# Activities and Participators

Related KMR pages:
Social Algebra

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Tänk och Känn: ///////

Community: $C = (A, P)$

Activities: $A = \{ A_1, A_2, \ldots, A_{n} \}$

Participators: $P = \{ P_1, P_2, \ldots, P_{m} \}$

Def: $P_{k} \in A_{i}$ if the participator $P_{k}$ takes part in the activity $A_{i}$.

Def: $A_{i} \in P_{k}$ if the activity $A_{i}$ includes the participator $P_{k}$.

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Definition of boolean factors:

$(P_{k} \in A_{i}) = 1 \qquad\text{if}\qquad P_{k} \in A_{i}$

$(P_{k} \in A_{i}) = 0 \qquad\text{if}\qquad P_{k} \not\in A_{i}$

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Hence:

The number of participators of an activity:

$|A_{i}|_P = \sum_{k=1}^m (P_{k} \in A_{i})$

The number of activities of a participator:

$|P_{k}|_A = \sum_{i=1}^n (P_{k} \in A_{i})$

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Def: The participators-of-activities list:

$\sum_{i=1}^n P_{A_{i}} A_{i}$

Def: The activities-of-participators list:

$\sum_{k=1}^m A_{P_{k}} P_{k}$

Def: The activity-person bilinear form:

$\sum_{i=1}^n\sum_{k=1}^m A_{i} P_{k} (P_{k} \in A_{i})$

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Definition of ordinal numbers:

$0 = \emptyset$

$1 = \{\emptyset \} \ = \{0\}$

$2 = \{0, 1\}$

$3 = \{0, 1, 2\}$

$\ldots$

$n = \{0, 1, \ldots, n-1\}$

$\ldots$

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The possible participator grouping polynomial
(of an activity $A_{i}$):

$\prod_{k=1}^m (1-P_{k})(P_{k} \in A_{i}) A_{i}$

The participator grouping coefficient
(of an activity $A_{i}$):

$G_{A_{i}} = \prod_{j=1}^m \prod_{s \in {\prod_{}^j}m} P_{s}(P_{s} \in A_{i})$

The participator grouping term
(of an activity $A_{i}$):

$G_{A_{i}} A_{i}$

The group-involvement polynomial
(of the activities of A):

$\sum_{i=1}^n G_{A_{i}} A_{i}$

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The activity merger polynomial of the organizational schemes $A$ and $A'$ with respect to the re-organization $A''$ :

$\sum_{i''=1}^{n''} ( \sum_{i=1}^n G_{A_{i}} A_{i})( \sum_{i'=1}^{n'} G_{A_{i'}} A_{i'}) A_{i''} = \sum_{i''=1}^{n''}\sum_{i=1}^n \sum_{i'=1}^{n'} G_{A_{i}} A_{i} G_{A_{i'}} A_{i'} R(A_{i}, A_{i'}, A_{i''}) A_{i''}$

The activity merger possibilities (= combinatorial combinations) of organizational schemes $A$ and $A'$ with respect to the new activity $A_{i''}$ of the re-organization $A''$:

$(\sum_{i=1}^n G_{A_{i}} A_{i})( \sum_{i'=1}^{n'} G_{A_{i'}} A_{i'}) = \sum_{i=1}^n \sum_{i'=1}^{n'} G_{A_{i}} A_{i} G_{A_{i'}} A_{i'}$

The activity merger condition of $A_{i}$ and $A_{i'}$ with respect to $A_{i''}$ :

$R(A_{i}, A_{i'}, A_{i''})$

The activity merger coefficient of $A_{i}$ and $A_{i'}$ with respect to $A_{i''}$ :

$\sum_{i=1}^n \sum_{i'=1}^{n'} G_{A_{i}} A_{i} G_{A_{i'}} A_{i'} R(A_{i}, A_{i'}, A_{i''})$

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Possible activity grouping polynomial (of a participator $P_{k}$):

$\prod_{i=1}^n (1-A_{i})(A_{i} \in P_{k}) P_{k}$

Activity grouping coefficient (of a participator $P_{k}$):

$G_{P_{k}} = \prod_{i=1}^n \prod_{s \in {\prod_{}^i}n} A_{s}(A_{s} \in P_{k})$

Activity grouping term (of a participator $P_{k}$):

$G_{P_{k}} P_{k}$

Activity grouping polynomial (of the participators of $\, P \,$):

$\sum_{k=1}^m G_{P_{k}} P_{k}$

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Cardinality formulas:

Total number of participators in the community $\, C \,$:

$|P| = \sum_{k=1}^n (-1)^{k-1} \sum_{s \in {\prod_{}^k}n}|\cap P_{A_{s}}| = m$

where

$\cap P_{A_{(1,2)}} = P_{A_1} \cap P_{A_2}$

Total number of activities in the community $\, C \,$:

$|A| = \sum_{k=1}^m (-1)^{k-1} \sum_{s \in {\prod_{}^k}m}|\cap A_{P_{s}}| = n$

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