### Regularity of Linear Hypersubstitutions for Algebraic Systems of Type $$((n),(m))$$

Thodsaporn Kumduang, Sorasak Leeratanavalee

#### Abstract

An algebraic system consisting a nonempty set together with a sequence of operations and a sequence of relations on this set. The properties of this structure are expressed by terms and formulas. In this paper we study on linear terms of type $$(n)$$ for a natural number $n\geq1$ and linear formulas of type $$((n),(m))$$ for natural numbers $$n,m\geq 1$$. Using the partial clone of linear terms and the partial clone of linear formulas, we define the new concept of linear hypersubstitutions for algebraic systems of type $$((n),(m))$$ and proved that the set of all linear hypersubstitutions for algebraic systems of type $$((n),(m))$$ with a binary operation on this set and the identity element forms a monoid. Finally, we also interest in studying the semigroup or monoid properties of its. In particular, we investigate the idempotency and regularity of linear hypersubstitutions for algebraic systems of this monoid.

#### Keywords

Algebraic systems; Linear terms; Linear formulas; Linear hypersubstitutions; Regular elements

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#### References

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DOI: http://dx.doi.org/10.26713%2Fcma.v10i1.1098

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