Centralising Monoids with Low-Arity Witnesses on a Four-Element Set

As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed...

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Bibliographic Details
Published in:Symmetry (Basel) 2021-08, Vol.13 (8), p.1471
Main Authors: Behrisch, Mike, Vargas-García, Edith
Format: Article
Language:English
Subjects:
Algebra
Boolean
Boolean algebra
centraliser clone
centralising monoid
Cloning
commutation
endomorphism monoid
endoprimal monoid
Investigations
maximal centralising monoid
Monoids
Permutations
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Summary:As part of a project to identify all maximal centralising monoids on a four-element set, we determine all centralising monoids witnessed by unary or by idempotent binary operations on a four-element set. Moreover, we show that every centralising monoid on a set with at least four elements witnessed by the Mal’cev operation of a Boolean group operation is always a maximal centralising monoid, i.e., a co-atom below the full transformation monoid. On the other hand, we also prove that centralising monoids witnessed by certain types of permutations or retractive operations can never be maximal.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13081471