 = 4$
• The dash $(i, n) = (1, 2)$ is placed in column 6 and indeed $(i + 1)(n + 1) = 2 \cdot 3 = 6$
• The dash $(i, n) = (1, 3)$ is placed in column 8 and indeed $(i + 1)(n + 1) = 2 \cdot 4 = 8$
• The dash $(i, n) = (2, 1)$ is placed in column 6 and indeed $(i + 1)(n + 1) = 3 \cdot 2 = 6$
• The dash $(i, n) = (2, 2)$ is placed in column 9 and indeed $(i + 1)(n + 1) = 3 \cdot 3 = 9$
• The dash $(i, n) = (2, 3)$ is placed in column 12 and indeed $(i + 1)(n + 1) = 3 \cdot 4 = 12$

We note that, in the definition of dashed line, the only restriction on the function $T$ is the following:

$$m < n \Rightarrow T(i, m) < T(i, n)$$

In other terms, this is equivalent to say that the dashes of a generic row $i$, taken in an orderly fashion:

$$(i, 0),\; (i, 1),\; (i, 2),\; (i, 3),\; \ldots \tag{3}$$

have values (that are column numbers) strictly increasing. So, in the table representing a dashed line, when reading any row $i$ from left to right, the exact sequence of dashes (3) is found (see Table 11).
Finally, the fact that in the definition there is $<$, instead of $\leq$, means that two dashes in the same row can never overlap, that is stay in the same column: if the dash $(i, n)$ has value $v$, the dash $(i, n + 1)$ – that is the first dash you can see after $(i, n)$ when reading the row towards right – must have a value of at least $v + 1$, it cannot have value $v$ too. This implies two consequences:

• A cell cannot contain more than one dash
• A column cannot contain more than $k$ dashes, where $k$ is the order of the dashed line

We saw that a particular kind of dashed lines are linear ones. Here is their mathematical definition.

Linear dashed line

Let $T$ be a linear dashed line of order $k$. If the function $T: I \times \mathbb{N} \rightarrow \mathbb{N}$ is of the kind

$$T(i, n) = n_i n \tag{4}$$

where $n_i$ is an integer greater than 1 that depends only on $i$, then T is said linear, and we write $T = (n_1, n_2, \dots, n_k)$ (assuming $I = \{1, \ldots, k\}$). The integers $n_1, n_2, \dots, n_k$ are said components of the dashed line $T$, and they are called respectively the first component, the second component, and so on.

All the examples seen in the post From prime numbers to dashed lines, until Table 9, are representations of linear dashed lines. For example we can rewrite Table 7 this way, highlighting the dashes from a mathematical point of view:

Table 12: graphical representation of the linear dashed line $(2, 3, 5)$ shown in Table 7, highlighting the dashes from a mathematical point of view
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
$(1, 0)$ $(1, 1)$ $(1, 2)$ $(1, 3)$ $(1, 4)$ $(1, 5)$ $(1, 6)$ $(1, 7)$ $(1, 8)$ $(1, 9)$ $(1, 10)$
$(2, 0)$ $(2, 1)$ $(2, 2)$ $(2, 3)$ $(2, 4)$ $(2, 5)$ $(2, 6)$
$(3, 0)$ $(3, 1)$ $(3, 2)$ $(3, 3)$ $(3, 4)$

In this case, in a linear dashed line, we can know the column of a dash $(i, n)$, that is $T(i, n)$ by Definition T.1, by simply multiplying $n_i$ by $n$. In the dashed line $(2, 3, 5)$ we have that $n_1 = 2$, $n_2 = 3$ and $n_3 = 5$, so for example:

• All the dashes of the kind $(i, 0)$ are placed in column $n_i \cdot 0 = 0$
• The dash $(i, n) = (1, 1)$ is placed in column $n_1 n = 2 \cdot 1 = 2$
• The dash $(i, n) = (1, 2)$ is placed in column $n_1 n = 2 \cdot 2 = 4$
• The dash $(i, n) = (1, 3)$ is placed in column $n_1 n = 2 \cdot 3 = 6$
• The dash $(i, n) = (2, 1)$ is placed in column $n_2 n = 3 \cdot 1 = 3$
• The dash $(i, n) = (2, 2)$ is placed in column $n_2 n = 3 \cdot 2 = 6$
• The dash $(i, n) = (3, 4)$ is placed in column $n_3 n = 5 \cdot 4 = 20$

