Distributive lattices with strong endomorphism kernel property as direct sums

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.


Introduction
The concept of the (strong) endomorphism kernel property for a universal algebra has been introduced by Blyth, Fang and Silva in [1] and [3] as follows.
Definition 1.1. An algebra A has the endomorphism kernel property (EKP) if every congruence relation θ on A different from the universal congruence ι A = A × A is the kernel of an endomorphism on A.
Let θ ∈ Con(A) be a congruence on A. A mapping f : A → A is said to be compatible with θ if a ≡ b(θ) implies f (a) ≡ f (b)(θ), it means if it preserves the congruence θ. An endomorphism of A is called strong, if it is compatible with every congruence θ ∈ Con(A).
The notion of compatibility of functions with congruences has been studied in various contexts by many authors. We refer to the monograph [13] for an overview. Compatible functions are sometimes called "congruence preserving functions" or "functions with substitution property". Definition 1.2. An algebra A has the strong endomorphism kernel property (SEKP) if every congruence relation θ on A different from the universal congruence ι A is the kernel of a strong endomorphism on A.
If the algebra A has two or more nullary operations and corresponding elements are different in A, the universal congruence ι A can not be the kernel of an endomorphism and that is the reason why the universal congruence ι A is excluded from the definition of both EKP and SEKP. It is not necessary to exclude it for algebras with one-element subalgebras.
Blyth and Silva considered the case of Ockham algebras and in particular of MS-algebras in [3]. For instance, a Boolean algebra has SEKP if and only if it has exactly two elements. A full characterization of MS-algebras having SEKP is provided in this paper. A full characterization of finite distributive double p-algebras and finite double Stone algebras having SEKP was proved by Blyth, J. Fang and Wang in [2]. SEKP for distributive p-algebras and Stone algebras has been studied and fully characterized by G. Fang and J. Fang in [6]. Semilattices with SEKP were fully described by J. Fang and Z.-J. Sun in [7]. Guričan and Ploščica described unbounded distributive lattices with SEKP in [11]. Halušková described monounary algebras with SEKP in [12]. Double MS-algebras with SEKP were described by J. Fang in [8].

Preliminary results
The following notion is from [10]. Let V be a variety. Let A i , i ∈ I be algebras from V such that they all have one element subalgebra and we have chosen (distinguished) elements e A i ∈ A i such that {e A i } is one element subalgebra of A i . (The situation is easier if the one element algebra is given by a nullary operation in V -no choice is needed in this case.) It is easy to check that B is a subalgebra of a direct product (A i , i ∈ I). We shall denote it as ((A i , e A i ); i ∈ I) and call it the direct sum of A i 's with distinguished elements e A i . All other notions and results in this section are from [11].
First, let us recall the full characterization of distributive {1}-lattices (it means distributive lattices in which only top element is considered as a part of its signature, bottom element need not exist and if it exists, it need not be preserved by endomorphims/homomorhisms) and {0}-lattices It is clear that these lattices are isomorphic to the sublattices of {0, 1} Z consisting of all (x i ) i∈Z with {i ∈ Z; x i = 1} finite (a direct sum of Z copies of {0, 1} with the distinguished elements 1).
For the bounded case we have the following theorem. In this note we shall deal with distributive lattices considered as unbounded lattices (that is, the top and/or bottom elements -if they existare not a part of the signature and therefore need not be preserved by homomorphisms). Let L be an unbounded distributive lattice in what follows.
The main tool which we will use is Priestley duality for unbounded distributive lattices. We follow [4, Section 1.2] to introduce its basic elements.
The bounded Priestley space assigned to a distributive lattice L is where Spec(L) is the set of all prime ideals of L, including ∅ and L, 0 = ∅, 1 = L, the set inclusion ⊆ is the order relation on Spec(L) and τ is the topology on Spec(L) with the subbasis consisting of all sets A x = {P ∈ Spec(L); x / ∈ P } and their complements B y = {P ∈ Spec(L); y ∈ P } for x, y ∈ L. This means that D(L) is an ordered topological space. This space as an ordered set is bounded, it is a compact topological space and it is totally order-disconnected. In general, a bounded Priestley space is X = (X, 0 X , 1 X , ≤ X , τ X ) with the mentioned structures (X being a nonempty set) which have just mentioned properties.
Let O(D(L)) be the set all nonempty proper clopen down sets of D(L), ordered by the set inclusion (a set U ⊆ Spec(L) is a down set if x ∈ U , y ∈ Spec(L) and y ⊆ x implies y ∈ U , up sets are defined dually). The representation theorem is The topology τ Z is given as a factor topology, that is, unique, strongest topology which makes inclusion maps of X and Y into Z embedings, 0 Z is the block {0 X , 0 Y } of equivalence relation R, 1 Z is the block {1 X , 1 Y } and the ordering ≤ Z is defined by a "union" of ≤ X and ≤ Y , that is, x ≤ Z y if and only if x, y ∈ X and x ≤ X y or x, y ∈ Y and x ≤ Y y, and block It is known, that the category of bounded Priestley spaces is closed under direct unions and therefore, by [4, Lemma 6.3.2], coproducts are given exactly by direct unions (in spite of the fact that there is a binary relation in D ∼ ). We are mostly interested in clopen down sets of Z (and induced clopen down sets of "components" X, Y). As τ Z is a factor topology, A is a nonempty proper clopen down set of Z if and only if (A ∩ X) ∪ {0 X } is a nonempty proper clopen down set in X, and (A ∩ Y ) ∪ {0 Y } is a nonempty proper clopen down set in Y.
We shall summarize most importat properties of Priestley spaces of distributive lattices with SEKP, see [11]. Let us start with the description of an order relation ⊆ of Spec(L).
We shall keep the notation A 0 , A 1 , A 2 in the paper. Here are most important properties of D(L) and of O(D(L)).
Lemma 2.5. Let L be any distributive lattice. Let L have SEKP, P ∈ Spec(L), P = ∅, P = L. Then P is a discrete point in the topology τ .  The existence of c from the item 2 of this theorem follows from Lemma 2.7, the clopen down set C = A c = {P ∈ Spec(L); c / ∈ P } ∈ O(D(L)), which is the O(D(L)) "form" of such c can be fully characterized by the fact that it is a clopen down set and that A 1 ⊆ C and A 2 ∩ C = ∅ (one implication of this characterization also follows from Lemma 2.7 and the second one is proved in [11] within the proof of Theorem 2.8).
Definition 2.9. Let L be an unbounded distributive lattice which has SEKP. The element c ∈ L is called strong if for every Every strong element C (the O(D(L)) form) can be written as C = A 1 ∪(C ∩A 0 )∪{∅} and therefore it is uniquely determined by its intersection with the set A 0 .

