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The monoid of generalized hypersubstitutions of type τ = (n)

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A (usual) hypersubstitution of type τ is a function which takes each operation symbol of the type to a term of the type, of the same arity. The set of all hypersubstitutions of a fixed type τ forms a monoid under composition, and semigroup properties of this monoid have been studied by a number of authors. In particular, idempotent and regular elements, and the Green's relations, have been studied for type (n) by S.L. Wismath.
A generalized hypersubstitution of type τ=(n) is a mapping σ which takes the n-ary operation symbol f to a term σ(f) which does not necessarily preserve the arity. Any such σ can be inductively extended to a map σ̂ on the set of all terms of type τ=(n), and any two such extensions can be composed in a natural way. Thus, the set $Hyp_{G}(n)$ of all generalized hypersubstitutions of type τ=(n) forms a monoid. In this paper we study the semigroup properties of $Hyp_{G}(n)$. In particular, we characterize the idempotent and regular generalized hypersubstitutions, and describe some classes under Green's relations of this monoid.
  • Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
  • Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
  • [1] K. Denecke, D. Lau, R. Pöschel and D. Schweigert, Hyperidentities, Hyperequational Classes, and Clone Congruences, Verlag Hölder-Pichler-Tempsky, Wien, Contributions to General Algebra 7 (1991), 97-118.
  • [2] S. Leeratanavalee and K. Denecke, Generalized Hypersubstitutions and Strongly Solid Varieties, p. 135-145 in: General Algebra and Applications, Proc. of the '59 th Workshop on General Algebra', '15 th Conference for Young Algebraists Potsdam 2000', Shaker Verlag 2000.
  • [3] S. Leeratanavalee, Submonoids of Generalized Hypersubstitutions, Demonstratio Mathematica XL (1) (2007), 13-22.
  • [4] W. Puninagool and S. Leeratanavalee, All Regular Elements in $Hyp_{G](2)$, preprint 2009.
  • [5] W. Puninagool and S. Leeratanavalee, Green's Relations on $Hyp_{G](2)$, preprint 2009.
  • [6] W. Puninagool and S. Leeratanavalee, The Order of Generalized Hypersubstitutions of Type τ =(2), International Journal of Mathematics and Mathematical Sciences, Vol 2008 (2008), Article ID 263541, 8 pages. doi: 10.1155/2008/263541
  • [7] W. Taylor, Hyperidentities and Hypervarieties, Aequationes Mathematicae 23 (1981), 111-127.
  • [8] S.L. Wismath, The monoid of hypersubstitutions of type (n), Southeast Asian Bull. Math. 24 (1) (2000), 115-128.
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