All 2-potent Elements in \(\mathit{Hyp}_G(2)\)

Apatsara Sareeto, Sorasak Leeratanavalee

Abstract


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 \(Hyp_{G}(2)\).


Keywords


Generalized hypersubstitution; \(m\)-potent elements; 2-potent elements

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References


G. Ayik, H. Ayik, Y. Ünlü 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.




DOI: http://dx.doi.org/10.26713%2Fcma.v11i2.1332

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