On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations
Abstract
:1. Introduction
2. Hypercompositional Structures with Inverted Associativity
- i.
- Ifis a LAC-hypergroup and , thenis a weak join left almost-hypergroup.
- ii.
- Ifis a LA-hypergroup,for alland , thenis a weak join left almost-hypergroup.
3. Enumeration and Structure Results
- The 65,955 non-trivial left almost-hypergroups are partitioned in 11,067 isomorphism classes. 10,920 of them consist of 6 members, 142 have 3 members, 4 have 2 members and the last one is a one-member class. The same are valid for the 65,955 non-trivial right almost-hypergroups.
- The 7036 both left and right almost-hypergroups are partitioned in 1174 isomorphism classes. 1172 of them consist of 6 members, while the other 2 have 2 members.
- The 16,044 noncommutative structures which are simultaneously left almost-hypergroups, right almost-hypergroups and hypergroups are partitioned in 2733 isomorphism classes. 2617 of them consist of 6 members, 110 have 3 members and the last 6 have 2 members.
4. Algebraic Properties
- i.
- and, for allin
- ii.
- and, for allin
- iii.
- the non-empty result of the induced hypercompositions is equivalent to the reproductive axiom.
- i.
- and
- ii.
- i.
- and
- ii.
- i.
- ii.
- iii.
- i.
- ii.
- iii.
- i.
- and
- ii.
- .
- i.
- and
- ii.
- .
- i.
- In any left almost-hypergroup the following property is valid:
- ii.
- In any right almost-hypergroup the following property is valid:
- i.
- If are non-empty subsets of a left almost-hypergroup, then:
- ii.
- Ifare non-empty subsets of a right almost-hypergroup, then:
- i.
- In any left almost-hypergroup the right inverted associativity of the induced hypercompositions is valid:
- ii.
- In any right almost-hypergroup the left inverted associativity of the induced hypercompositions is valid:
- i.
- If are non-empty subsets of a left almost-hypergroup, then:
- ii.
- Ifare non-empty subsets of a right almost-hypergroup, then:
5. Identities and Symmetric Elements
- i.
- and , for all
- ii.
- , for all
- i.
- for all
- ii.
- for all
- i.
- ifthenfor all
- ii.
- ifthenfor all
6. Substructures of the Left/Right Almost-Hypergroups
- i.
- and
- ii.
- .
- i.
- is a right closed sub-LA-hypergroup of, if and only if, for every
- ii.
- is a left closed sub-LA-hypergroup of, if and only if, for every
- iii.
- is a closed sub-LA-hypergroup of, if and only ifand, for every
- i.
- A sub-LA-hypergroup ofis right closed in H, if and only if
- ii.
- A sub-LA-hypergroup ofis left closed in H, if and only if
- iii.
- A sub-LA-hypergroup ofis closed in H, if and only if
- i.
- Ifis a sub-LA-hypergroup of H and
- ii.
- Ifis a sub-LA-hypergroup of H and
- i.
- If K is a right closed sub-LA-hypergroup in H,and, then
- ii.
- If K is a left closed sub-LA-hypergroup in H,and, then
- i.
- If K is a right closed sub-LA-hypergroup in H,and, then
- ii.
- If K is a left closed sub-LA-hypergroup in H,and, then
- i.
- is a right invertible sub-LA-hypergroup of, if and only if:
- ii.
- is a left invertible sub-LA-hypergroup of, if and only if:
7. Fortification in Transposition Left Almost-Hypergroups
- i.
- is a left identity and
- ii.
- for everythere exists a uniquesuch thatand
- i
- If, then.
- ii
- .
- i.
- and, if x is attractive
- ii.
- , if e is left strong identity and x is attractive
- iii.
- , if x is non attractive.
- i.
- and, if X contains an attractive element
- ii.
- , if e is left strong identity and X contains an attractive element
- iii.
- , if X consists of non-attractive elements.
- i.
- Ifis a right attractive element of a transposition polysymmetrical left almost-hypergroup, then all the elements ofare right attractive.
- ii.
- Ifis a left attractive element of a transposition polysymmetrical left almost-hypergroup, then all the elements ofare left attractive.