Strong elements and direct sums
We shall describe the structure of strong elements of an unbounded distributive lattice with SEKP and show that an unbounded distributive lattice with SEKP 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. All lattices in this section are considered as unbounded lattices. Proof. By the definition of a strong element c, for we know that A 1 ⊆ C and A 2 ∩C = ∅. These properties are clearly preserved by the union and the intersection and they also ensure the convexity in O(D(L)).
Lemma 3.2. Let L be a distributive lattice with SEKP. Let B, C ∈ O(D(L)).
Then sets B, C and also B 0 , C 0 differ only in a finite number of elements (that is, the symmetrical differences B∆C and B 0 ∆C 0 are finite).
Proof. We know that (B∆C) ∩ A 0 = B 0 ∆C 0 , therefore it is enough to prove that B∆C is finite. Now, B∆C = (B ∪ C) \ (B ∩ C), so that B∆C is the difference of two clopen sets, therefore it is clopen, and hence compact. By Lemma 2.5, it consists of discrete points and by compactness it is finite.
Now we can formulate the first "decomposition" therorem  (ii) L 2 is a direct sum of U copies of {0, 1} with the distinguished elements 1 for some set U .
(iii) L 3 is a direct sum of V copies of {0, 1} with the distinguished elements 0 for some set V .
Sets U, V can be choosen in such a way that one of the following holds: (a) both U, V are infinite, (b) one of U, V is empty.
Proof. By the proof of Theorem 3.3, we know that A 1 ∪ A 2 is empty for L , i.e. O(D(L )) = T of Theorem 3.3. As the order on A 0 is trivial, we can, for example, decompose T as follows. Take C a nonempty proper clopen down set of T. Let us denote X = ((A 0 ∩ C) ∪ {∅, L }, ∅, L , ⊆, τ X ) and Y = ((A 0 \ C) ∪ {∅, L }, ∅, L , ⊆, τ Y ) with topologies induced from T.
It is clear that T is the direct union of X and Y , so that it is a coproduct and denoting L 2 a lattice corresponding to X , L 3 a lattice corresponding to Y , we see that L is isomorphic to L 2 × L 3 .
X and Y have orders and topologies corresponding to what is described in Lemmas 2.4 -2.7 (with A 1 = A 2 = ∅) and therefore both have SEKP. Let us discuss at first the most general case, when both U = C ∩ A 0 and V = A 0 \ C are infinite.
Applying Lemma 2.5 -by removing a finite number of discrete points from a clopen set we obtain a clopen set -and Lemma 3.2 to O(D(L 2 )) we see that L 2 is isomorphic to the lattice of all cofinite subsets of the set U , which is isomorphic to the direct sum of U copies of {0, 1} with the distinguished elements 1.
Applying Lemma 2.5 -by adding a finite number of discrete points to a clopen set we obtain a clopen set -and Lemma 3.2 to O(D(L 3 )), we see that L 3 is isomorphic to the lattice of all finite subsets of the set V , which is isomorphic to the direct sum of V copies of {0, 1} with the distinguished elements 0.
If one of U, V is finite and the other one is infinite, the finite one can be "made" empty by Lemma 2.5, because by removing/adding finite number of discrete points from/to a clopen set we get a clopen set.
If both U, V are finite, we can make one of them empty.
Summarizing these results we get (v) the product L 2 × L 3 is isomorphic to the sublattice of all strong elements of L.
Sets U, V can be choosen in such a way that one of the following holds: (a) both U, V are infinite, (b) one of U, V is empty.