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Related KMR pages:
• Business Algebra
• Social Algebra

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Tänk och Känn:

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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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