20Nxx Other generalizations of groups
This subtopic studies other generalizations of groups, including quasigroups, loops, and related systems that weaken group axioms.
Specific topics
20N02 Sets with a single binary operation (groupoids)
Overview
20N02 studies sets with a single binary operation (groupoids) in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for sets with a single binary operation (groupoids)
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks
20N05 Loops, quasigroups
Overview
20N05 studies loops, quasigroups in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for loops, quasigroups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks
20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
Overview
20N10 studies ternary systems (heaps, semiheaps, heapoids, etc.) in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for ternary systems (heaps, semiheaps, heapoids, etc.)
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks
20N15 $n$-ary systems
Overview
20N15 studies $n$-ary systems in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for $n$-ary systems
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks
20N20 Hypergroups
Overview
20N20 studies hypergroups in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for hypergroups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks
20N25 Fuzzy groups
Overview
20N25 studies fuzzy groups in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for fuzzy groups
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks
20N99 None of the above
Overview
20N99 studies none of the above in other generalizations of groups. It studies algebraic systems generalizing groups, including quasigroups, loops, hypergroups, and multiary operations.
Related Wikipedia Page
Other Generalizations Of Groups (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for none of the above
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in combinatorics, algebraic design, and the study of non-group symmetry systems.
Applications
- Quasigroups and loops
- Multiary and hypergroup structures
- Generalized binary-operation systems
References
Recommended Textbooks