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Year 2022, Volume: 51 Issue: 2, 501 - 508, 01.04.2022
https://doi.org/10.15672/hujms.994459

Abstract

References

  • [1] T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
  • [2] I. Chajda, Congruences on semilattices with section antitone involutions, Discuss. Math. Gen. Algebra Appl. 30 (2), 207-215, 2010.
  • [3] R.A. Dean and R.H. Ochmke, Idempotent semigroups with distributive right congruence lattices, Pacific J. Math. 14, 1187-1209, 1964.
  • [4] Jie Fang and Zhongju Sun, Semilattices with the strong endomorphism kernel property, Algebra Universalis, 70 (4), 393-401, 2013.
  • [5] G. Grätzer, General Lattice Theory, 2nd edn, Birkhäuser, Basel, 1998.
  • [6] J. Hyndman, J.B. Nation and J. Nishida, Congruence lattices of semilattices with operations, Studia Logica, 104 (2), 305-316, 2016.
  • [7] Marcel Jackson, Semilattices with closure, Algebra Universalis, 52 (1), 1-37, 2004.
  • [8] J. Ježek, Subdirectly irreducible semilattices with an automorphism, Semigroup Forum, 43 (2), 178-186, 1991.

Subdirectly irreducible semilattices with endomorphism

Year 2022, Volume: 51 Issue: 2, 501 - 508, 01.04.2022
https://doi.org/10.15672/hujms.994459

Abstract

In this paper we initiate an investigation into the class of meet semilattices endowed with an endomorphism. A consideration of the subdirectly irreducible algebras leads to a description of a subclass of those algebras (S;,k)(S;∧,k) in which (S;)(S;∧) is a meet semilattice and kk is an endomorphism on SS characterised by the property kidSk⩾idS. We particularly show that such an algebra is subdirectly irreducible if and only if it is a chain with one of the following forms

  1. <aj<aj1<<a0⋯<aj<aj−1<⋯<a0;
  2. 0<aj<aj1<<a00⋯<aj<aj−1<⋯<a0

in which k(aj)=aj1k(aj)=aj−1 for j1j⩾1, k(0)=0k(0)=0 and k(a0)=a0k(a0)=a0.

References

  • [1] T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
  • [2] I. Chajda, Congruences on semilattices with section antitone involutions, Discuss. Math. Gen. Algebra Appl. 30 (2), 207-215, 2010.
  • [3] R.A. Dean and R.H. Ochmke, Idempotent semigroups with distributive right congruence lattices, Pacific J. Math. 14, 1187-1209, 1964.
  • [4] Jie Fang and Zhongju Sun, Semilattices with the strong endomorphism kernel property, Algebra Universalis, 70 (4), 393-401, 2013.
  • [5] G. Grätzer, General Lattice Theory, 2nd edn, Birkhäuser, Basel, 1998.
  • [6] J. Hyndman, J.B. Nation and J. Nishida, Congruence lattices of semilattices with operations, Studia Logica, 104 (2), 305-316, 2016.
  • [7] Marcel Jackson, Semilattices with closure, Algebra Universalis, 52 (1), 1-37, 2004.
  • [8] J. Ježek, Subdirectly irreducible semilattices with an automorphism, Semigroup Forum, 43 (2), 178-186, 1991.
There are 8 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Jie Fang 0000-0002-6331-4360

Publication Date April 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 2

Cite

APA Fang, J. (2022). Subdirectly irreducible semilattices with endomorphism. Hacettepe Journal of Mathematics and Statistics, 51(2), 501-508. https://doi.org/10.15672/hujms.994459
AMA Fang J. Subdirectly irreducible semilattices with endomorphism. Hacettepe Journal of Mathematics and Statistics. April 2022;51(2):501-508. doi:10.15672/hujms.994459
Chicago Fang, Jie. “Subdirectly Irreducible Semilattices With Endomorphism”. Hacettepe Journal of Mathematics and Statistics 51, no. 2 (April 2022): 501-8. https://doi.org/10.15672/hujms.994459.
EndNote Fang J (April 1, 2022) Subdirectly irreducible semilattices with endomorphism. Hacettepe Journal of Mathematics and Statistics 51 2 501–508.
IEEE J. Fang, “Subdirectly irreducible semilattices with endomorphism”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, pp. 501–508, 2022, doi: 10.15672/hujms.994459.
ISNAD Fang, Jie. “Subdirectly Irreducible Semilattices With Endomorphism”. Hacettepe Journal of Mathematics and Statistics 51/2 (April 2022), 501-508. https://doi.org/10.15672/hujms.994459.
JAMA Fang J. Subdirectly irreducible semilattices with endomorphism. Hacettepe Journal of Mathematics and Statistics. 2022;51:501–508.
MLA Fang, Jie. “Subdirectly Irreducible Semilattices With Endomorphism”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 2, 2022, pp. 501-8, doi:10.15672/hujms.994459.
Vancouver Fang J. Subdirectly irreducible semilattices with endomorphism. Hacettepe Journal of Mathematics and Statistics. 2022;51(2):501-8.