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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

A new view of relationship between atomic posets and complete (algebraic) lattices

Bin Yu / Qingguo Li / Huanrong Wu
Published Online: 2017-03-19 | DOI: https://doi.org/10.1515/math-2017-0027

Abstract

In the context of the atomic poset, we propose several new methods of constructing the complete lattice and the algebraic lattice, and the mutual decision of relationship between atomic posets and complete lattices (algebraic lattices) is studied.

Keywords: Atomic poset; C(D)-operator; Complete lattice; Algebraic lattice; Mutual decision

MSC 2010: 20N20; 20N25; 03E72

1 Introduction

Order theory can formally be seen as a subject between lattice theory [2325, 34, 48] and graph theory [6, 22, 36]. Indeed, one can say with good reason that lattices are special types of ordered sets, which are in turn special types of directed graphs. Yet this would be much too simplistic an approach. In each theory the distinct strengths and weaknesses of the given structure can be explored. This leads to general as well as discipline specific questions and results. Of the three research areas mentioned, order theory undoubtedly is the youngest. In recent years, as order and partial ordered set theory were widely applied in the combinatorics [1, 9, 13, 37, 43], fuzzy mathematics [7, 32, 40, 42, 44], computer science [2, 39], and even in the social science [14, 15] etc.

A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements needs to be related: for some pairs, it may be that neither element precedes the other in the poset. Thus, partial orders generalize the more familiar total orders, in which every pair is related. A finite poset can be visualized through its Hasse diagram (discrete graphs), which depicts the ordering relation [35]. This area of order theory was investigated in a series of papers by Erné [16, 18] and independently by Chajda, Halaš, Larmerová, Rachånek, Niederle [8, 2629, 31], and later by Joshi, Kharat, Mokbel, Mundlik, Waphare [33, 45, 46] and many others. In [19], they are mainly interested in ideal-theoretic properties and various degrees of (finite or infinite) distributivity in atomic posets. However, we are more interested in atoms of atomic posets. And it is conceivable that the role of the atomic elements is very important (each element in the boolean lattice can be expressed by atomic elements i.e. a = ⋁{x ∈ 𝒜(B)|xa}) in the Boolean lattice [11]. Similarly, atoms in atomic posets also deserve a keen attention.

In this paper, we stress the importance of the two kinds of operators (C-operator and D-operator) in the study of the theoretical aspect of atomic posets. Specifically, we first define two relation operators (C-operator and D-operator) between the non-atomic element and the atomic element, and get series of related properties. Almost immediately, two kinds of operators above are combined to construct complete (algebraic) lattices, and used to study the relation between atomic posets and complete (algebraic) lattices.

The work of this paper is organized as follows. We shall first briefly introduce poset and related concepts. In Section 3, two kinds of operators above are combined to construct complete lattices, and used to study the relation between atomic posets and complete lattices. In Section 4, two kinds of operators above are combined to construct algebraic lattices, and used to study the relation between atomic posets and algebraic lattices.

2 Preliminaries

By a partial order on set P we mean a binary relation ≤ on P which is reflexive, antisymmetric and transitive, and by a partially ordered set we mean a non-empty set P together with a partial ≤ on P. Less familiar is the symbol ∥ used to denote non-comparability: we write xy if xy and yx. We say P has a bottom element if there exist 0 ∈ P (called bottom) with the property that 0 ≤ x for all xP. An element xP is an upper bound of S if sx for all sL. A lower bound is defined dually. The set of all upper bounds of L is denoted by Su (read as “L upper”) and the set of all lower bounds by Ll (read as “L lower”).

Throughout this article, 0 denotes the least element in a poset.

Definition 2.1 ([11])

Let P be an ordered set and x, yP. We say x is covered by y (y covers x), and write xy or yx, if x < y and xz < y implies z = x. The latter condition is demanding that there is no element z of P with x < z < y.

Observe that if P is finite, x < y if and only if there exist a finite sequence of covering relations x = x0x1 ≺ … ≺ xn = y. Thus, in the finite case, the order relation determines, and is determined by the covering relation.

Definition 2.2 ([25])

A subset D of a poset P is directed provided it is nonempty and every finite subset of D has an upper bound in D.

Definition 2.3 ([4, 11])

Let P be a non-empty ordered set.

