Antiassociative Algebra

Methods to deal with the free antiassociative algebra over the reals with an arbitrary number of indeterminates. Antiassociativity means that (xy)z = -x(yz). Antiassociative algebras are nilpotent with nilindex four (Remm, 2022, <doi:10.48550/arXiv.2202.10812>) and this drives the design and philosophy of the package. Methods are defined to create and manipulate arbitrary elements of the antiassociative algebra, and to extract and replace coefficients. A vignette is provided.

Details

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Functionality to work with the free antiassociative algebra in R. The hex sticker features an image taken from Hoffnung (1959) in which musical concepts [pizzicato, crescendo, etc] are given whimsical visual form. The character on the hex sticker is captioned “A Discord”: Hoffnung's interpretation of the musical concept of dissonance. In the book, the preceding image was a “chord”, evoking harmony. The discord, on the other hand, embodies–for me at least–antiassociativity: everything is wrong, wrong, wrong.

Author

Robin K. S. Hankin [aut, cre] (ORCID: <https://orcid.org/0000-0001-5982-0415>)

Maintainer: Robin K. S. Hankin <hankin.robin@gmail.com>

References

Hoffnung G (1959). Hoffnung's Acoustics. Dobson.

See also

aaa

Examples

x <- raaa()
x
#> free antiassociative algebra element:
#> +3b +5d +2c.d +3d.d +1(b.a)b +3(b.d)d +1(c.a)d
y <- raaa()

x+y
#> free antiassociative algebra element:
#> +3a +6b +7d +1b.b +2b.c +2c.d +2d.c +3d.d +4(a.c)c +1(b.a)b +2(b.d)c +3(b.d)d
#> +1(c.a)d +2(d.b)c
x*y
#> free antiassociative algebra element:
#> +9b.a +9b.b +6b.d +15d.a +15d.b +10d.d -3(b.b)b -6(b.b)c -6(b.d)c +6(c.d)a
#> +6(c.d)b +4(c.d)d -5(d.b)b -10(d.b)c +9(d.d)a +9(d.d)b -10(d.d)c +6(d.d)d