{"Package":"freealg","Type":"Package","Title":"The Free Algebra","Version":"1.0-0","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 ","Description":"The free algebra in R; multivariate polynomials with non-commuting indeterminates.","License":"GPL (>= 2)","Imports":"Rcpp (>= 0.12.3)","LinkingTo":"Rcpp","SystemRequirements":"C++11","Suggests":"knitr,testthat","VignetteBuilder":"knitr","URL":"https://github.com/RobinHankin/freealg.git","BugReports":"https://github.com/RobinHankin/freealg/issues","NeedsCompilation":"yes","Packaged":"2019-09-15 20:52:47 UTC; rhankin","Author":"Robin K. S. Hankin [aut, cre] ()","Repository":"CRAN","Date/Publication":"2019-09-23 14:50:02 UTC","repoType":"cran","tarballUrl":"ftp://cran.r-project.org/pub/R/src/contrib/freealg_1.0-0.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[![Build\nStatus](https://travis-ci.org/RobinHankin/freealg.svg?branch=master)](https://travis-ci.org/RobinHankin/freealg)\n[![CRAN\\_Status\\_Badge](https://www.r-pkg.org/badges/version/freealg)](https://cran.r-project.org/package=freealg)\n[![Rdoc](http://www.rdocumentation.org/badges/version/freealg)](http://www.rdocumentation.org/packages/freealg)\n[![Codecov test\ncoverage](https://codecov.io/gh/RobinHankin/freealg/branch/master/graph/badge.svg)](https://codecov.io/gh/RobinHankin/freealg/branch/master)\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.\n\nThe package uses `C++`’s STL `map` class for efficiency, which has the\ndownside that the order of the terms is undefined. This does not matter\nas the mathematical value is unaffected by reordering; and the print\nmethod does a good job in producing human-readable output.\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\n\"\\\\{x,y,z\\\\}\") and\n\n \n![\nA=\\\\alpha x^2yx + \\\\beta zy\n](https://latex.codecogs.com/png.latex?%0AA%3D%5Calpha%20x%5E2yx%20%2B%20%5Cbeta%20zy%0A\n\"\nA=\\\\alpha x^2yx + \\\\beta zy\n\") \n\nand\n\n \n![\nB=\\\\gamma z + \\\\delta y^4\n](https://latex.codecogs.com/png.latex?%0AB%3D%5Cgamma%20z%20%2B%20%5Cdelta%20y%5E4%0A\n\"\nB=\\\\gamma z + \\\\delta y^4\n\") \n\nwe would have\n\n \n![\nA\\\\cdot B=\\\\left(\\\\alpha x^2yx+\\\\beta zy\\\\right)\\\\cdot\\\\left(\\\\gamma\nz+\\\\delta y^4\\\\right)=\\\\alpha\\\\gamma x^2yxz+\\\\alpha\\\\delta\nx^2yxy^4+\\\\beta\\\\gamma zyz+\\\\beta\\\\delta zy^5\n](https://latex.codecogs.com/png.latex?%0AA%5Ccdot%20B%3D%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%5Ccdot%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%0A\n\"\nA\\\\cdot B=\\\\left(\\\\alpha x^2yx+\\\\beta zy\\\\right)\\\\cdot\\\\left(\\\\gamma z+\\\\delta y^4\\\\right)=\\\\alpha\\\\gamma x^2yxz+\\\\alpha\\\\delta x^2yxy^4+\\\\beta\\\\gamma zyz+\\\\beta\\\\delta zy^5\n\") \n\nNote that multiplication is not commutative, but it is associative. A\nnatural and easily implemented extension is to use upper-case symbols to\nrepresent multiplicative inverses of the lower-case equivalents. Thus if\n\n \n![\nC=\\\\epsilon\nX^2](https://latex.codecogs.com/png.latex?%0AC%3D%5Cepsilon%20X%5E2 \"\nC=\\\\epsilon X^2\") \n\nwe would have\n\n \n![\nA\\\\cdot C=\\\\left(\\\\alpha x^2yx+\\\\beta zy\\\\right)\\\\cdot\\\\epsilon X^2=\n\\\\alpha\\\\epsilon x^2yX + \\\\beta\\\\epsilon zyX^2\n](https://latex.codecogs.com/png.latex?%0AA%5Ccdot%20C%3D%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%5Ccdot%5Cepsilon%20X%5E2%3D%0A%5Calpha%5Cepsilon%20x%5E2yX%20%2B%20%5Cbeta%5Cepsilon%20zyX%5E2%0A\n\"\nA\\\\cdot C=\\\\left(\\\\alpha x^2yx+\\\\beta zy\\\\right)\\\\cdot\\\\epsilon X^2=\n\\\\alpha\\\\epsilon x^2yX + \\\\beta\\\\epsilon zyX^2\n\") \n\nand\n\n \n![\nC\\\\cdot A=\\\\epsilon X^2\\\\cdot\\\\left(\\\\alpha x^2yx+\\\\beta zy\\\\right)=\n\\\\alpha\\\\epsilon yx + \\\\beta\\\\epsilon X^2zy.\n](https://latex.codecogs.com/png.latex?%0AC%5Ccdot%20A%3D%5Cepsilon%20X%5E2%5Ccdot%5Cleft%28%5Calpha%20x%5E2yx%2B%5Cbeta%20zy%5Cright%29%3D%0A%5Calpha%5Cepsilon%20yx%20%2B%20%5Cbeta%5Cepsilon%20X%5E2zy.%0A\n\"\nC\\\\cdot A=\\\\epsilon X^2\\\\cdot\\\\left(\\\\alpha x^2yx+\\\\beta zy\\\\right)=\n\\\\alpha\\\\epsilon yx + \\\\beta\\\\epsilon X^2zy.\n\") \n\nThe system inherits power associativity from distributivity and\nassociativity of concatenation, but 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#> + 1*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 - 1*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 - 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 + 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 + 1*a + 5*aCBB + 2*bc\n(x2 <- rfalg(inc=TRUE))\n#> free algebra element algebraically equal to\n#> + 6 + 1*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 + 1*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":"1.0-0","parsingStatus":"success","parserVersion":1,"parsedAt":"2021-04-19T07:29:58+0000"}}

Description

Usage

Arguments

Details

See Also

Examples