Published in Volume XXXIII, Issue 1, 2023, pages 79-92, doi: 10.7561/SACS.2023.1.79

Authors: H. Machida

Abstract

A centralizing monoid M is a set of unary operations which commute with some set F of operations. Here, F is called a witness of M . On a 3-element set, a centralizing monoid is maximal if and only if it has a constant operation or a majority minimal operation as its witness.

In this paper, we take one such majority operation, which corresponds to a maximal centralizing monoid, on a 3-element set and obtain its generalization, called mb , on a k-element set for any k ≥ 3. We explicitly describe the centralizing monoid M(mb ) with mb as its witness and then prove that it is not maximal if k > 3, contrary to the case for k = 3.

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References

[1] B. Csákány. All minimal clones on the three-element set. Acta Cybernetica, 6(3):227–238, 1983.

[2] Hajime Machida. Search for some majority operation and studies of its centralizing monoid. In 53rd IEEE International Symposium on MultipleValued Logic, ISMVL 2023. IEEE Computer Society, 2023 (accepted, to appear).

[3] Hajime Machida and Ivo G. Rosenberg. Maximal centralizing monoids and their relation to minimal clones. In Jaakko Astola and Radomir S. Stankovic, editors, 41st IEEE International Symposium on Multiple-Valued Logic, ISMVL 2011, pages 153–159. IEEE Computer Society, 2011. doi:10.1109/ISMVL.2011.36.

[4] Hajime Machida and Ivo G. Rosenberg. Maximal centralizing monoids in connection with minimal clones. 2022 (submitted).

[5] Tamás Waldhauser. Minimal clones generated by majority operations. Algebra Universalis, 44:15–26, 2000. doi:10.1007/s000120050167.

Bibtex

@article{sacscuza:machida23ascmmow,
  title={A Study on Centralizing Monoids with Majority Operation Witnesses},
  author={Hajime Machida},
  journal={Scientific Annals of Computer Science},
  volume={33},
  number={1},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  year={2023},
  pages={79-92},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},
  doi={10.7561/SACS.2023.1.79}
}