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The Monoid of Hypersubstitutions of Type (n)

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A hypersubstitution of type (n) is a map σ which takes the n-ary operation symbol f to an n-ary term σ (f). Any such σ can be inductively extended to a map ς on the set of all terms of type (n), and any two such extensions can be composed in a natural way. Thus, the set Hyp(n) of all hypersubstitutions of type (n) forms a monoid. For n = 2, many properties of this monoid were described by Denecke and Wismath [5]. In this paper, we study the semigroup properties of Hyp(n) for arbitrary n ≥ 2. In particular, we characterize the projection, dual and idempotent hypersubstitutions, and describe the classes of these elements under Green’s relations.

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Correspondence to S. L. Wismath.

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AMS Subject Classification (1991): 08B15 20M07

Research supported by NSERC, Canada

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Wismath, S.L. The Monoid of Hypersubstitutions of Type (n). SEA bull. math. 24, 115–128 (2000). https://doi.org/10.1007/s10012-000-0115-5

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  • hypersubstitution
  • monoid
  • idempotent