Subdirectly irreducible semilattices with endomorphism

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 k⩾idSk⩾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 ⋯<aj<aj−1<⋯<a0⋯<aj<aj−1<⋯<a0;0⋯<aj<aj−1<⋯<a00⋯<aj<aj−1<⋯<a0 in which k(aj)=aj−1k(aj)=aj−1 for j⩾1j⩾1, k(0)=0k(0)=0 and k(a0)=a0k(a0)=a0.

Keywords:

Semilattice, endomorphism subdirectly irreducible,

PDF

___

[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.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

Arşiv