Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E) is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be a bifull sublattice of E.


Preliminaries
Effect algebras, introduced by D. J. Foulis and M. K. Bennett [3], have their importance in the investigation of uncertainty.Lattice ordered effect algebras generalize orthomodular lattices and MValgebras.Thus they may include non-compatible pairs of elements as well as unsharp elements.Definition 1 (Foulis and Bennett [3]) An effect algebra is a system (E; ⊕, 0, 1) consisting of a set E with two different elements 0 and 1, called zero and unit, respectively and ⊕ is a partially defined binary operation satisfying the following conditions for all p, q, r ∈ E: (E1) If p ⊕ q is defined, then q ⊕ p is defined and p ⊕ q = q ⊕ p. (E2) If q ⊕ r is defined and p ⊕ (q ⊕ r) is defined, then p ⊕ q and (p ⊕ q) ⊕ r are defined and p ⊕ (q ⊕ r) = (p ⊕ q) ⊕ r. (E3) For every p ∈ E there exists a unique q ∈ E such that p ⊕ q is defined and p ⊕ q = 1.(E4) If p ⊕ 1 is defined then p = 0.
The element q in (E3) will be called the supplement of p, and will be denoted as p .
In the whole paper, for an effect algebra (E, ⊕, 0, 1), writing a ⊕ b for arbitrary a, b ∈ E will mean that a ⊕ b exists.On an effect algebra E we may define another partial binary operation by The operation induces a partial order on E.
Namely, for a, b ∈ E b ≤ a if there exists a c ∈ E such that a b = c.If E with respect to ≤ is lattice ordered, we say that E is a lattice effect algebra.For the sake of brevity we will write just LEA.Further, in this article we often briefly write 'an effect algebra E' skipping the operations.S. P. Gudder ( [5,6]) introduced the notion of sharp elements and sharply dominating lattice effect algebras.Recall that an element x of the LEA E is called sharp if x ∧ x = 0. Jenča and Riečanová in [7] proved that in every lattice effect algebra E the set S(E) = {x ∈ E; x ∧ x = 0} of sharp elements is an orthomodular lattice which is a sub-effect algebra of E, meaning that if among x, y, z ∈ E with x ⊕ y = z at least two elements are in S(E) then x, y, z ∈ S(E).Moreover S(E) is a full sublattice of E, hence a supremum of any set of sharp elements, which exists in E, is again a sharp element.Further, each maximal subset M of pairwise compatible elements of E, called a block of E, is a sub-effect algebra and a full sublattice of E and E = {M ⊆ E; M is a block of E} (see [16,17]).Central elements and centers of effect algebras were defined in [4].In [14,15] it was proved that in every lattice effect algebra E the center where Recall that an element p of an effect algebra E is called an atom if and only if p is a minimal nonzero element of E and E is atomic if for each x ∈ E, x = 0, there exists an atom p ≤ x.
Definition 2 Let (E, ⊕, 0) be an effect algebra.To each a ∈ E we define its isotropic index, notation ord(a), as the maximal positive integer n such that na := a ⊕ . . .⊕ a n-times exists.We set ord(a) = ∞ if na exists for each positive integer n.We say that E is Archimedean, if for each a ∈ E, a = 0, ord(a) is finite.
An element u ∈ E is called finite, if there exists a finite system of atoms a 1 , . . ., a n (which are not necessarily distinct We say that for a finite system in E (briefly we will write n j=1 x j ).We define also It is known that if E is a distributive effect algebra (i.e., the effect algebra E is a distributive lattice -e.g., if E is an MV-effect algebra) then C(E) = S(E).If moreover E is Archimedean and atomic then the set of atoms of C(E) = S(E) is the set {n a a; a ∈ E is an atom of E}, where n a = ord(a) (see [20]).Since S(E) is a bifull sublattice of E if E is an Archimedean atomic LEA (see [13]), we obtain that for every Archimedean atomic distributive lattice effect algebra E. In [8] it was shown that there exists an LEA E for which this property fails to be true.Important properties of Archimedean atomic lattice effect algebras with an atomic center were proven by Riečanová in [21].
Theorem 1 (Riečanová [21]) Let E be an Archimedean atomic lattice effect algebra with an atomic center C(E).Let A E be the set of all atoms of E and A C(E) the set of all atoms of C(E).The following conditions are equivalent: 1.
2. For every atom a ∈ A E there exists an atom In this case E is isomorphic to a subdirect product of Archimedean atomic irreducible lattice effect algebras.
