# Distributive lattices with strong endomorphism kernel property as direct sums

Document Type : Research Paper

Author

Department of Algebra and Geometry, Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, Slovakia.

Abstract

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem  2.8}). We shall determine the structure of special elements (which are introduced after  Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.

Keywords

#### References

[1] Blyth, T.S., Fang, J., and Silva, H.J., The endomorphism kernel property in finite distributive lattices and de Morgan algebras, Comm. Algebra 32(6) (2004), 2225- 2242.
[2] Blyth, T.S., Fang, J., and Wang, L.-B., The strong endomorphism kernel property in distributive double p-algebras, Sci. Math. Jpn. 76(2) (2013), 227-234.
[3] Blyth, T.S. and Silva, H.J., The strong endomorphism kernel property in Ockham algebras, Comm. Algebra 36(5) (2008), 1682-1694.
[4] Clark, D.M. and Davey, B.A., Natural Dualities for the Working Algebraist", Cambridge University Press, 1998.
[5] Davey, B.A. and Priestley, H.A., Introduction to Lattices and Order", 2nd edn. Cambridge University Press, 2002.
[6] Fang, G. and Fang, J., The strong endomorphism kernel property in distributive p-algebras, Southeast Asian Bull. Math. 37(4) (2013), 491-497.
[7] Fang, J. and Sun, Z.-J., Semilattices with the strong endomorphism kernel property, Algebra Universalis 70(4) (2013), 393-401.
[8] Fang, J., The Strong endomorphism kernel property in double MS-algebras, Studia Logica 105(5) (2017), 995-1013.
[9] Gratzer, G., Lattice theory: Foundation", Birkhauser, 2011.
[10] Gurican, J., Strong endomorphism kernel property for Brouwerian algebras, JP J. Algebra Number Theory Appl. 36(3) (2015), 241-258.
[11] Gurican, J. and Ploscica M., The strong endomorphism kernel property for modular p-algebras and distributive lattices, Algebra Universalis 75(2) (2016), 243-255.
[12] Haluskova, E., Strong endomofphism kernel property for monounary algebras, Math. Bohem. 143(2) (2018), 161-171.
[13] Kaarli, K. and Pixley, A.F., Polynomial completeness in algebraic system", Chapman & Hall/CRC, 2001.
[14] Ploscica, M., Afine completions of distributive lattices, Order 13(3) (1996), 295-311.