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Congeneric Intervals Chains

Congeneric Intervals Chains is a stack of a compatible congeric interval chain that represents Intervals Chain or Partial Intervals Chain. The congeneric intervals chains are sorted in the stack by the first position with a non-empty element in it.

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Mathematical Definition

Let \(IC_{c}\) is Congeneric Interval Chain

Define a Congeric intervals chains

as m-tuple of congeneric intervals chain

Let \(compatible(IC1_c, IC2_c)\) is compatibility of Partial Sequences function

\(CIC\) is m-tuple of \(IC_c\) intervals chain

\[CIC = <cic_1, cic_2, ..., cic_m>,\]
\[\forall j \in \{1, ..., m\} \exists cic_j \in \{IC_c\}, \]

all congeneric intervals chains are compatible

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

and sorted by first apperance of non-empty element

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

where:

  • \(cic_j\)​ is called the \(j\)-th congeneric intervals chain.
  • \(l := |CIC(1)|\) is length, \(l \in N\)
  • \(m := |CIC|\) is power, \(m \in N\)

as matrix (m,l)

\(CIC\) is (m,l) matrix of \(\{1,..,, l\} \cup \{-\}\)

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

columns are compatible

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

and sorted by first apperance of non-empty element

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

where:

  • \(\Delta_j\)​ is called the \(j\)-th congeneric intervals chain or \(j\)-th row.
  • \(\Delta_{j,i}\)​ is called the \(i\)-th element of \(j\)-th congeneric intervals chain.
  • \(l\) is length, \(l \in N\)
  • \(m\) is power, \(m \in N\)

from Congeneric Sequences

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

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

Let \(Binging_p\) is Partial Binding

Let \(Intervals_p\) is Partial Intervals function described as function

\[Intervals_p : \big\{Binding_p\big\} \times \big\{S_p\} \longrightarrow \big\{ IC_p \big\}\]
\[ CS = \begin{pmatrix} cs_{1,1} & cs_{1,2} & \cdots & cs_{1,l} \\ cs_{2,1} & cs_{2,2} & \cdots & cs_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ cs_{m,1} & cs_{m,2} & \cdots & cs_{m,l} \end{pmatrix} \]

Define

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

from Congeneric Orders

Let \(CO\) is Congeneric Orders - (m,l) matrix of \(\{1,...,l\} \cup \{-\}\)

Let \(Binging_p\) is Partial Binding

Let \(Intervals_p\) is Partial Intervals function described as function

\[Intervals_p : \big\{Binding_p\big\} \times \big\{S_p\} \longrightarrow \big\{ IC_p \big\}\]
\[ CO = \begin{pmatrix} co_{1,1} & co_{1,2} & \cdots & co_{1,l} \\ co_{2,1} & co_{2,2} & \cdots & co_{2,l} \\ \vdots & \vdots & \ddots & \vdots \\ co_{m,1} & co_{m,2} & \cdots & co_{m,l} \end{pmatrix} \]

Define

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