Distributive lattices with strong endomorphism kernel property as direct sums

Abstract. Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [۳] were
fully characterized in [۱۱] using Priestley duality (see Theorem ۲.۸). We
shall determine the structure of special elements (which are introduced after
Theorem ۲.۸ 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 ۲ element lattices with
distinguished elements ۱ and a lattice which is a direct sum of ۲ element
lattices with distinguished elements ۰, and the sublattice of strong elements
is isomorphic to a product of last two mentioned lattices.

Keywords:

unbounded distributive lattice , strong endomorphism kernel property , congruence relation , bounded Priestley space , Priestley duality , strong element , direct sum