alternative treatment of concatenation

It is possible to define words and concatenationMathworldPlanetmath in terms of ordered sets. Let A be a set, which we shall call our alphabet. Define a word on A to be a map from a totally ordered setMathworldPlanetmath into A. (In order to have words in the usual sense, the ordered set should be finite but, as the definition presented here does not require this condition, we do not impose it.)

Suppose that we have totally ordered sets (u,<) and (v,) and words f:uA and g:vA. Let uv denote the disjoint unionMathworldPlanetmath of u and v and let p:uuv and q:uuv be the canonical maps. Then we may define an order on uv as follows:

  • If xu and yu, then p(x)p(y) if and only if x<y.

  • If xu and yv, then p(x)q(y).

  • If xv and yv, then q(x)q(y) if and only if xy.

We define the concatenation of f and g, which will be denoted fg, to be map from uv to A defined by the following conditions:

  • If xu, then (fg)(p(x))=f(x).

  • If yu, then (fg)(q(x))=g(x).

Title alternative treatment of concatenation
Canonical name AlternativeTreatmentOfConcatenation
Date of creation 2013-03-22 17:24:10
Last modified on 2013-03-22 17:24:10
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Definition
Classification msc 68Q70
Classification msc 20M35
Related topic Word