It’s worth observing that the function of a linear dashed line, that in (4) was defined in a compact way, is indeed defined by cases. For example, concerning the dashed line $(2, 3, 5)$ the function defined as $T(i, n) := n_i n$ can be rewritten in extended manner like this:

$$T(i, n) := \begin{cases} 2n & \text{if i = 1}\\3n & \text{if i = 2}\\5n & \text{if i = 3}\end{cases} \tag{5}$$

Since linear dashed lines constitute most of the subject of study of dashed line theory, we can say that this theory studies for the most part functions like (5), that can differ by the number of provided cases (that is the order of the dashed line) or by the coefficients of the variable $n$ (that are the components).

Talking about the connection between dashed lines and prime numbers, we introduced the concept of space. In mathematical terms, a space is a value that is never assumed by the dashed line function:

Space

Let $T: I \times \mathbb{N} \rightarrow \mathbb{N}$ a dashed line of order $k$. We define space an element of $\mathbb{N} \setminus T\left(I \times \mathbb{N}\right)$, that is a natural number $s$ such that no dash of $T$ has $s$ as value.
We agree on denoting a space with the letter $s$.

In the representation of a dashed line, spaces are therefore the numbers of columns (possible values) which don’t contain any dash: this is the intuitive definition of space we stated previously.

The spaces of a dashed line, being simple natural numbers, have a total ordering, that is we can say which is the first, the second, the third, and so on. We can do the same thing with dashes: since the dashed line has a finite number of rows, we can read its table from up to bottom and from left to right (that is by columns), and enumerate dashes in the order we run into them. By convention, in a linear dashed line, dashes of column 0 are excluded from this enumeration. For example, in Table 12 the order of the dashes is the following:

Table 13: graphical representation of the dashed line $(2, 3, 5)$, where the ordinal numbers of dashes are written
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 3 5 7 9 11 13 16 17 19
2 6 8 12 14 18
4 10 15 20

This ordering is mathematically defined as follows:

Dash ordering

Let $T$ be a dashed line. Given two dashes of $T$, $(i, m)$ and $(j, n)$, we say that $(i, m) \leq (j, n)$ if and only if one of the following conditions is satisfied:

• the first dash has a lower value than the second dash: $T(i, m) < T(j, n)$
• the two dashes have the same value but the first one has a row number not greater than that of the second dash: $T(i, m) = T(j, n)$ and $i \leq j$

If $T$ is linear, the least dash having a value greater than 0 is defined as the first dash of $T$ and hence, following to the just defined ordering, the second dash, the third dash, and so on, are defined. In general, if $t$ is the $x$-th dash of $T$, the number $x$ is said ordinal of $t$.

Having defined the relation $\leq$ like above, the relations $=$ (that is $t = u \iff \text{t \leq u and u \leq t}$) and $<$ (that is $t < u \iff \text{t \leq u and t \neq u}$) are defined too. You can verify that these two “derived” relations are the following:

• $(i, m) = (j, n) \iff \text{i = j and m = n}$
• $(i, m) < (j, n) \iff \text{(i, m) and (j, n)}$ satisfy Definition T.4 with $i < j$ in place of $i \leq j$ in the second case

The passages for getting to the latter definitions starting from Definition T.4 are strictly technical and of little interest for dashed line theory, so we disregard them. However we note that the definition of $=$ is rather obvious (essentially it says that a dash is equal to itself), while we can verify in practice, by means of Table 12, that the definition of $<$ corresponds to reading dashes by columns. In fact, when reading by columns the dashed of that Table, the following ordering is obtained:

$$(1, 1) < (2, 1) < (1, 2) < (3, 1) < (1, 3) < (2, 2) < (1, 4) < (2, 3) < (1, 5) < (3, 2) < \dots$$

The values of these dashes are, respectively:

$$2,\; 3,\; 4,\; 5,\; 6,\; 6,\; 8,\; 9,\; 10,\; 10,\; \dots$$

Observing the values, you can immediately see that, when passing from a dash to the next one, either the value is strictly increasing (first case of Definition T.4), or it remains constant and, in such a case, the row number is increasing (second case of Definition T.4), as for the dashes $(1, 3)$ and $(2, 2)$ and the dashes $(1, 5)$ and $(3, 2)$.

