Transformation (combinatorics)
Encyclopedia
In combinatorial
mathematics, the notion of transformation is used with several slightly different meanings. Informally, a transformation of a set of N values is an arrangement of those values into a particular order, where values may be repeated, but the ordered list is N elements in length. Thus, there are 27 transformations of the set {1,2,3}, namely [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1],[1,3,2], [1,3,3], [2,1,1], [2,1,2],[2,1,3],[2,2,1], [2,2,2], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,1], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2], and [3,3,3]. In general there are NN transformations for a set of N elements.
Analogous to a permutation group
having elements that are permutation
s, a transformation semigroup
has elements that are transformations. For N > 1, the set of permutations on N values is a proper subset of the set of transformations on N values.
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
mathematics, the notion of transformation is used with several slightly different meanings. Informally, a transformation of a set of N values is an arrangement of those values into a particular order, where values may be repeated, but the ordered list is N elements in length. Thus, there are 27 transformations of the set {1,2,3}, namely [1,1,1], [1,1,2], [1,1,3], [1,2,1], [1,2,2], [1,2,3], [1,3,1],[1,3,2], [1,3,3], [2,1,1], [2,1,2],[2,1,3],[2,2,1], [2,2,2], [2,2,3], [2,3,1], [2,3,2], [2,3,3], [3,1,1], [3,1,2], [3,1,3], [3,2,1], [3,2,2], [3,2,3], [3,3,1], [3,3,2], and [3,3,3]. In general there are NN transformations for a set of N elements.
Analogous to a permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
having elements that are permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s, a transformation semigroup
Transformation semigroup
In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup is associated with the composite of the two corresponding...
has elements that are transformations. For N > 1, the set of permutations on N values is a proper subset of the set of transformations on N values.