Theorem 2 (Paseka, Riečanová [13]) Let E be an atomic Archimedean lattice effect algebra.Then the set S(E) of all sharp elements of E is a bifull sublattice of E.
We will deal only with atomic Archimedean lattice effect algebras E. We have C(E) ⊂ S(E) ⊂ E. Because of this inclusion and Theorem 2, considering the bifullness of the center C(E) in E is equivalent to considering the bifullness of C(E) in S(E).And S(E) is an orthomodular lattice.For this reason, in the rest of the paper we will restrict our attention to atomic orthomodular lattices L and their centers C(L).For the sake of completeness, we give the definition of an orthomodular lattice.
Definition 4 Let L be a bounded lattice with a unary operation (called complementation) satisfying the following conditions 1. for all a ∈ L (a Then L is said to be an orthomodular lattice (OML for brevity).
Remark 1 Though in OML's we have just latticetheoretical operations ∨ and ∧, we will use also effect algebraic operations ⊕ and with the meaning 2 Orthomodular lattice L whose center is not a bifull sublattice Let us have the following sequences of atoms (sets): For such a choice of atoms, q 1 = q 2 are compatible if and only if q 1 ∩q 2 = ∅.Fig. 1 shows the compatibility among atoms.For their non-compatibility (denoted by ↔) the following rules hold q q q s s s s s s s s q q q p 1 p 2 p 3 p 4 p 5 p 6 p n p n+1 q q q s s s s s s s s q q q a 0 b 0 a For non-compatible atoms the following equalities hold Denote B0 , Bj (for j = 1, 2, . ..) complete atomic Boolean algebras with the corresponding sets of atoms A 0 , A j (j = 1, 2, . ..), given by Disjointness occurring among some atoms of the system ( 2) is equivalent to the fact that A 0 and A j (j = 1, 2, . ..) are unique maximal sets of pairwise compatible atoms.
Bi .Let L 1 be the complete OML generated by sets of atoms {c j , d j } and N the complete Boolean algebra generated by the set of atoms An element u ∈ Bl is finite if and only if u = q 1 ⊕ q 2 ⊕ . . .⊕ q n for an n ∈ N and q 1 , q 2 , . . ., q n ∈ A l .Set Q l = {u ∈ B l ; u is finite}, l = 0, 1, 2, . ... Then Q l is a generalized Boolean algebra, since B l = Q l ∪ Q * l is a Boolean algebra, where Q * l = {u * ; u * = 1 l u and u ∈ Q l } (see [22], or [2, pp. 18-19]).This means that B l is a Boolean subalgebra of finite and cofinite elements of Bl (l = 0, 1, 2, . ..).

Completion of the center of L
We are going to show that it is possible to extend the orthomodular lattice L from Theorem 4 to L, whose center, C( L), is a complete Boolean algebra which is not a bifull sublattice of L. Denote F a fixed non-trivial ultrafilter on N (the index set of atoms p j ).Then F has the following properties which will be important for our construction: Let Q L1 denote the set of all finite elements of L 1 .Further set and Theorem 5 Let L = G ∪ G ⊥ .Then the system L, ∨, ∧, 0, 1 is an orthomodular lattice.
The center and C( L) is a complete Boolean algebra which is not bifull in L.
Proof.First we show that L is a bounded lattice.Consider elements h 1 , h 2 ∈ G. Then there exist elements By the properties of the non-trivial ultrafilter F we get that This implies that G is closed under ∨ and ∧.Because G ⊥ consists of complements of elements of G, we have that also G ⊥ is closed under ∨ and ∧.Now assume that h 1 ∈ G and h 2 ∈ G ⊥ .Then h 2 ∈ G and we can write with the same meaning of f 1 , f 2 , g 1 , g 2 as in formula (6).This means that h 2 = f 2 g 2 .Then, because of the monotonicity of the ultrafilter F , we have Since G is a monotone system (meaning that with an arbitrary element δ 1 ∈ G it contains also all elements δ 2 ∈ L such that δ 2 ≤ δ 1 ), we get from the duality between G and G ⊥ that Obviously it is a bounded and orthocomplemented lattice.Showing that it is an OML is a matter of routine.We will omit the detailed proof.

Let us consider an element
Due to the fact that F is a non-trivial ultrafilter, C( L) is a complete Boolean algebra.
The only central element that is greater than all atoms p j for j = 1, 2, . .., is 1, hence we have that On the other hand, let us take an arbitrary atom α ∈ {c j , d j } and assume that L {p j ; j = 1, 2, . ..} does exist.Since α is orthogonal to all atoms from the set ∞ j=1 {p j }, we have that α is orthogonal to L {p j ; j = 1, 2, . ..} and hence L {p j ; j = 1, 2, . ..} = 1.