  1. If xy and xy exist for all x, yP, then P is called a lattice;

  2. IfS andS exist for all SP, then P is called a complete lattice.

Definition 2.4 ([24])

Let P and Q be ordered sets. A map φ : PQ is said to be

  1. order-preserving if if xy in P implies φ(x) ≤, φ(y) in Q;

  2. order-embedding (and we write φ : PQ) if xy in P if and only if φ(x) ≤ φ(y) in Q;

  3. order-isomorphism if φ is onto and xy in P if and only if φ(x) ≤ φ(y) in Q.

Definition 2.5 ([24])

Let L and K be lattices. A map f : LK is said to be a lattice homomorphism if f is join-preserving and meet-preserving, that is, for all a, bL, f(ab)=f(a)f(b)andf(ab)=f(a)f(b).

A bijective lattice homomorphism is a lattice isomorphism.

Proposition 2.6 ([11])

Let L and K be lattices and f : LK is a map. f is a lattice isomorphism if and only if it is an order-isomorphism.

Lemma 2.7 ([11])

Let X be a set andbe a family of subsets of X, ordered by inclusion, such that

  1. iIL for every non-empty family {Ai}iI ⊆ ℒ, and

  2. X ∈ ℒ.

    That is to say thatis a topped intersection structure on X. Thenis a complete lattice in which iIAi=iIAi,iIAi={BL|iIAiB}.

Lemma 2.8 ([11])

Let P and Q be ordered sets and φ : PQ be an order-isomorphism map. Then φ preserves all existing joins and meets.

Definition 2.9 ([11])

Let L be a complete lattice and let kL.

  1. k is called finite (in L), for every directed set D in L, kDkdforsomedD.

    The set of finite elements of L is denoted F(L)

  2. k is said to be compact if, for every subset S of L, kSkTforsomefinitesubsetTofS.

    The set of compact elements of L is denoted K(L).

Lemma 2.10 ([11])

Let L be a complete lattice. Then F(L) = K(L)

Definition 2.11 ([11])

A complete lattice L is said to be algebraic if, for each aL, a={kK(L)|ka}.

Definition 2.12 ([20])

A poset is said to be directed complete if every directed subset has a sup. A directed complete algebraic poset L is called an algebraic domain.

Definition 2.13 ([19])

Let P be a poset. If P has a least element 0, then xP is called an atom if 0 ≺ x. If P has no least element, then xP is called an atom when x is a minimal element in P.

All finite posets are atomic. The set of atoms of P is denoted by 𝒜(P) and let 𝒜0(P) = 𝒜(P) ∪ {0} and P0 = P\𝒜0(P). The poset P is called atomic if, given a(≠ 0) in P, there exists x ∈ 𝒜(P) such that xa.

In atomic posets, according to the relationship between the non-atomic element and the atomic element, We define the relation operator naturally, called C-operator and D-operator.

Definition 2.14

Let P be an atomic poset.

  1. (C-operator) ∀aP0, denote Ca = {x ∈ 𝒜(P)|xa};

  2. (D-operator) ∀b ∈ 𝒜(P), denote Db = {xP0|bx}.

C-operator and D-operator can be viewed as operators between 𝒜(P) and P0. Naturally, we can define CA = aACa for any AP0, and DB = bBDb for any BP0. Then we can define two kinds of operators between 𝒜(P) and P0 via Ca and Db.

Based on C-operators and D-operators, we generate several new relational operators as follows:

Definition 2.15

Let P be an atomic poset.

  1. (Co-operator) ∀ AP0, denote CAO={xA(P)|DxA};

  2. (Do-operator) ∀ B ⊆ 𝒜(P), denote DBo={xP0|CxB};

  3. (C¯-operator) ∀ AP0, denote C¯A={xA(P)|ADx};

  4. (D¯-operator) ∀ B ⊆ 𝒜(P), denote D¯B={xP0|BCx}.

It is obvious that C¯A=aACa,D¯B=bBDb. In Definition 2.14 and 2.15, these operators are very reasonable.

Then, we study some properties of several operators.