8. Conclusions and Open Problems
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A
BeginPackage["LRHtest`"];
Clear["LRHtest`*"];
LRHtest::usage = "LRHtest[groupoid] returns
1: for Left Asso
2: for Right Asso
3: for Left+Right Asso
4: for Hypergroup
5: for Hypergroup + Left Asso
6: for Hypergroup + Right Asso
7: for Hypergroup + Left+Right Asso"
Begin["`Private`"];
Clear["LRHtest`Private`*"];
LRHtest[groupoid_List] := Module[{r}, r = 0;
If[groupoid != Transpose[groupoid] && ReproductivityTest[groupoid],
If[LeftAs[groupoid], r = 1];
If[RightAs[groupoid], r = r + 2];
If[Asso[groupoid], r = r + 4]]; Return[r]];
LeftAs[groupoid_List] :=
Not[MemberQ[
Flatten[Table[
Union[Flatten[
Union[Extract[groupoid,
Distribute[{groupoid[[i, j]], {k}}, List]]]]] ==
Union[Flatten[
Union[Extract[groupoid,
Distribute[{groupoid[[k, j]], {i}}, List]]]]], {i, 1,
Length[groupoid]}, {j, 1, Length[groupoid]}, {k, 1,
Length[groupoid]}], 2], False]];
RightAs[groupoid_List] :=
Not[MemberQ[
Flatten[Table[
Union[Flatten[
Union[Extract[groupoid,
Distribute[{{i}, groupoid[[j, k]]}, List]]]]] ==
Union[Flatten[
Union[Extract[groupoid,
Distribute[{{k}, groupoid[[j, i]]}, List]]]]], {i, 1,
Length[groupoid]}, {j, 1, Length[groupoid]}, {k, 1,
Length[groupoid]}], 2], False]];
Asso[groupoid1_List] :=
Not[MemberQ[
Flatten[Table[
Union[Flatten[
Union[Extract[groupoid1,
Distribute[{groupoid1[[i, j]], {k}}, List]]]]] ==
Union[Flatten[
Union[Extract[groupoid1,
Distribute[{{i}, groupoid1[[j, k]]}, List]]]]], {i, 1,
Length[groupoid1]}, {j, 1, Length[groupoid1]}, {k, 1,
Length[groupoid1]}], 2], False]];
ReproductivityTest[groupoid_List] :=
Min[Table[
Length[Union[Flatten[Transpose[groupoid][[j]]]]], {j, 1,
Length[groupoid]}]] == Length[groupoid] &&
Min[Table[
Length[Union[Flatten[groupoid[[j]]]]], {j, 1,
Length[groupoid]}]] == Length[groupoid];
End[];
EndPackage[];
Use of the package:
for checking a magma, for instance the following one:
{1} {1} {2,3}
{1} {1} {2,3}
{1,2,3} {2,3} {1,3}
write in Mathematica:
In[1]:=LRHtest[{{{1}, {1}, {2, 3}}, {{1}, {1}, {2, 3}}, {{1, 2, 3}, {2,3}, {1, 3}}}]
And the output is:
Out[1]=2
where number 2 corresponds to «RA-hypergroup»
|
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◦ | 1 | 2 | 3 |
---|---|---|---|
1 | 1 | 2 | 3 |
2 | 3 | 1 | 2 |
3 | 2 | 3 | 1 |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | 1 | 3 | 2 |
2 | 2 | 1 | 3 |
3 | 3 | 2 | 1 |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1} | {1,2,3} |
2 | {1} | {1} | {2,3} |
3 | {2,3} | {2,3} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {2} | {3} | {1,3} |
2 | {1,2} | {1,3} | {1,2,3} |
3 | {1,3} | {1,2,3} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1,2} | {1,3} |
2 | {1,3} | {1,2,3} | {3} |
3 | {1,2} | {2} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1,3} | {1,2} |
2 | {1,2} | {1,2,3} | {2} |
3 | {1,3} | {3} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1,2,3} | {1,2,3} | {1,2,3} |
2 | {1,3} | {1,2} | {1,3} |
3 | {2,3} | {2,3} | {1} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1,2} | {1,2,3} | {1,3} |
2 | {1,2} | {1,3} | {1,2,3} |
3 | {1,2,3} | {1,2} | {1,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {2,3} | {1,2,3} | {1,3} |
2 | {1,2} | {1,3} | {1,2,3} |
3 | {1,3} | {2,3} | {1,2} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1,3} | {1,2} | {1,2,3} |
2 | {1,2,3} | {1,3} | {1,2} |
