Update 01 Background_and_Definitions.adoc

D01 Integration of INSEL Models and SimStadt/01 Background_and_Definitions.adoc

@@ -24,8 +24,9 @@ The C part is necessary to connect the workflow step to an existing workflow in
 In order to control the workflow in more detail, the C part can be configured e.q. to select the used level of detail in a multi-resolution building modeln or to exclude buildings with a small volume or unknown building function  from the workflow (filter). This configuration has to be implemented in each workflow environment seperatly. 
 
 Dependencies:  
+
 * The C part executes the functional part. The C part depends on the F part, but the F part has to be independend of the C part  
-* The C part shall have no functionality that is relevant for the purpose of the WFS. It’s purpose to to connect the functional part to a workflow in a given workflow environment.  
+* The C part shall have no functionality that is relevant for the purpose of the WFS. It’s purpose is to connect the functional part to a workflow in a given workflow environment.  
 
 
 An example of a SD_connector is given in the SimStadt documentation. The example shows an empty workflow step without any functionality. Examples of a WS_connectors are provided in the SimStadt documentation on redmine.
@@ -38,14 +39,12 @@ Math: The SimStadt data model is a heterogeneous algebra.
 
 Birkhoff and Lipson (1968) have defined a heterogeneous algebra as a system A=[Σ,F] in which
 
-	Σ={S_i} is a family  of non-void sets S_i of different types of elements, each called a phylum of the algebra A. The phyla S_i are indexed by some set I; i.e. S_i∈Σ for i∈I (or are called by appropriate names).
-	F={f_α} is a set of finitary operations, where each f_α is a mapping
-
-f_α:S_(i(1,α))×S_(i(2,α))×⋯ ×S_(i(n(α),α))→S_(r(α))
+1. Σ={S~i~} is a family  of non-void sets S~i~ of different types of elements, each called a *phylum* of the algebra A. The *phyla* S~i~ are indexed by some set I; i.e. S~i~∈Σ for i∈I (or are called by appropriate names).  
+2. F={f~α~} is a set of finitary operations, where each f~α~ is a mapping f~α~:S~i(1,α)~×S~i(2,α)~×⋯ ×S~i(n(α),α)~→S~r(α)~  
 
-for some non-negative interger n(α), function i_α:k→i(k,α) from {1,2,⋯,n(α)} to I, and r(α)∈I. The operations f_α  are indexed by some set Ω; i.e. f_α∈F for α∈Ω (or are called by appropriate names).
+for some non-negative interger n(α), function i~α~:k→i(k,α) from {1,2,⋯,n(α)} to I, and r(α)∈I. The operations f~α~  are indexed by some set Ω; i.e. f~α~∈F for α∈Ω (or are called by appropriate names).
 
-Thus each operation f_α assigns to each n(α)-tuple (x_1,⋯,x_n(α) ), where  x_j∈S_(i(j,α)), some value  f_α (x_1,⋯,x_n(α) ) in S_(r(α)). The operation f_α  is said to be n(α)-ary: unary when n(α)=1, binary when n(α)=2, ternary when n(α)=3, etc. When n(α)=0, it selects a fixed element (distinguished constant) of  S_(r(α)).
+Thus each operation f~α~ assigns to each n(α)-tuple (x~1~,⋯,x~n~(α) ), where  x~j~∈S_(i(j,α)), some value  f~α~ (x~1~,⋯,x~n(α)~ ) in S~r(α)~. The operation f_α  is said to be n(α)-ary: unary when n(α)=1, binary when n(α)=2, ternary when n(α)=3, etc. When n(α)=0, it selects a fixed element (distinguished constant) of  S_(r(α)).
 
 An algebra which has only one Phylum will be called a homogeneous algebra; an algebra having more than one phylum, a heterogeous algebra.