cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Arman Shamsgovara

Arman Shamsgovara's wiki page.

Arman Shamsgovara has authored 8 sequences.

A357295 Number of balanced quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 1, 9, 106, 1597, 29720, 663897, 17747907, 620659554
Offset: 1

Views

Author

Arman Shamsgovara, Sep 22 2022

Keywords

Comments

For definition of quantale see A354493.
A quantale is balanced if T * T = T, where T denotes the top element of the underlying lattice.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.
  • Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.

Crossrefs

Cf. A354493.

A357294 Number of integral quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 1, 2, 9, 49, 364, 3335, 37026, 496241
Offset: 1

Views

Author

Arman Shamsgovara, Sep 22 2022

Keywords

Comments

For definition of quantale see A354493.
A quantale is integral if the top element in the underlying lattice is a unit element. Equivalently, it is both unital and balanced. Equivalently, it is both strictly left-sided and strictly right-sided.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.
  • Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.

Crossrefs

Cf. A354493, A354495, A354497, A357295.

A354498 Number of two-sided quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 2, 8, 47, 354, 3277, 36506, 490983, 8301353
Offset: 1

Views

Author

Arman Shamsgovara, Aug 03 2022

Keywords

Comments

A quantale is two-sided if it is both left- and right-sided. See A354493 for the definition of a quantale and A354496 for the definition of left- and right-sided quantales.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract, to appear), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.

Crossrefs

Cf. A354493, A354496.

A354497 Number of strictly left-sided quantales on n elements, up to isomorphism. Also number of strictly right-sided quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 1, 4, 23, 164, 1482, 15838, 197262, 2830649
Offset: 1

Views

Author

Arman Shamsgovara, Aug 03 2022

Keywords

Comments

For definition of quantale see A354493.
A quantale is strictly left-sided if every element x satisfies T * x = x, and strictly right-sided if every element x satisfies x * T = x. Here T denotes the top element of the underlying lattice.
Transposition of the semigroup table yields a bijection between strictly left-sided and strictly right-sided quantales, so this sequence counts either type.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.

Crossrefs

Cf. A354493, A354496.

A354496 Number of left-sided quantales on n elements, up to isomorphism. Also number of right-sided quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 2, 9, 60, 497, 4968, 58507, 807338, 13341730
Offset: 1

Views

Author

Arman Shamsgovara, Aug 03 2022

Keywords

Comments

For definition of quantale see A354493.
A quantale is left-sided if every element x satisfies T * x <= x, and right-sided if every element x satisfies x * T <= x. Here T denotes the top element of the underlying lattice.
Transposition of the semigroup table yields a bijection between left-sided and right-sided quantales, so this sequence counts either type.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.

Crossrefs

Cf. A354493.

A354495 Number of unital quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 1, 3, 20, 149, 1488, 18554, 295292, 6105814
Offset: 1

Views

Author

Arman Shamsgovara, May 28 2022

Keywords

Comments

For definition of quantale see A354493.
A quantale is unital if it contains a unit element e such that x * e = e * x = x for all elements x.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract, to appear), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.
  • Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.

Crossrefs

Cf. A354493, A354494.

A354494 Number of semi-unital quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 1, 6, 64, 939, 17578, 403060, 11327795, 440735463
Offset: 1

Views

Author

Arman Shamsgovara, May 28 2022

Keywords

Comments

For definition of quantale see A354493.
A quantale is semi-unital if for all elements x: x <= x * T and x <= T * x, where T is the top element of the underlying lattice.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract, to appear), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.

Crossrefs

Cf. A354493, A354495.

A354493 Number of quantales on n elements, up to isomorphism.

Original entry on oeis.org

1, 2, 12, 129, 1852, 33391, 729629, 19174600, 658343783
Offset: 1

Views

Author

Arman Shamsgovara, May 28 2022

Keywords

Comments

A quantale is an algebraic structure (X,*,v) composed of a set X of elements, a semigroup operator "*" and a supremum operator "v" (in the sense of lattices) such that * distributes over v: x * (y v z) = (x * y) v (x * z) and (x v y) * z = (x * z) v (y * z) for all elements x,y,z in X. In addition the bottom element corresponding to v, denoted 0, must satisfy x * 0 = 0 * x = 0.

References

  • P. Eklund, J. G. García, U. Höhle, and J. Kortelainen, (2018). Semigroups in complete lattices. In Developments in Mathematics (Vol. 54). Springer Cham.
  • K. I. Rosenthal, Quantales and their applications. Longman Scientific and Technical, 1990.
  • Arman Shamsgovara, A catalogue of every quantale of order up to 9 (abstract, to appear), LINZ2022, 39th Linz Seminar on Fuzzy Set Theory, Linz, Austria.
  • Arman Shamsgovara and P. Eklund, A Catalogue of Finite Quantales, GLIOC Notes, December 2019.

Links

Crossrefs

Related algebraic structures: A027851, A006966.

Programs

  • Mace4
    assign(max_models,-1).
assign(domain_size,4).
formulas(assumptions).
% Comment: This will find all quantales on 4 elements, fixing
% 0 as the bottom and 3 as the top. Elements will be numbered
% 0-3. Results *must* be run through the companion program
% isofilter that is included with the downloads for mace4,
% otherwise the output will contain isomorphic duplicates!
% By changing the domain size, this file should be sufficient
% for up to 6 elements, but will crash for higher numbers.
(x*y)*z = x*(y*z).
(x v y) v z = x v (y v z).
x v y = y v x.
x v x = x.
x*(y v z) = (x*y) v (x*z).
(x v y)*z = (x*z) v (y*z).
0*x = 0.
x*0 = 0.
0 v x = x.
3 v x = 3.
end_of_list.
formulas(goals).
end_of_list.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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