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Congeneric Orders

Congeneric orders is a stack of a compatible congeric orders that represents Order or Partial Order. The congeneric orders are sorted in the stack by the first position with a non-empty element in it.

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Mathematical Definition

Let \(X_{-}\) is Partial Carrier set

Let \(O_{c}\) is Congeneric Order

Define a Congeric orders

as m-tuple of congeneric order

Let \(compatible(O1_c, O2_c)\) is compatibility of Partial Sequences function

\(CO\) is m-tuple of \(O_c\) orders

\[CO = <co_1, co_2, ..., co_m>,\]
\[\forall j \in \{1, ..., m\} \exists co_j \in \{O_c\}, \]

all congeneric orders are compatible

\[compatible(CO(k), CO(j)) \bigg| \forall k,j \in \{1, ..., m\}, k \ne j,\]

and sorted by first apperance of non-empty element

\[\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, CO(k)(x) \ne 1 \bigg| CO(j)(i) = 1 \land CO(j)(x) \ne 1\]

where:

  • \(co_j\)​ is called the \(j\)-th congeneric order.
  • \(l := |CO(1)|\) is length, \(l \in N\)
  • \(m := |CO|\) is power, \(m \in N\)

as matrix (m,l)

\(CO\) is (m,l) matrix of \(\{-, 1\}\)

\[ C0 = \begin{pmatrix} co_{1,1} & co_{1,2} & \cdots & co_{1,l} \\ co_{2,1} & co_{2,2} & \cdots & co_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ co_{m,1} & co_{m,2} & \cdots & co_{m,l} \end{pmatrix} \]
\[\forall j \in \{1, ..., m\}, i \in \{1, ..., l\} \exists co_{j,i} \in \{-, 1\}, \]

columns are compatible

\[\ co_{k,i} \ne 1 \big | \forall i \in \{1,...,l\}, \forall j \in \{1,...,m\}, \forall k \ne j,\ co_{j,i} = 1 \]

and sorted by first apperance of non-empty element

\[\forall k > j \in \{1,...,m\},\ x < i \in \{1,...,l\}, co_{k,x} \ne 1 \bigg| co_{j,i} = 1 \land co_{j,x} \ne 1\]

where:

  • \(co_j\)​ is called the \(j\)-th congeneric order or \(j\)-th _row.
  • \(co_{j,i}\)​ is called the \(i\)-th element of \(j\)-th congeneric order.
  • \(l\) is length, \(l \in N\)
  • \(m\) is power, \(m \in N\)

from Order

Let \(O_p\) is Partial Order defined as function

\[O_p : \{1,...,l\} \longrightarrow \{1,...,l\} \cup \{-\}\]
\[alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}\]

Define

\[congenerics\_orders : \{O_p\} \longrightarrow \{CO\}\]
\[ congenerics\_orders(O_p)(j)(i) = \Bigg\{\begin{array}{l} 1 & O_p(i)=j \\ -, & O_p(i) \in \{-\} \lor O_p(i) \ne j \end{array} \]
\[\exists congenerics\_orders^{-1} : \{CO\} \longrightarrow \{S_p\},\]
\[ congenerics\_orders^{-1}(CO)(i) = \Bigg\{\begin{array}{l} j, & \exists j \in \{1,...,m\}, CS(j)(i) = 1 \\ -, & otherwise \end{array} \]

from Congeneric Sequences

Let \(CS\) is Congeneric Sequences - (m,l) matrix of \(X_{-}\)

\[ CS = \begin{pmatrix} cs_{1,1} & cs_{1,2} & \cdots & cs_{1,l} \\ cs_{2,1} & cs_{2,2} & \cdots & cs_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ cs_{m,1} & cs_{m,2} & \cdots & cs_{m,l} \end{pmatrix} \]

Define

\[congenerics\_orders : \{CS\} \longrightarrow \{CO\}\]
\[congenerics\_orders(CS)(j)(i) = \Bigg\{\begin{array}{l} 1, & CS(i) \notin \{-\} \\ -, & CS(i) \in \{-\} \end{array}\]
\[\exists congenerics\_orders^{-1} : \{CO\} \longrightarrow \{CS\},\]
\[congenerics\_orders^{-1}(CO)(i) = \Bigg\{\begin{array}{l} j, & \exists j \in \{1,...,m\}, CO(j)(i) = 1 \\ -, & otherwise \end{array}\]

Congeneric Orders product Alphabet

Let \(X_{-}\) is a Parial carrier set

Let \(A\) is a Aphabet \(A : \{1, ..., m\} \longrightarrow X,\)

Let \(alphabet_p\) is Alphabet of Partial Sequence function

\[alphabet_p : \big\{\{1,...,l\} \longrightarrow X_{-} \big\} \longrightarrow \big\{\{1,...,m\} \longrightarrow X \big\}\]

Let \(CS\) is a Congeneric Sequeneces

the following equations are true

\[CO = congenerics\_orders(CS),\]
\[A = alphabet_p(CS),\]
\[CS = ( CO \odot A ),\]
\[ CS(j)(i) = \Bigg\{\begin{array}{l} A(j), & \exists j \in \{1,...,m\}, CO(j)(i) = 1 \\ -, & otherwise \end{array} \]