Clones of inductive superpositions of terms

Pongsakorn Kitpratyakul; Bundit Pibaljommee; Pongsakorn Kitpratyakul; Bundit Pibaljommee

doi:10.3934/math.2023389

AIMS Mathematics

2023, Volume 8, Issue 4: 7747-7765. doi: 10.3934/math.2023389

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Research article

Clones of inductive superpositions of terms

Department of Mathematics, Faculty of Science Khon Kaen University, Khon Kaen 40002, Thailand

Received: 22 September 2022 Revised: 27 December 2022 Accepted: 12 January 2023 Published: 18 January 2023
MSC : 08A35, 08A40, 08A70, 08B20

A superposition is an operation of terms by which we substitute each variable within a term with other forms of terms. With more options of terms to be replaced, an inductive superposition is apparently more general than the superposition. This comes with a downside that it does not satisfy the superassociative property on the set of all terms of a given type while the superposition does. A derived base set of terms on which the inductive superposition is superassociative is given in this paper. A clone-like algebraic structure involving such base set and superposition is the main topic of this paper. Generating systems of the clone-like algebra are characterized and it turns out that the algebra is only free with respect to itself under the certain selections of fixed terms concerning its inductive superposition or the specific type of its base set.
- term,
- inductive composition of terms,
- inductive superposition of terms,
- clone of terms,
- unitary Menger algebra of rank $ n $
Citation: Pongsakorn Kitpratyakul, Bundit Pibaljommee. Clones of inductive superpositions of terms[J]. AIMS Mathematics, 2023, 8(4): 7747-7765. doi: 10.3934/math.2023389

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Abstract

A superposition is an operation of terms by which we substitute each variable within a term with other forms of terms. With more options of terms to be replaced, an inductive superposition is apparently more general than the superposition. This comes with a downside that it does not satisfy the superassociative property on the set of all terms of a given type while the superposition does. A derived base set of terms on which the inductive superposition is superassociative is given in this paper. A clone-like algebraic structure involving such base set and superposition is the main topic of this paper. Generating systems of the clone-like algebra are characterized and it turns out that the algebra is only free with respect to itself under the certain selections of fixed terms concerning its inductive superposition or the specific type of its base set.

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