primitive: Primitive elements of the free algebra

In RobinHankin/freegroup: The Free Group

Primitive elements of the free algebra

Description

A primitive word is one that is not of the form a^m for any m>1. The concept is used in Lyndon and Schutzenberger 1962.

Usage

is.primitive(x)
is.power(d,n)

Arguments

x	Freegroup object, coerced to Tietze form
d	Numeric vector
n	Integer

Details

Function is.primitive() returns a boolean vector indicating whether the elements of its argument are primitive.

Function is.power() is a lower-level helper function. is.power(d,n) determines whether d is an n-th power (that is, d may be written as n copies of some numeric vector).

Thus is_power(c(4,5,7,4,5,7,4,5,7,4,5,7),4) returns TRUE because its primary argument is indeed a fourth power (of c(4,5,7)).

Value

Returns a boolean.

Author(s)

Robin K. S. Hankin. The code for finding the factors of an integer was (somewhat more than) inspired by the numbers package.

References

R. C. Lyndon and M. P. Schutzenberger 1962. “The equation a^M=b^Nc^P in a free group”. Michigan Mathematical Journal, 9(4): 289–298.

Examples

is.primitive(as.free(c("a","aaaa", "aaaab", "aabaab", "aabcaabcaabcaabc")))

is.power(c(7,8,4,7,8,4,7,8,4,7,8,4),4)


table(is.primitive(rfree(100)))


## primitive with >1 symbols is rare:
x <- rfree(1000)
x[!is.primitive(x) & number(x)>1]
##  try x <- rfree(10000) [but this takes a long time]

RobinHankin/freegroup documentation built on April 12, 2025, 11:46 p.m.