Proposition 2.16

Let P be an atomic poset. A, AiP0, B, Bi ⊆ 𝒜(P), for iI, a, bP0, c ∈ 𝒜(P). Then

  1. abCaCb;

  2. A1A2CA1CA2, B1B2DB1DB2;

  3. A1A2CA1oCA2,oB1B2DB1oDB2o;

  4. A1A2C¯A1C¯A2,B1B2D¯B1D¯B2;

  5. aD¯Ca,cC¯Dc,AD¯C¯A,BC¯D¯B;

  6. ADCAo,CDBoB;

  7. C¯A=C¯D¯C¯A,D¯B=D¯C¯D¯B;

  8. DCDBo=DBo,CDCAo=CA;

  9. C¯iIAi=iIC¯Ai,D¯iIBi=iID¯Bi;

  10. AD¯BC¯AB.

Proof

We consider cases of (6), (7), (8), (10), and the other proofs are similar.

(6)

  1. DCAo={xP0|CxCA}. If aA, then there must be CxCA.SoaDCAo and therefore ADCAo.

  2. DBo={xP0|CxB},CDBo=xDBoCx.IfbCDBo, then there exists x0DBo such that bCx0B. So bB and therefore CDBB.

(7) Since AD¯C¯A,thenC¯AC¯D¯C¯A. Since BC¯D¯B,thenC¯AC¯D¯C¯A. Therefore C¯A=C¯D¯C¯A.

(8) Since CDBB,thenDCDBOoDBo. Since DCAoA,thenDCDBOoDBo. Therefore DCDBOo=DBo.

(10) (⇒) Since AD¯B={xP0|BCx}, then ∀aA, BCa and so BaACa=C¯A.

(⇐) Since BC¯A={xA(P)|ADx}, then ∀ bB, ADb and so AbBDb=D¯B. □

3 Constructing complete lattices and mutual decision

The construction of complete lattice [11, 12, 17, 30, 38, 49] is very essential branch in the research of various order structures. In 3.1 and 3.2, several operators (CA,C¯A,D¯BandDBo) are worked on atomic posets, and then a complete lattice is generated. Subsequently, we find that complete lattices and posets are mutually corresponding. Thus, in the theoretical study of posets, we can see the crucial role of the content of this section.

3.1 Complete lattices via(C¯A,D¯B,=)

Let P be an atomic poset. A pair (A, B) satisfies AP0,BA(P),C¯A={xA(P)|ADx}=B and D¯B={aP0|BCx}=A. This implies A=D¯C¯A,B=C¯D¯B. The set of all those pairs of P is denoted by ℬ(P).

For pairs (A1, B1) and (A2, B2) in ℬ(P) we write (A1, B1) ≤ (A2, B2) if A1A2. Also A1A2 implies C¯A1C¯A2, and the reverse implication is also valid, so A1=D¯C¯A1andA2=D¯C¯A2. We therefore have (A1,B1)(A2,B2)A1A2B1B2.

We can then see easily that the relation ≤ is an order on ℬ(P) . As we can see in Theorem 3.1, < ℬ(P); ≤ > is a complete lattice.

Theorem 3.1

Let P be an atomic poset. Then < ℬ(P); ≤ > is a complete lattice in which join and meet are given by iI(Ai,Bi)=(D¯iIC¯Ai,iIBi)iI(Ai,Bi)=(iIAi,C¯iID¯Bi).

Proof

Define B(P0):={AP0|A=D¯C¯A}. The map μ : (A, B) → A gives an order-isomorphism between ℬ(P) and ℬ(P0).

We shall prove that ℬ(P0) is a topped intersection structure. Let Ai ∈ ℬ(P0) for iI. Then D¯C¯Ai=Ai for each i. By (5) in Proposition 2.16 iIAiD¯c¯iIAi

Also iIAiAi for all iI, which, by (4) in Proposition 2.16, implies that D¯c¯iIAiD¯C¯Ai=AiforalliI,

whence D¯c¯iIAiiIAi.

Therefore D¯c¯iIAi=iIAi and hence iIAjB(P0).Also,P0D¯C¯PO. So that P0=D¯C¯P0, which shows that ℬ(P0) is topped.

By Theorem 2.7, ℬ(P0) is a complete lattice in which meet is given by intersection. A formula for the join is given in Theorem 2.7 but we shall proceed more directly. We claim that iIAi=D¯c¯iIAi=D¯iIC¯Ai.

Let A=D¯iIC¯Ai. Certainly A=D¯C¯A, by (7) in Proposition 2.16, and iIAiA, by (2) in Proposition 2.16. Hence A is an upper bound for {Ai}iI in ℬ(P0). Also, if X is an upper bound in ℬ(P0) for {Ai}iI, then iIAiXAD¯C¯X=X.

