On lexicographic representatives in braid monoids
The language of maximal lexicographic representatives of elements in the positive braid monoid A n with n generators is a regular language. We describe with great detail the smallest finite-state automaton accepting such language and study the proportion of elements of length k whose maximal lexicog...
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| Veröffentlicht in: | Journal of algebraic combinatorics 2020-12, Vol.52 (4), p.561-597 |
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| container_title | Journal of algebraic combinatorics |
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| creator | Flores, Ramón González-Meneses, Juan |
| description | The language of maximal lexicographic representatives of elements in the positive braid monoid
A
n
with
n
generators is a regular language. We describe with great detail the smallest finite-state automaton accepting such language and study the proportion of elements of length
k
whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number
P
n
,
1
, as
k
tends to infinity, and we show that
P
n
,
1
≥
3
16
=
0.1875
for every
n
≥
1
. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton. Finally, we present the pseudocode of an algorithm which computes the adjacency matrix of the finite-state automaton. |
| doi_str_mv | 10.1007/s10801-019-00913-7 |
| format | Article |
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A
n
with
n
generators is a regular language. We describe with great detail the smallest finite-state automaton accepting such language and study the proportion of elements of length
k
whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number
P
n
,
1
, as
k
tends to infinity, and we show that
P
n
,
1
≥
3
16
=
0.1875
for every
n
≥
1
. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton. Finally, we present the pseudocode of an algorithm which computes the adjacency matrix of the finite-state automaton.</description><identifier>ISSN: 0925-9899</identifier><identifier>EISSN: 1572-9192</identifier><identifier>DOI: 10.1007/s10801-019-00913-7</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithms ; Braiding ; Combinatorics ; Computer Science ; Convex and Discrete Geometry ; Fibonacci numbers ; Finishes ; Group Theory and Generalizations ; Lattices ; Mathematics ; Mathematics and Statistics ; Monoids ; Order ; Ordered Algebraic Structures</subject><ispartof>Journal of algebraic combinatorics, 2020-12, Vol.52 (4), p.561-597</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2019</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2019.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-46b12f1ecc2ddb1aea8fbde0b47e651cfaa1b770502725ae118c49e23badf78b3</cites><orcidid>0000-0002-2520-2755 ; 0000-0002-4315-9957</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10801-019-00913-7$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10801-019-00913-7$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Flores, Ramón</creatorcontrib><creatorcontrib>González-Meneses, Juan</creatorcontrib><title>On lexicographic representatives in braid monoids</title><title>Journal of algebraic combinatorics</title><addtitle>J Algebr Comb</addtitle><description>The language of maximal lexicographic representatives of elements in the positive braid monoid
A
n
with
n
generators is a regular language. We describe with great detail the smallest finite-state automaton accepting such language and study the proportion of elements of length
k
whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number
P
n
,
1
, as
k
tends to infinity, and we show that
P
n
,
1
≥
3
16
=
0.1875
for every
n
≥
1
. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton. Finally, we present the pseudocode of an algorithm which computes the adjacency matrix of the finite-state automaton.</description><subject>Algorithms</subject><subject>Braiding</subject><subject>Combinatorics</subject><subject>Computer Science</subject><subject>Convex and Discrete Geometry</subject><subject>Fibonacci numbers</subject><subject>Finishes</subject><subject>Group Theory and Generalizations</subject><subject>Lattices</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Monoids</subject><subject>Order</subject><subject>Ordered Algebraic Structures</subject><issn>0925-9899</issn><issn>1572-9192</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LxDAQR4MoWFe_gKeC5-hM2jTNURb_wcJe9BySdLpm2W1r0hX99lYrePM0l_d-A4-xS4RrBFA3CaEG5ICaA2gsuDpiGUoluEYtjlkGWkiua61P2VlKW5ioGmXGcN3lO_oIvt9EO7wGn0caIiXqRjuGd0p56HIXbWjyfd_1oUnn7KS1u0QXv3fBXu7vnpePfLV-eFrerrgvsBx5WTkULZL3omkcWrJ16xoCVyqqJPrWWnRKgQShhLSEWPtSkyicbVpVu2LBrubdIfZvB0qj2faH2E0vjSglyrJSBUyUmCkf-5QitWaIYW_jp0Ew32nMnMZMacxPGqMmqZilNMHdhuLf9D_WF8TSZvg</recordid><startdate>20201201</startdate><enddate>20201201</enddate><creator>Flores, Ramón</creator><creator>González-Meneses, Juan</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-2520-2755</orcidid><orcidid>https://orcid.org/0000-0002-4315-9957</orcidid></search><sort><creationdate>20201201</creationdate><title>On lexicographic representatives in braid monoids</title><author>Flores, Ramón ; González-Meneses, Juan</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-46b12f1ecc2ddb1aea8fbde0b47e651cfaa1b770502725ae118c49e23badf78b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algorithms</topic><topic>Braiding</topic><topic>Combinatorics</topic><topic>Computer Science</topic><topic>Convex and Discrete Geometry</topic><topic>Fibonacci numbers</topic><topic>Finishes</topic><topic>Group Theory and Generalizations</topic><topic>Lattices</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Monoids</topic><topic>Order</topic><topic>Ordered Algebraic Structures</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Flores, Ramón</creatorcontrib><creatorcontrib>González-Meneses, Juan</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of algebraic combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Flores, Ramón</au><au>González-Meneses, Juan</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On lexicographic representatives in braid monoids</atitle><jtitle>Journal of algebraic combinatorics</jtitle><stitle>J Algebr Comb</stitle><date>2020-12-01</date><risdate>2020</risdate><volume>52</volume><issue>4</issue><spage>561</spage><epage>597</epage><pages>561-597</pages><issn>0925-9899</issn><eissn>1572-9192</eissn><abstract>The language of maximal lexicographic representatives of elements in the positive braid monoid
A
n
with
n
generators is a regular language. We describe with great detail the smallest finite-state automaton accepting such language and study the proportion of elements of length
k
whose maximal lexicographic representative finishes with the first generator. This proportion tends to some number
P
n
,
1
, as
k
tends to infinity, and we show that
P
n
,
1
≥
3
16
=
0.1875
for every
n
≥
1
. We also provide an explicit formula, based on the Fibonacci numbers, for the number of states of the automaton. Finally, we present the pseudocode of an algorithm which computes the adjacency matrix of the finite-state automaton.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10801-019-00913-7</doi><tpages>37</tpages><orcidid>https://orcid.org/0000-0002-2520-2755</orcidid><orcidid>https://orcid.org/0000-0002-4315-9957</orcidid><oa>free_for_read</oa></addata></record> |
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| issn | 0925-9899 1572-9192 |
| language | eng |
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| source | SpringerNature Journals |
| subjects | Algorithms Braiding Combinatorics Computer Science Convex and Discrete Geometry Fibonacci numbers Finishes Group Theory and Generalizations Lattices Mathematics Mathematics and Statistics Monoids Order Ordered Algebraic Structures |
| title | On lexicographic representatives in braid monoids |
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