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2. Domain Entities 2.3. Parts2.3.2. Properties2.3.2.2. Mereology93Concrete Models of MereologyThe concrete mereology example models above illustrated maps and sequences assuch models.• In general we can model mereologies in terms of– (i) sets: A-set,– (ii) Cartesians: A 1 ×A 2 ×...×A m ,– (iii) lists: A ∗ , and– (iv) maps: A → m B,where A, A 1 , A 2 ,...,A m and B are types [we assume that they are type names] andwhere the A 1 , A 2 ,...,A m type names need not be distinct.• Additional concrete types, say D, can be defined by concrete type definitions,D=E, where E is either of the type expressions (i–iv) given above or (v) E i |E j , or(vi) (E i ). where E k (for suitable k) are either of (i–vi).• Finally it may be necessary to express well-formedness predicates for concretelymodelled mereologies.A Precursor for Requirements Engineering 93 c○ Dines Bjørner 2012, DTU Informatics, Techn.Univ.of Denmark – July 20, 2012: 12:33
942. Domain Entities 2.3. Parts2.3.2. Properties2.3.2.2. MereologyAbstract Models of MereologyAbstractly modelling mereology of parts, to us, means the following.• With part types P 1 , P 2 , . . . , P n– is associated the unique part identifier types, Π 1 , Π 2 , . . . , Π n ,– that is uid Π i : P i →Π i for i ∈ {1..n},• and with each part type, P i ,– is then associated a mereology observer,– mereo P i : P i → Π j -set×Π k -set×...×Π l -set,• such that for all p:Pi we have that– if mereo P i (p) = ({..., π ja , ...},{..., π kb , ...},...,{..., π lc , ...})– for i,j,k,...l ∈ {1..n}– then part p:P i is connected (related) to the parts identified by..., π ja , ... π kb , ..., π lc , ....• Finally it may be necessary to express axioms for abstractly modelled mereologies.c○ Dines Bjørner 2012, DTU Informatics, Techn.Univ.of Denmark – July 20, 2012: 12:33 94 Domain Science & Engineering
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942. Domain Entities 2.3. Parts2.3.2. Properties2.3.2.2. MereologyAbstract Models of MereologyAbstractly modelling mereology of parts, to us, means the following.• With part types P 1 , P 2 , . . . , P n– is associated the unique part identifier types, Π 1 , Π 2 , . . . , Π n ,– that is uid Π i : P i →Π i for i ∈ {1..n},• and with each part type, P i ,– is then associated a mereology observer,– mereo P i : P i → Π j -set×Π k -set×...×Π l -set,• such that for all p:Pi we have that– if mereo P i (p) = ({..., π ja , ...},{..., π kb , ...},...,{..., π lc , ...})– for i,j,k,...l ∈ {1..n}– then part p:P i is connected (related) to the parts identified by..., π ja , ... π kb , ..., π lc , ....• Finally it may be necessary to express axioms for abstractly modelled mereologies.c○ Dines Bjørner 2012, <strong>DTU</strong> <strong>Informatics</strong>, Techn.Univ.of Denmark – July 20, 2012: 12:33 94 Domain Science & Engineering
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