3 | {1,3} | {1,2} | {2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | a11 | a12 | a13 |
2 | a21 | a22 | a23 |
3 | a31 | a32 | a33 |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1,2} | {1,2,3} |
2 | {1,2} | {3} | {1,3} |
3 | {1,2,3} | {1,2,3} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1} | {1,2,3} |
2 | {1} | {1} | {2,3} |
3 | {1,2,3} | {1,2,3} | {1,2,3} |
Total Number | Isomorphism Classes | Classes with 1 Member (Rigid) | Classes with 2 Members | Classes with 3 Members | Classes with 4 or 5 Members | Classes with 6 Members | |
---|---|---|---|---|---|---|---|
non-trivial left almost-hypergroups | 65,955 | 11,067 | 1 | 4 | 142 | 0 | 10,920 |
non-trivial right almost-hypergroups | 65,955 | 11,067 | 1 | 4 | 142 | 0 | 10,920 |
non-trivial both left and right almost-hypergroups | 7036 | 1174 | 0 | 2 | 0 | 0 | 1172 |
simultaneously left almost-hypergroups, right almost-hypergroups and non-commutative hypergroups | 16,044 | 2733 | 0 | 6 | 110 | 0 | 2617 |
non-commutative hypergroups (satisfying the associativity only) | 4628 | 800 | 0 | 1 | 56 | 0 | 723 |
simultaneously left almost-hypergroups and non-commutative hypergroups | 0 | ||||||
simultaneously right almost-hypergroups and non-commutative hypergroups | 0 | ||||||
commutative hypergroups (trivial left almost-hypergroups and trivial right almost-hypergroups) | 2520 | 466 | 6 | 3 | 78 | 0 | 399 |
hypergroups | 23,192 | 3999 | 6 | 10 | 244 | 0 | 3739 |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1,2,3} | {2,3} | {2,3} |
2 | {1,3} | {1,2,3} | {1,3} |
3 | {1,2} | {1,2} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1,2,3} | {1,3} | {1,2} |
2 | {2,3} | {1,2,3} | {1,2} |
3 | {2,3} | {1,3} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {2} | {3} |
2 | {3} | {1,2,3} | {1,3} |
3 | {2} | {1,2} | {1,2,3} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {3} | {2} |
2 | {2} | {1,2,3} | {1,3} |
3 | {3} | {1,2} | {1,2,3} |
◦ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|
1 | {1,2} | {1,2} | {1,2,3,4} | {1,2,3,4} | {1,2,5,6,7} | {1,2,5,6,7} | {1,2,5,6,7} |
2 | {1} | {1,2} | {1,2,3,4} | {1,2,3,4} | {1,2,5,6,7} | {1,2,5,6,7} | {1,2,5,6,7} |
3 | {1,2,3,4} | {1,2,3,4} | {3,4} | {4} | {3,4,5,6,7} | {3,4,5,6,7} | {3,4,5,6,7} |
4 | {1,2,3,4} | {1,2,3,4} | {3} | {3,4} | {3,4,5,6,7} | {3,4,5,6,7} | {3,4,5,6,7} |
5 | {1,2,5,6,7} | {1,2,5,6,7} | {3,4,5,6,7} | {3,4,5,6,7} | {5,7} | {7} | {6,7} |
6 | {1,2,5,6,7} | {1,2,5,6,7} | {3,4,5,6,7} | {3,4,5,6,7} | {6,7} | {7} | {7} |
7 | {1,2,5,6,7} | {1,2,5,6,7} | {3,4,5,6,7} | {3,4,5,6,7} | {6,7} | {6,7} | {6,7} |
◦ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1,2} | {1,3} |
2 | {1,3} | {1,2,3} | {1,3} |
3 | {1,2} | {1,2} | {1,2,3} |
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Massouros, C.G.; Yaqoob, N. On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations. Mathematics 2021, 9, 1828. https://doi.org/10.3390/math9151828
Massouros CG, Yaqoob N. On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations. Mathematics. 2021; 9(15):1828. https://doi.org/10.3390/math9151828
Chicago/Turabian StyleMassouros, Christos G., and Naveed Yaqoob. 2021. "On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations" Mathematics 9, no. 15: 1828. https://doi.org/10.3390/math9151828
APA StyleMassouros, C. G., & Yaqoob, N. (2021). On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations. Mathematics, 9(15), 1828. https://doi.org/10.3390/math9151828