reduce

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Reduction of a word to reduced form

Given a word, remove redundant zero-power terms, and consolidate adjacent like terms into a single power

Usage
reduce(a)
is_reduced(a)
remove_zero_powers(a)
consolidate(a)
is_proper(a)
Arguments
a

An object of class free

Details

A word is reduced if no symbol appears next to its own inverse and no symbol has zero power. The essence of the package is to reduce a word into a reduced form. Thus \(a^2b^{-1}ba\) will transformed into \(a^3\).

In the package, reduction happens automatically at creation, in function free().

Apart from is_proper(), the functions all take a free object, but the meat of the function operates on a single two-row matrix.

Reduction is carried out by repeatedly consolidating adjacent terms of identical symbol (function consolidate()), and removing zero power terms (function remove_zero_power()) until the word is in reduced form (function is_reduced()).

Function is_proper() checks to see whether a matrix is suitably formed for passing to reduce().

A free object is cyclically reduced iff every cyclic permutation of the word is reduced. A reduced word is cyclically reduced iff the first letter is not the inverse of the last one. A reduced word is cyclically reduced if the first and last symbol differ (irrespective of power) or, if identical, have powers of opposite sign. For example, abac and abca are cyclically reduced but abca^{-1} is not. Function is.cyclically.reduced() tests for this, documented at cycred.Rd.

Whether the identity should be regarded as cyclically reduced is problematic. On the one hand the identity cannot be shortened by a combination of cyclic permutation followed by reduction; but on the other, I cannot exhibit a symbol at the start of the identity which can be reduced by juxtaposition with a symbol at the end (because there are no symbols). Currently it returns NA but I am open to suggestions.

See Also

cycred

Aliases
  • reduce
  • remove_zero_powers
  • is_reduced
  • consolidate
  • is_proper
Examples
# NOT RUN {
## create a matrix:
M <- rbind(c(1,2,3,3,2,3,2,1),c(1,2,3,-3,5,0,7,0))

## call the print method (note non-reduced form):
as.character_free(M)

## show the effect of reduce():
as.character_free(reduce(M))

## free() calls reduce() automatically:
free(M)
# }
Documentation reproduced from package freegroup, version 1.1-0, License: GPL-2

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