Centralising Monoids with Low-Arity Witnesses on a Four-Element Set
Abstract
:1. Introduction
2. Preliminaries
2.1. Notation and Basic Concepts
2.2. Fundamental Results on Clones, Centralisers and Centralising Monoids
- (a)
- M is a centralising monoid, i.e., there is such that .
- (b)
- , i.e., M is witnessed by .
- (I)
- and is a permutation of prime order or a retractive operation ;
- (II)
- is idempotent, i.e., ;
- (III)
- is a ternary (minority) Mal’cev operation arising as for from a Boolean group ;
- (IV)
- is a ternary majority operation;
- (V)
- is a proper semiprojection of arity n where .
- (a)
- F is a clone of idempotent operations, i.e., .
- (b)
- For every and all we have .
- (c)
- .
- (a)
- .
- (b)
- .
- (c)
- .
- (B)
- for all pairwise distinct there is such that ;
- (D)
- for all pairwise distinct there is such that .
- (B’)
- for all pairwise distinct there is such that ,
- (D’)
- for all pairwise distinct there is such that and ,
- (B”)
- for all pairwise distinct there is such that or or ,
- (D”)
- for all pairwise distinct there is satisfying one of the conditions or or ,
3. Monoids Witnessed by Unary Operations
3.1. Computational Results for
3.2. Monoids Witnessed by Permutations
- (a)
- , and
- (b)
- for all the value is arbitrary, but for all .
3.3. Monoids Witnessed by Retractive Operations
3.4. Results
- and is a transposition of just two elements,
- is even and is a product of disjoint transpositions without fixed points,
- is any (non-identical) retraction,
- and is a transposition of just two elements,
- and has a single fixed point,
- and is a product of two disjoint transpositions (without fixed points),
- f is a retraction with for some ,
4. Monoids Witnessed by Binary Idempotent Operations
5. Monoids Witnessed by Mal’cev Operations of Boolean Groups
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Appendix A.1
Appendix A.2
Appendix B
Appendix C
References
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Behrisch, M.; Vargas-García, E. Centralising Monoids with Low-Arity Witnesses on a Four-Element Set. Symmetry 2021, 13, 1471. https://doi.org/10.3390/sym13081471
Behrisch M, Vargas-García E. Centralising Monoids with Low-Arity Witnesses on a Four-Element Set. Symmetry. 2021; 13(8):1471. https://doi.org/10.3390/sym13081471
Chicago/Turabian StyleBehrisch, Mike, and Edith Vargas-García. 2021. "Centralising Monoids with Low-Arity Witnesses on a Four-Element Set" Symmetry 13, no. 8: 1471. https://doi.org/10.3390/sym13081471
APA StyleBehrisch, M., & Vargas-García, E. (2021). Centralising Monoids with Low-Arity Witnesses on a Four-Element Set. Symmetry, 13(8), 1471. https://doi.org/10.3390/sym13081471