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The order of generalized hypersubstitutions of type \(\tau =(2)\). (English) Zbl 1159.08002

Summary: The order of hypersubstitutions, all idempotent elements on the monoid of all hypersubstitutions of type \(\tau =(2)\) were studied by K. Denecke and Sh. L. Wismath and all idempotent elements on the monoid of all hypersubstitutions of type \(\tau =(2,2)\) were studied by Th. Changpas and K. Denecke. We want to study similar problems for the monoid of all generalized hypersubstitutions of type \(\tau =(2)\). In this paper, we use similar methods to characterize idempotent generalized hypersubstitutions of type \(\tau =(2)\) and determine the order of each generalized hypersubstitution of this type. The main result is that the order is 1, 2 or infinite.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
08B15 Lattices of varieties
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References:

[1] S. Leeratanavalee and K. Denecke, “Generalized hypersubstitutions and strongly solid varieties,” in General Algebra and Applications, Proceedings of the “59th Workshop on General Algebra,” “15th Conference for Young Algebraists Potsdam 2000”, pp. 135-146, Shaker, Aachen, Germany, 2000.
[2] K. Denecke and S. L. Wismath, “The monoid of hypersubstitutions of type (2),” in Contributions to General Algebra, 10 (Klagenfurt, 1997), pp. 109-126, Heyn, Klagenfurt, Austria, 1998. · Zbl 1080.20503
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