The "natural" language of graphs in model theory fails when we consider multigraphs

 1. Preliminaries

   Briefly speaking, a language consists of a set of constant symbols, a set of relations and a set of function symbols. Moreover, each relation or function symbol is associated with a natural number indicating the (fixed) number of members in that relation/function. In other words, a relation/function in a language is forced to involve a fixed number of members. A real-life example of a such relation is the (normal) marriage relation, which must be between exactly two people. The formal definition of a language is as follows. 



The formal definition of a language (taken from the book
"Model Theory: An Introduction" written by David Marker)

   Based on a given language L, we can construct an L-structure, which is defined as follows.  

   If there exists a bijective mapping n from an L-structure M to an L-structure N such that the following holds

   then we say that M and N are L-isomorphic. 

 2. The "natural" language of graphs in model theory and its failure when we want to talk about multigraphs

   In various textbooks in model theory, the language of graph only consists of a binary relation R. For every two elements/vertices x, y, R(x, y) means that x is adjacent to y. To characterize simple graphs, the following axioms can be used: 
   1) For all x, R(x, x) is not true. This indicates that the graph is loopless. 
   2) For all x, y, R(x, y) if and only if R(y, x). This indicates that adjacency is mutual. 
   The relation R only indicates whether two vertices are adjacent or not. It gives no information about the number of edges between two adjacent vertices. In other words, in this language, a multigraph and its underlying graph is the same (although they might be graph-theoretically non-isomorphic) since they are L-isomorphic, where L is the "natural" language of graphs that only consists of a binary relation R. 
   Instead of using only a single binary relation R, we might introduce the binary relation symbols R_0, R_1, ..., R_n if we only consider graphs in which every pair of vertices are connected by at most n edges. The following axioms are sufficient: 
   i) For all x, y, the composition of R_0(x, y), R_1(x, y), R_2(x, y), ..., R_n(x, y) is true
  ii) If R_0(x, y) is true, then the composition of R_1(x, y), R_2(x, y), ..., R_n(x, y) is false
 iii) If R_1(x, y) is true, then the composition of R_0(x, y), R_2(x, y), ..., R_n(x, y) is false
  ...
   What we are doing is to introduce additional symbols to the language. The symbol R in the old language corresponds to the symbol R_1 in the new language. The story is similar when we add operations to a given set of numbers. There are, of course, cases in which two initially isomorphic structures become non-isomorphic when additional operations are added. For example, the additive group of the field F_8 with eight elements and the additive group of the ring Z_2 x Z_2 x Z_2 are isomorphic, but F_8 and Z_2 x Z_2 x Z_2 are not isomorphic rings since F_8 is a field while Z_2 x Z_2 x Z_2 is not. 

   

   



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