Free Algebras of Full Terms Generated by Order-Preserving Transformations

Full terms, which serve as tools for classifying algebras into subclasses, can be studied using an algebraic approach. For a natural number n, this paper introduces the algebra of order-preserving full terms under the $(n+1)$-superposition operation satisfying the superassociativity using mappings on the set $O_n$ of all order-preserving transformations on a finite chain $\{1\le \cdots \le n\}$. We prove the freeness property of such algebra with respect to the variety of superassociative algebras. Additionally, binary operations on the powerset of order-preserving full terms whose elements are called tree languages are discussed. To define order-preserving identities and order-preserving varieties, the left-seminearring of full hypersubstitutions is determined. The required characteristics for any identity to be an order-preserving identity are considered. Furthermore, we also discuss the homomorphism of full hypersubstitutions with other algebraic structures.