Another concept which has to be formally defined, among those introduced in the post From prime numbers to dashed lines, is the addition of a row to a dashed line. For example we saw by a graphical point of view that, in Table 8, by adding a row to the dashed line $(2,3)$ the dashed line $(2,3,5)$ is obtained.
The definition of dashed line contains indeed an explicit reference to the rows, identified by the integers in the set $I$ (usually $\{1, \dots, k\}$): in fact a dash is a couple of the kind $(i, n)$, where $i$ is just the row number. Coming back to the cited definition, we have that:

• A dashed line is a function $T: I \times \mathbb{N} \rightarrow \mathbb{N}$
• The domain of the function $T$, $I \times \mathbb{N}$, is the set of the possible dashes, that is the possible couples given by a row number and an integer number
• The set of all the possible dashes of row $i$ is $\{i\} \times \mathbb{N} = \{(i,0),(i,1),(i,2),\dots\}$, that is the set of all dashes having $i$ as first element

So adding a row to a dashed line $T$ of order $k$, imagining to leave the existing rows in their places and adding the new row as the last one, it means to define the function $T$ also on dashes of the kind $(k+1,n)$, where $k+1$ is the number of the new row. The new row could be also added “in the middle” of the other rows, by shifting the indexes of the next rows.
On the other side, removing row $k$ from the dashed line $T$ means to restrict the domain of $T$, eliminating the definition of the function on dashes of the kind $(k,n)$. In this case too, if we removed a row that is not the last one, we should shift the indexes of the next rows.
Extrapolating the mechanism underlying these conceptual operations, the concept of dashed subline is obtained:

Dashed subline

Let $T$ be a dashed line of order $k$ with indexes set $I$. Let $\{i_1, \dots, i_h\}$ be a subset of $I$. We define dashed subline of $T$ with respect to the indexes $i_1, \dots, i_h$ the dashed line $U: \{i_1, \dots, i_h\} \times \mathbb{N} \rightarrow \mathbb{N}$ of order $h$ defined as follows:

$$U(i_1, n) := T(i_1, n)$$
$$\dots$$
$$U(i_h, n) := T(i_h, n)$$

that is, in other terms, $U$ is the restriction of $T$ over the set $\{i_1, \dots, i_h\} \times \mathbb{N}$.

The dashed line $U$ so defined is denoted as $T[i_1, \dots, i_h]$.

If $U \neq T$, or equivalently if $\{i_1, \dots, i_h\} \neq I$, $U$ is said to be a proper dashed subline of $T$.

The dashed line $T[i_1, \dots, i_h]$ is so, in simple words, the dashed line obtained from $T$ by extracting the rows $i_1, \dots, i_h$. The resulting dashed line has the row $i_1$ of T as first row, …, the row $i_h$ of $T$ as last row ($h$-th row). For example, the dashed sublines of the dashed line $(2,3,5)$ are the following:

• $(2,3,5)[1] = (2)$
• $(2,3,5)[2] = (3)$
• $(2,3,5)[3] = (5)$
• $(2,3,5)[1,2] = (2,3)$
• $(2,3,5)[1,3] = (2,5)$
• $(2,3,5)[2,3] = (3,5)$
• $(2,3,5)[1,2,3] = (2,3,5)$

They are all proper dashed sublines, except the last one.

With the definition of dashed subline, the fact that the dashed line $(2,3,5)$ is obtained by adding a row to the dashed line (2,3), as shown in Table 8, can be expressed mathematically with the expression $(2,3,5)[1,2] = (2,3)$, stating that the two dashed lines coincide in their first two rows: so the first one, which has three rows, has the last row extra with respect to the second.

Though the notation $T[i_1, \dots, i_h]$ may suggest that the indexes form an ordered list, indeed what matters is only which indexes are present, not the order in which they are listed. In fact the definition of the function $U$ in Definition T.6 does not depend on the order of $i_1, \dots, i_h$. Indexes are normally listed in increasing order, like in $(2,3,5)[1,2]$, but occasionally some alternative notations like $(2,3,5)[2,1]$ may occur: in these cases we have to remind that the dashed subline is always the same, no matter how the indexes are ordered. It would be more correct to write something like $T[\{1,2\}]$, but we prefer to avoid complicating the notation this way.