Classification of All Associative Mono-n-ary Algebras with 2 Elements

Classification of All Associative Mono-n-ary Algebras with 2 Elements

We consider algebras with a single 𝑛-ary operation and a certain type of associativity. We prove that (up to isomorphism) there are exactly 5 of these associative mono-𝑛-ary algebras with 2 elements for even 𝑛≥2 and 6 for odd 𝑛≥3. These algebras are described explicitly. It is shown that a similar r...

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Main Author: Stephan Dominique Andres
Format: Article
Language:English
Published: Hindawi Limited 2009-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Online Access:http://dx.doi.org/10.1155/2009/678987
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spelling doaj-975ba42e7f1043a0931f5b9e31de27142020-11-24T22:47:40ZengHindawi LimitedInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252009-01-01200910.1155/2009/678987678987Classification of All Associative Mono-n-ary Algebras with 2 ElementsStephan Dominique Andres0Department of Mathematics and Computer Science, Fern Universität in Hagen, Lützowstraße 125, 58084 Hagen, GermanyWe consider algebras with a single 𝑛-ary operation and a certain type of associativity. We prove that (up to isomorphism) there are exactly 5 of these associative mono-𝑛-ary algebras with 2 elements for even 𝑛≥2 and 6 for odd 𝑛≥3. These algebras are described explicitly. It is shown that a similar result is impossible for algebras with at least 4 elements. An application concerning the assignment of a control bit to a string is given.http://dx.doi.org/10.1155/2009/678987
collection DOAJ
language English
format Article
sources DOAJ
author Stephan Dominique Andres
spellingShingle Stephan Dominique Andres
Classification of All Associative Mono-n-ary Algebras with 2 Elements
International Journal of Mathematics and Mathematical Sciences
author_facet Stephan Dominique Andres
author_sort Stephan Dominique Andres
title Classification of All Associative Mono-n-ary Algebras with 2 Elements
title_short Classification of All Associative Mono-n-ary Algebras with 2 Elements
title_full Classification of All Associative Mono-n-ary Algebras with 2 Elements
title_fullStr Classification of All Associative Mono-n-ary Algebras with 2 Elements
title_full_unstemmed Classification of All Associative Mono-n-ary Algebras with 2 Elements
title_sort classification of all associative mono-n-ary algebras with 2 elements
publisher Hindawi Limited
series International Journal of Mathematics and Mathematical Sciences
issn 0161-1712
1687-0425
publishDate 2009-01-01
description We consider algebras with a single 𝑛-ary operation and a certain type of associativity. We prove that (up to isomorphism) there are exactly 5 of these associative mono-𝑛-ary algebras with 2 elements for even 𝑛≥2 and 6 for odd 𝑛≥3. These algebras are described explicitly. It is shown that a similar result is impossible for algebras with at least 4 elements. An application concerning the assignment of a control bit to a string is given.
url http://dx.doi.org/10.1155/2009/678987
work_keys_str_mv AT stephandominiqueandres classificationofallassociativemononaryalgebraswith2elements
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