Generalized arithmetic progression

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In mathematics, a multiple arithmetic progression, generalized arithmetic progression, k-dimensional arithmetic progression or a linear set, is a set of integers or tuples of integers constructed as an arithmetic progression is, but allowing several possible differences. So, for example, we start at 17 and may add a multiple of 3 or of 5, repeatedly. In algebraic terms we look at integers

where and so on are fixed, and and so on are confined to some ranges

and so on, for a finite progression. The number  , that is the number of permissible differences, is called the dimension of the generalized progression.

More generally, let

be the set of all elements in of the form

with in , in , and in . is said to be a linear set if consists of exactly one element, and is finite.

A subset of is said to be semilinear if it is a finite union of linear sets. The semilinear sets are exactly the sets definable in Presburger arithmetic.[1]

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References[edit]

  1. ^ Ginsburg, Seymour; Spanier, Edwin Henry (1966). "Semigroups, Presburger Formulas, and Languages". Pacific Journal of Mathematics 16: 285–296.