All 2-potent Elements in \(\mathit{Hyp}_G(2)\)
DOI:
https://doi.org/10.26713/cma.v11i2.1332Keywords:
Generalized hypersubstitution, \(m\)-potent elements, 2-potent elementsAbstract
A generalized hypersubstitution of type \(\tau = (2)\) is a function which takes the binary operation symbol \(f\) to the term \(\sigma(f)\) which does not necessarily preserve the arity. Let \(Hyp_{G}(2)\) be the set of all these generalized hypersubstitutions of type \((2)\). The set \(Hyp_{G}(2)\) with a binary operation and the identity generalized hypersubstitution forms a monoid. The index and period of an element \(a\) of a finite semigroup are the smallest values of \(m\geq1\) and \(r\geq1\) such that \(a^{m+r}=a^m\). An element with the index \(m\) and period 1 is called an $m$-potent element. In this paper we determine all \(2\)-potent elements in \(Hyp_{G}(2)\).
Downloads
References
G. Ayik, H. Ayik, Y. íœnlü and J.M. Howie, The structure of elements in finite full transformation semigroups, Bulletin of the Australian Mathematical Society 71(1) (2005), 69 – 74, DOI: 10.1017/S0004972700038028.
K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, Hyperequational Classes, and Clone Congruences, Contributions to General Algebra 7, Verlag Hölder-Pichler-Tempsky, Wien (1991), pp. 97 – 118, http://www.math.tu-dresden.de/~poeschel/poePUBLICATIONSpdf/Hyperidentities.pdf.
S. Leeratanavalee, Universal Algebra (in Thai), Jarus Business Printing, Chiang Mai, Thailand (2017).
K.D. Denecke and S. Leeratanavalee, Generalized hypersubstitutions and strongly colid varieties, in General Algebra and Applications, Proceedings of the 59th Workshop on General Algebra, 15th Conference for Young Algebraists Potsdam 2000/ Hrsg.: Klaus Denecke; Hans-Jürgen Vogel. - Aachen: Shaker, 2000, pp. 135 – 146, https://publishup.uni-potsdam.de/opus4-ubp/frontdoor/index/index/docId/19978.
W. Puninagool, Monoids of Generalized Hypersubstitutions of Type (pi=(n)), Doctor's Thesis, The Graduate School, Chiang Mai University (2010).
W. Puninagool and S. Leeratanavalee, The order of generalized hypersubstitutions of type (pi=(2)), International Journal of Mathematics and Mathematics Sciences 2008 (2008), Article ID 263541, 8 pages, DOI: 10.1155/2008/263541.
P. Zhao, T. You and H. Hu, On the m-potent ranks of certain semigroups of orientation preserving transformations, Bulletin of the Korean Mathematical Society 51 (2014), 1841 – 1850, DOI: 10.4134/BKMS.2014.51.6.1841.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