Therefore A is indeed the required join. We may now appeal to Theorem 2.8 to deduce that ℬ(P) is a complete lattice in which joins and meets are given by iI(Ai,Bi)=(D¯iIC¯Ai,iIBi)iI(Ai,Bi)=(iIAi,C¯iID¯Bi) .  □

Theorem 3.2

Let P be an atomic poset and ℬ(P) be the complete lattice by Theorem 3.1 Then φ(P0) is join-dense in ℬ(P) and φ(𝒜(P)) is meet-dense in ℬ(P).

Proof

Let (A, B) ∈ ℬ(P). Then φ(A)=gAφ(g)=gA(D¯Cg,Cg)=(D¯gAC¯D¯C¯g,gACg)=(D¯gACg,gACg)

However, according to Definition 2.15 gACg=C¯gA{g}=C¯A=B.

Since (A, B) and ⋁ φ(A) are elements of ℬ(P) with the same second coordinate, ⋁ φ(A) = (A, B). Consequently φ(P0) is join-dense in ℬ(P) and φ(𝒜(P)) is meet-dense in ℬ(P). □

Theorem 3.3

For every complete lattice L, there is an atomic poset P such that L is order-isomorphic to ℬ(P).

Proof

Suppose (L, ⊑) is a complete lattice. Define the atomic poset P= {0} ⋃ 𝒜(P)⋃ P0, where 𝒜(P) = L and P0 = L. Further, in P, let ab for ∀ a, b ∈ 𝒜(P) ; let ab iff ab for ∀ a ∈ 𝒜(P), bP0; let ab iff ab for ∀ a, bP0. As L is a lattice, it is easy to see that P is an atomic poset. We want to show that (L, ⊑) is order-isomorphic to ℬ(P).

First note that for any XP0, we have C¯X={bA(P)|XDb}={bA(P)|xX,bx}={bL|xX,bx}=xX{L(x)}=L(X)

Among them, ↓D(x) means the upper set of x in L. On the other hand, for any Y ⊆ 𝒜P, D¯Y={aP0|YCa}={aP0|yY,ya}={aL|yY,ya}=yY{L(y)}=L(Y)

Therefore, X ∈ ℬ(P0) iff D¯C¯P=X, or L((L(X)))=X

Since ↑L ⋁(↓LX) = ↑L(∧ X), hence, X ⊆ ℬ(P0) iff X = ↑L(∧ X). In other words, ℬ(P0) are precisely the up-closed subsets of L generated by a single element. Hence, a subset of P0 belongs to ℬ(P0) if and only if it is a principal filter.

The mapping x ↦ ↑x provides an order-isomorphism between L and ℬ(P0). Since ℬ(P0) is isomorphic to ℬ(P), therefore (L, ⊑) is order-isomorphic to ℬ(P). □

Example 3.4

Let L = {a, b, c, d} be a complete lattice, the Hasse diagram of L is illustrated by Figure 1. We can get an atomic poset P by Theorem 3.3, whose Hasse diagram is illustrated by Figure 1, and can also get a complete lattice ℬ(P) which is isomorphic to L by Theorem 3.1. In Figure 1, a1 and a2 in P is a in L, b1 and b2 in P is b in L, c1 and c2 in P is C in L, d1 and d2 in P is d in L. In ℬ(P), A = ({d2}, {a1, b1, c1, d1}), B = ({b2, d2}, {a1, b1}), C = ({c2, d2}, {a1, c1}), D = ({a2, b2, c2, d2}, {a1, b1, c1, d1}).

The figure in Example 3.4
Fig. 1

The figure in Example 3.4

3.2 Complete lattices via(CA,DBo,=)

Let P be an atomic poset. A set A satisfies AP0, DCAo=A. The set of all those sets of P is denoted by ℬo(P).

For set A1 and A2 in ℬo(P) we write A1A2 if A1A2. We can then see easlly that the relation ≤ is an order on ℬo(P). As we see in Theorem 3.5, ℬo(P);≤> is a complete lattice.

Theorem 3.5

Let P be an atomic poset. Then < ℬo(P);≤> is a complete lattice in which join and meet are given by iIAi=DiICAio,iIAi=iIAi.

