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Mathematical systems, like the relations <math>L\!</math> and operational structures <math>\underline{X}\!</math> encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems. Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures. This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations. Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.
Mathematical systems, like the relations <math>L\!</math> and operational structures <math>\underline{X}\!</math> encountered above, are seldom comprehended in perfect isolation, but need to be viewed in relation to each other, as belonging to families of comparable systems. Systems are compared by finding or making correspondences between them, and this can be formalized as a task of setting up and probing various types of mappings between the sundry appearances of their objective structures. This requires techniques for exploring the spaces of mappings that exist between families of systems, for inquiring into and demonstrating the existence of specified types of functions between them, plus technical concepts for classifying and comparing their diverse representations. Therefore, in order to compare the structures of different objective systems and to recognize the same objective structure when it appears in different phenomenal or syntactic disguises, it helps to develop general forms of comparison that can organize the welter of possible associations between systems and single out those that represent a preservation of the designated forms.
The next series of definitions develops the mathematical concepts of ''homomorphism'' and ''isomorphism'', special types of mappings between systems that serve to formalize the intuitive notions of structural analogy and abstract identity, respectively. In very rough terms, a ''homomorphism'' is a ''structure-preserving mapping'' between systems, but only in the sense that it preserves some part of some aspect of the structure mapped, whereas an ''isomorphism'' is a correspondence that preserves all of the relevant structure.
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The "induced action" of a function f : X >Y on the cartesian power Xn is the function f' : Xn >Yn defined by:
The "induced action" of a function f : X >Y on the cartesian power Xn is the function f' : Xn >Yn defined by:
