{"Package":"freealg","Type":"Package","Title":"The Free Algebra","Version":"1.1-8","Authors@R":"person(given=c(\"Robin\", \"K. S.\"), family=\"Hankin\", role = c(\"aut\",\"cre\"), email=\"hankin.robin@gmail.com\", comment = c(ORCID = \"0000-0001-5982-0415\"))","Maintainer":"Robin K. S. Hankin ","Depends":"R (>= 3.5.0), methods","Description":"The free algebra in R with non-commuting indeterminates.\nUses 'disordR' discipline\n(Hankin, 2022, ). To cite the\npackage in publications please use Hankin (2022)\n.","License":"GPL (>= 2)","LazyData":"yes","Imports":"Rcpp (>= 1.0-7), partitions (>= 1.9-22), disordR (>= 0.9-5-1)","LinkingTo":"Rcpp","Suggests":"knitr,testthat,magrittr,markdown,rmarkdown,covr","VignetteBuilder":"knitr","URL":"https://github.com/RobinHankin/freealg,\nhttps://robinhankin.github.io/freealg/","BugReports":"https://github.com/RobinHankin/freealg/issues","NeedsCompilation":"yes","Packaged":"2024-08-26 20:39:27 UTC; rhankin","Author":"Robin K. S. Hankin [aut, cre] ()","Repository":"CRAN","Date/Publication":"2024-08-26 23:30:02 UTC","repoType":"cran","tarballUrl":"ftp://cran.r-project.org/pub/R/src/contrib/freealg_1.1-8.tar.gz","jsonAuthors":[{"name":["Robin Hankin","K. S. Hankin"],"email":"hankin.robin@gmail.com","maintainer":true}],"readme":"The Free Algebra in R\n================\n\n\n\n

\n\n\n\n[![Total\nDownloads](https://cranlogs.r-pkg.org/badges/grand-total/freealg)](https://CRAN.R-project.org/package=freealg)\n[![CRAN_Status_Badge](https://www.r-pkg.org/badges/version/freealg)](https://cran.r-project.org/package=freealg)\n\n\n# Overview\n\nThe free algebra is an interesting and useful object. Here I present the\n`freealg` package which provides some functionality for free algebra in\nthe R programming environment. The package uses the `C++ map` class for\nefficiency and conforms to `disordR` discipline. Several use-cases are\nprovided.\n\n# Installation\n\nYou can install the released version of `freealg` from\n[CRAN](https://CRAN.R-project.org) with:\n\n``` r\n# install.packages(\"freealg\") # uncomment this to install the package\nlibrary(\"freealg\")\n```\n\n# The free algebra\n\nThe free algebra is the free R-module with a basis consisting of all\nwords over an alphabet of symbols with multiplication of words defined\nas concatenation. Thus, with an alphabet of\n![\\\\x,y,z\\\\](https://latex.codecogs.com/png.latex?%5C%7Bx%2Cy%2Cz%5C%7D \"\\{x,y,z\\}\")\nand\n\n![A=\\alpha x^2yx + \\beta zy](https://latex.codecogs.com/png.latex?A%3D%5Calpha%20x%5E2yx%20%2B%20%5Cbeta%20zy \"A=\\alpha x^2yx + \\beta zy\")\n\nand\n\n![B=\\gamma z + \\delta y^4](https://latex.codecogs.com/png.latex?B%3D%5Cgamma%20z%20%2B%20%5Cdelta%20y%5E4 \"B=\\gamma z + \\delta y^4\")\n\nwe would have\n\n![AB=\\left(\\alpha x^2yx+\\beta zy\\right)\\left(\\gamma z+\\delta y^4\\right)=\\alpha\\gamma x^2yxz+\\alpha\\delta x^2yxy^4+\\beta\\gamma zyz+\\beta\\delta zy^5](https://latex.codecogs.com/png.latex?AB%3D%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%5Cleft%28%5Cgamma%20z%2B%5Cdelta%20y%5E4%5Cright%29%3D%5Calpha%5Cgamma%20x%5E2yxz%2B%5Calpha%5Cdelta%20x%5E2yxy%5E4%2B%5Cbeta%5Cgamma%20zyz%2B%5Cbeta%5Cdelta%20zy%5E5 \"AB=\\left(\\alpha x^2yx+\\beta zy\\right)\\left(\\gamma z+\\delta y^4\\right)=\\alpha\\gamma x^2yxz+\\alpha\\delta x^2yxy^4+\\beta\\gamma zyz+\\beta\\delta zy^5\")\n\nand\n\n![BA=\\left(\\gamma z+\\delta y^4\\right)\\left(\\alpha x^2yx+\\beta zy\\right)=\\alpha\\gamma zx^2yx + \\alpha\\delta y^4 x^2yx + \\beta\\gamma z^2y + \\beta\\delta y^4zy.](https://latex.codecogs.com/png.latex?BA%3D%5Cleft%28%5Cgamma%20z%2B%5Cdelta%20y%5E4%5Cright%29%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%3D%5Calpha%5Cgamma%20zx%5E2yx%20%2B%20%5Calpha%5Cdelta%20y%5E4%20x%5E2yx%20%2B%20%5Cbeta%5Cgamma%20z%5E2y%20%2B%20%5Cbeta%5Cdelta%20y%5E4zy. \"BA=\\left(\\gamma z+\\delta y^4\\right)\\left(\\alpha x^2yx+\\beta zy\\right)=\\alpha\\gamma zx^2yx + \\alpha\\delta y^4 x^2yx + \\beta\\gamma z^2y + \\beta\\delta y^4zy.\")\n\nA natural and easily implemented extension is to use upper-case symbols\nto represent multiplicative inverses of the lower-case equivalents\n(formally we would use the presentation\n![xX=1](https://latex.codecogs.com/png.latex?xX%3D1 \"xX=1\")). Thus if\n\n![C=\\epsilon\\left(x^{-1}\\right)^2=\\epsilon X^2](https://latex.codecogs.com/png.latex?C%3D%5Cepsilon%5Cleft%28x%5E%7B-1%7D%5Cright%29%5E2%3D%5Cepsilon%20X%5E2 \"C=\\epsilon\\left(x^{-1}\\right)^2=\\epsilon X^2\")\n\nwe would have\n\n![AC=\\left(\\alpha x^2yx+\\beta zy\\right)\\epsilon X^2=\n\\alpha\\epsilon x^2yX + \\beta\\epsilon zyX^2](https://latex.codecogs.com/png.latex?AC%3D%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%5Cepsilon%20X%5E2%3D%0A%5Calpha%5Cepsilon%20x%5E2yX%20%2B%20%5Cbeta%5Cepsilon%20zyX%5E2 \"AC=\\left(\\alpha x^2yx+\\beta zy\\right)\\epsilon X^2=\n\\alpha\\epsilon x^2yX + \\beta\\epsilon zyX^2\")\n\nand\n\n![CA=\\epsilon X^2\\left(\\alpha x^2yx+\\beta zy\\right)=\n\\alpha\\epsilon yx + \\beta\\epsilon X^2zy.](https://latex.codecogs.com/png.latex?CA%3D%5Cepsilon%20X%5E2%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%3D%0A%5Calpha%5Cepsilon%20yx%20%2B%20%5Cbeta%5Cepsilon%20X%5E2zy. \"CA=\\epsilon X^2\\left(\\alpha x^2yx+\\beta zy\\right)=\n\\alpha\\epsilon yx + \\beta\\epsilon X^2zy.\")\n\nThe system inherits associativity from associativity of concatenation,\nand distributivity is assumed, but it is not commutative.\n\n# The `freealg` package in use\n\nCreating a free algebra object is straightforward. We can coerce from a\ncharacter string with natural idiom:\n\n``` r\nX <- as.freealg(\"1 + 3a + 5b + 5abba\")\nX\n#> free algebra element algebraically equal to\n#> + 1 + 3*a + 5*abba + 5*b\n```\n\nor use a more formal method:\n\n``` r\nfreealg(sapply(1:5,seq_len),1:5)\n#> free algebra element algebraically equal to\n#> + a + 2*ab + 3*abc + 4*abcd + 5*abcde\n```\n\n``` r\nY <- as.freealg(\"6 - 4a +2aaab\")\nX+Y\n#> free algebra element algebraically equal to\n#> + 7 - a + 2*aaab + 5*abba + 5*b\nX*Y\n#> free algebra element algebraically equal to\n#> + 6 + 14*a - 12*aa + 6*aaaab + 2*aaab + 30*abba - 20*abbaa + 10*abbaaaab + 30*b\n#> - 20*ba + 10*baaab\nX^2\n#> free algebra element algebraically equal to\n#> + 1 + 6*a + 9*aa + 15*aabba + 15*ab + 10*abba + 15*abbaa + 25*abbaabba +\n#> 25*abbab + 10*b + 15*ba + 25*babba + 25*bb\n```\n\nWe can demonstrate associativity (which is non-trivial):\n\n``` r\nset.seed(0)\n(x1 <- rfalg(inc=TRUE))\n#> free algebra element algebraically equal to\n#> + 7*C + 6*Ca + 4*B + 3*BC + a + 5*aCBB + 2*bc\n(x2 <- rfalg(inc=TRUE))\n#> free algebra element algebraically equal to\n#> + 6 + CAAA + 2*Ca + 3*Cbcb + 7*aaCA + 4*b + 5*c\n(x3 <- rfalg(inc=TRUE))\n#> free algebra element algebraically equal to\n#> + 3*C + 5*CbAc + BACB + 2*a + 10*b + 7*cb\n```\n\n(function `rfalg()` generates random `freealg` objects). Then\n\n``` r\nx1*(x2*x3) == (x1*x2)*x3\n#> [1] TRUE\n```\n\n# Further information\n\nFor more detail, see the package vignette\n\n`vignette(\"freealg\")`","jobInfo":{"package":"freealg","version":"1.1-8","parsingStatus":"success","parserVersion":2,"parsedAt":"2024-08-27T02:29:58+0000"}}

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