It can be shown (see [8]) that L {p j ; j = 1, 2, . ..} does not exist.This implies that C( L) is not a bifull sublattice of L. 24 σ-complete orthomodular lattice Lσ whose center is not a bifull sublattice Let I denote the set of all ordinal numbers less than Ω (the first uncountable ordinal number).Further, denote E the set of all limit ordinal numbers up to Ω and J = I \ E. Assume sets of elements {p i ; i ∈ I}, {a i ; i ∈ I}, {b i ; i ∈ I}, {c i ; i ∈ I}, {d i ; i ∈ I}, where the corresponding elements for i ∈ J will act as atoms.We will have a partial relation ↔ modelling noncompatibility.This partial relation will have the following form among atoms c j ↔ a i , c j ↔ b i for all j ∈ J and i ≤ j, Sets of elements {p i ; i ∈ I}, {a i ; i ∈ I}, {b i ; i ∈ I}, {c i ; i ∈ I}, {d i ; i ∈ I} will present atoms for i ∈ J and for κ ∈ E we will have As a possible model for the just presented sets of elements fulfilling the non-compatibility relation we may have the following: Let us choose a good order of positive real numbers of type Ω, i.e., positive real numbers will be enumerated by ordinal numbers from J .For i ∈ J and r > 0, r ∈ R, we denote r i the i-th number in the chosen good order.Then we identify the set {p i ; i ∈ J } with the set of all positive real numbers, i.e., p i = r i .Further we put for i, j ∈ J For κ ∈ E we define the corresponding elements p κ , a κ , b κ , c κ , d κ by equalities 7, 8, 9, 10, respectively.Compatibility among different atoms is given by disjointness of the corresponding sets.This implies that the uniquely given maximal sets of pairwise compatible atoms are for j ∈ J .Sets of atoms Ã0 and Ãj for j ∈ J , generate complete Boolean algebras B0 and Bj for j ∈ J , respectively.For κ ∈ E we get complete atomic Boolean algebras Bκ generated by sets of atoms This means that for κ ∈ E Bκ ⊂ B0 .The union of all complete atomic Boolean algebras, L = B0 ∪ i∈I Bi , is a complete OML.An element f ∈ L will be called countable if there exists an at most countable set of atoms (an at most countable set of indices K) By definition of elements p i , a i , b i , c i , d i for i ∈ I we get that each of these elements is countable.Let K denote the set of all countable elements of L and K ⊥ = {f ∈ L; f ∈ K}.Further, let P denote the set of all countable elements generated by {p i , i ∈ J }, and P ⊥ = {f ∈ L; f ∈ P}.
Proof.Each of the atoms p i , a i , b i , c i , d i for i ∈ J (and hence also each of the elements p i , a i , b i , c i , d i for i ∈ I) is countable.This implies that Lσ is an OML.Since it is by definition closed under countable meets and joins, it is σ-complete.
Elements p i for i ∈ I are central because each of the elements p i is compatible with all atoms of Lσ .This implies that P ∪P ⊥ ⊂ C( Lσ ).On the other hand, let f be a countable element, f / ∈ P. Then there exists c i such that c i ≤ f for i ∈ J and an atom out of e ∈ {a j , b j , c j , d j } for j < i, e ≤ f .Then c i ↔ e and hence c j ↔ f .Similarly, if f ∈ K ⊥ , there exists c i ≤ f and an atom out of e ∈ {a j , b j , c j , d j } for j < i such that e ≤ f .In this case e ↔ c i and hence also e ↔ f .We conclude that C( Lσ ) = P ∪P ⊥ .
We show that C( Lσ ) is not a bifull sublattice of Lσ .Obviously This means that C( Lσ ) is not a bifull sublattice of Lσ . 2

C
( Lσ) {p i , i ∈ I} = 1.Assume thatLσ {p i , i ∈ I} does exist.Then all elements e ∈ i∈I {a i , b i , c i , d i } are orthogonal with all elements from the set I {p j } and consequently also with Lσ {p i , i ∈ I}.This implies Lσ {p i , i ∈ I} = 1.
is an orthomodular lattice and B(E) is an MV-effect algebra, we obtain that C(E) is a Boolean algebra.Note that E is an orthomodular lattice if and only if E = S(E) and E is an MV-effect algebra if and only if E = B(E).Thus E is a Boolean algebra if and only if E