the relation EQ  as defined by Randell  Cui and Cohn and analogously by Egenhofer becomes identity under the unique name assumption  Geometry and and Franzosa topology  conversely  allow  other kinds of equivalence relations  These relations are weaker than the identity  namely the equivalence classes they induce are larger than singletons The simplest weakening we can define is the congruence relation  Even though  the relation has been studied  Borgo  Guarino   
Masolo Cristani Mutinelli the complexity of reasoning in subalgebras has not yet been deeply investigated The reason to introduce this new relation  and to provide an algebraic structure to host relations based on it  in particular relations in which we compare regions  which  even if they are not congruent  can be overlapped by roto translating one into the interior of the other is that in many cases the knowledge we have to represent in our systems necessarily includes internal structures Consider the following example

## Example 1

A GIS is dedicated to the representation of geographical data about industrial sites In the system the structure of factories can be described by means of various attributes including size and shape One of the users wants to query the system about the opportunity of moving the factory  where he works  from the present site to a new one  In the new site prefabricated facilities already exist  and the problem consists of deciding how to preserve topological layout of the factory and minimise the costs of buying new engines to substitute those which cannot fit with the new facilities In order to reason about these problems the system can be made able to represent the qualitative relations between the old and new site parts  establishing  in particular whether the spatial regions occupied by the engines are  congruent"  or even  congruent to a part" of the ones which can be chosen for hosting them in the new factory s site Clearly we cannot use topological information  since a spatial region presently occupied by an engine is surely disjoint from the region that this engine is going to occupy in the new site  but this information is insufficient for deciding whether the new site is able to host the engine itself Though metric information can be involved in final decision about moving the factory qualitative reasoning can be used in the initial design phase In Figure we give a pictorial representation of one possible situation in which to make decisions we need to represent spatial information which is not topological
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The algebra MC can be used to represent the four basic relations which can be built on the equivalence relation of congruence   if we consider the roto translation of a region x with respect to a region y  only four possible situations arise
- There is at least one roto translation in which x coincides with y The two regions are congruent

- There is at least one roto translation in which x coincides with a proper part of y The region x is congruent to a proper part of y