Notes on symmetries
| Abstract | It is helpful to begin with an abstract characterisation of symmetries. A structure consists of: a set, D, of objects together with a set, R = {Ri}i∈I, of relations defined upon D (no restrictions are placed on the cardinality of D or on that of the index set I). If (D, {Ri}i∈I) and (D′, {R′i}i∈I) are structures, then we say that a map φ : D → D′ fixes the n-ary relation Ri if: Ri(x1, . . . , xn) iff R′i(φ(x1), . . . , φ(xn)), for every n-tuple of objects in D. The automorphisms of the structure (D, {Ri}i∈I) are bijections φ : D → D that fix each Ri ∈ R. The set of automorphisms forms a group under composition of functions. | |||||||||
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