Proof

We shall prove that ℬo(P) is a topped intersection structure. Let Ai ∈ ℬo(P) for iI. Then DCAio=Ai for each i. By (6) in Proposition 2.16 iIAiDciIAio

Also iIAiAi for all iI, which, by (3) in Proposition 2.16, implies that DcoiIAiDCAio=AiforalliI,

whence DciIAioiIAi.

Therefore DciIAio=iIAi and hence iIAiBo(P).Also,P0DCP0o. So taht P0=DCP0o, which shows that ℬo(P) is topped.

By Theorem 2.7, ℬo(P) is a complete lattice in which meet is given by intersection. A formula for the join is given in Theorem 2.7 but we shall proceed more directly. We claim that iIAi=DciIAio=DiICAio.

Let A=DiICAio. Certainly A=DCAo, by (8) in Proposition 2.16, and iIAiA, by (2) in Proposition 2.16.

Hence A is an upper bound for {Ai}iI in ℬo(P). Also, if X is an upper bound in ℬo(P) for {Ai}iI, then iIAiXADCXo=X.

Therefore A is indeed the required join. We may now appeal to Theorem 2.8 to deduce that ℬo(P) is a complete lattice in which joins and meets are given by iIAi=DiICAio,iIAi=iIAi. □

Theorem 3.6

For every complete lattice L, there is an atomic poset P such that L is order-isomorphic too(P).

Proof

Suppose (L, ≤) is a complete lattice. Define the atomic poset P = 𝒜(P)∪ P0, where 𝒜(P) = L and P0 = L. Further, in P,

  1. let ab for ∀ a, b ∈ 𝒜(P);

  2. let ab for ∀ a ∈ 𝒜(P), bP0 iff a = b in case that a is the least element in L, ab in case that b is the largest element in L, in other cases ab in L;

  3. let ab for ∀ a, bP0 iff ab in L.

Under the order relation defined in P, it is easy to see that P is an atomic poset. We want to show that (L, ≤) is order-isomorphic to ℬ(P).

First note that for any XP0 which does not contain the least and largest elements, we have CX={bA(P)|xX,bx}={bL|xX,bx}=L{bL|xX,bx}=LxXL(x)=LL(X)

Among them, ↑L(x) means the upper set of x in L. On the other hand, DCXo={aP0|CaCX}={aL|LaLL(x)}={aL|a⊇↑(x)}={aL|ax}=L(x)

If X(⊆ P0) which contains the least or largest elements, we can easily check that DCXo=(X). Therefore, we have {DCXo|XP0}={x|xL}. Hence, X ⊆ ℬo(P) iff X = ↓L(⋁ X). In other words, ℬo(P) are precisely the lower sets of L generated by a single element. It is obvious that ℬo(P) is isomorphic to L. □

Example 3.7

Let L = {a, b, c, d} be a complete lattice, the Hasse diagram of L is illustrated by Figure 2. We can get an atomic poset P by Theorem 3.6, whose Hasse diagram is illustrated by Figure 2, and can also get a complete latticeo(P) which is isomorphic to L by Theorem 3.5. In Figure 2, a1 and a2 in P is a in L, b1 and b2 in P is b in L, c1 and c2 in P is C in L, d1 and d2 in P is d in L. Ino(P), A = {a2}, B = {a2, b2}, C = {a2, c2}, D = {a2, b2, c2, d2}.

The figure in Example 3.7
Fig. 2

The figure in Example 3.7

4 Constructing algebraic lattices and mutual decision

The construction of algebraic lattice [21, 41, 47] is very essential branch in the research of various order structures. In 4.1 and 4.2, Several operators (CA,C¯A,D¯BandDBo) are worked on atomic posets, and then a algebraic lattice is generated. Subsequently, we find that algebraic lattices and posets are mutually corresponding. Thus, in the theoretical study of posets, we can see the crucial role of the content of this section.

4.1 Algebraic lattices via (C¯A,D¯B,)

Let P be an atomic poset. A set A satisfies AP0 and for every finite subset XA, D¯C¯XA. The set of all those sets of P is denoted by ℱ(P).

For A1, A2 in ℱ(P) we write A1A2 iff A1A2. We can see easily that the relation ≤ is an order on ℱ(P). As we can see in Theorem 4.1, < ℱ(P); ≤> is an algebraic lattice.

Theorem 4.1

Let P be an atomic poset. Then < ℱ(P);≤> forms an algebraic lattice.

