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Groups of negations on the unit square.
1Department of Mathematics, Shandong Normal University, Jinan 250014, China.
Thescientificworldjournal
|September 9, 2014
Summary
This study explores group structures within negations on the unit square, identifying specific groups of mappings and their relationships. Findings reveal normal subgroups and generator sets for these algebraic structures.
Area of Science:
- Lattice theory
- Abstract algebra
- Mathematical analysis
Background:
- The unit square and interval-valued sets are fundamental in various mathematical domains.
- Understanding algebraic structures like groups and subgroups is crucial for analyzing mappings and transformations.
Purpose of the Study:
- To investigate the group properties of negations and isomorphisms on the unit square considered as a bilattice.
- To identify and characterize specific groups formed by different types of mappings.
- To analyze the algebraic structure of interval-valued sets.
Main Methods:
- Utilizing concepts from group theory, specifically focusing on automorphisms, isomorphisms, and negations.
- Applying the operator 'composition' to define group operations.
- Examining the unit square as a bilattice and interval-valued sets as its subset.
Main Results:
- Proving that automorphisms on the unit square form a group.
- Identifying groups G₂, G₃, G₄, and G₅ based on monotonic isomorphisms and strict negations.
- Demonstrating that G₂, G₃, and G₄ are normal subgroups of G₅.
- Providing a generator set for G₅.
- Discovering two groups related to interval-valued sets, with one being a normal subgroup of the other.
- Identifying a generator set for the latter group.
Conclusions:
- The study establishes significant group structures within negations on the unit square and interval-valued sets.
- The identified groups and their relationships (normal subgroups, generator sets) offer insights into the algebraic properties of these mathematical objects.
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