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 <i>n</i>, this paper introduces the algebra of order-preserving full terms under the <inline-formula><math xmlns="http://www.w3.org/19...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-04-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/9/1433 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | Full terms, which serve as tools for classifying algebras into subclasses, can be studied using an algebraic approach. For a natural number <i>n</i>, this paper introduces the algebra of order-preserving full terms under the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-superposition operation satisfying the superassociativity using mappings on the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>O</mi><mi>n</mi></msub></semantics></math></inline-formula> of all order-preserving transformations on a finite chain <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>1</mn><mo>≤</mo><mo>⋯</mo><mo>≤</mo><mspace width="4pt"></mspace><mi>n</mi><mo>}</mo></mrow></semantics></math></inline-formula>. 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. |
|---|---|
| ISSN: | 2227-7390 |