Proof

We first show that < ℱ(P);≤> is a complete lattice. To show that ℱ(P);≤> is a complete lattice it suffices to show that ℱ(P);≤>is a topped intersection structure by Lemma 2.7. Given any subset T ⊆ ℱ(P), it suffices to show that ⋂ T ∈ ℱ(P). Suppose X is a finite subset of ⋂ T. Then Xt for each tT. Since each t ∈ ℱ(P), we have D¯C¯Xt for each tT. This implies D¯C¯XT and so ⋂ T ∈ ℱ(P). It is easy to see L0 ∈ ℱ(P) and so < ℱ(P);≤ > is a topped intersection structure.

To show that < ℱ(P);≤> is algebraic, note that D¯C¯X is a compact element for each finite X in L0. To see this, we first show D¯C¯XF(P). Let X1 be a finite subset of D¯C¯X,thenD¯C¯X1D¯C¯(D¯C¯X)=D¯C¯X by Proposition 2.16, which implies D¯C¯XF(P). Then let {Ai|iI} be a directed subset of ℱ(P) such that D¯C¯XiIAi.

By Proposition 2.16, XD¯C¯X. Therefore XiIAi. Since X is finite and {Ai|iI} is directed, XAk for some kI. But Ak ∈ ℱ(P), therefore D¯C¯XAk. By Definition 2.9 and Lemma 2.10, D¯C¯X is a compact element for each finite X.

Next we will show that for any T ∈ ℱ(P), T={D¯C¯X|XfinT}. For any Xfin T, as T ∈ ℱ(P), we have D¯C¯XT.Then{D¯C¯X|XfinT}T.AsXD¯C¯X,SoT=X{D¯C¯X|XfinT}. Therefore T={D¯C¯X|XfinT}. Therefore < ℱ(P);≤> forms an algebraic lattice by Definition 2.11. □

Corollary 4.2

Let P be a finite atomic poset. Then F(P)={D¯C¯X|XL}.

Proof

First we will show F(P){D¯C¯X|XP}.AF(P), we have AP0 and for every finite subset XA,D¯C¯XA. As P is finite, we have that A is finite and D¯C¯AA.SinceAD¯C¯A by Proposition 2.16, therefore A=D¯C¯A.SoF(P){D¯C¯X|XP}. Then we will show {D¯C¯X|XP}F(P). Since we show in Theorem 4.1, D¯C¯X is a compact element in < ℱ(P);≤> for each finite X in P0. Since P is finite, so {D¯C¯X|XL}F(P).ThereforeF(P)={D¯C¯X|XP}. □

Theorem 4.3

For every algebraic lattice D, there is an atomic poset P such that D is order-isomorphic to ℱ(P).

Proof

Suppose (D, ⊑) is an algebraic lattice. Define a poset (P, ≤) = {0} ⋃ 𝒜(P)⋃ P0 with 𝒜(P) = D, P0 = K(D), where K(D) stands for the set of compact elements of D. Further, in P, let ab for ∀a, b ∈ 𝒜(P); let ab iff ba for ∀ a ∈ 𝒜(P), bP0; let ab iff ba for ∀ a, bP0. As D is an algebraic lattice, it is easy to see that P is an atomic poset. We want to show that (D, ⊑) is order-isomorphic to ℱ(P).

First note that for any XP0, we have C¯X={bA(P)|XDb}={bA(P)|xX,xb}={bD|xX,xb}=xX{D(x)}=D(x)

Among them, ↑D(⋁ X) means the upper set of ⋁ X in D. On the other hand, D¯Y={aP0|YCa}={aP0|yY,ya}={aK(D)|yY,ay}={aK(D)|yY,aK(D)(y)}=yY{K(D)(y)}=K(D)(Y)

Therefore, I ∈ ℱ(P) iff D¯C¯XI for any finite subset Xfin I, or K(D)((D(X)))I

for each Xfin I. Since K(D)((D(X)))=K(D)(X),

this is equivalent to say that I is a downward closed, directed subset of compact elements of D. A downward closed, directed subset is called an ideal. Hence, a subset of P0 belongs to ℱ(P) if and only if it is an ideal.

At last, by the classical result about algebraic domains [3]: an algebraic domain is isomorphic to the ideal completion of the poset of its compact elements through the isomorphism d{aK(D)|ad}

that is, ℱ(P) is isomorphic to D. □

Example 4.4

Let D ={a, b, c, d, e} be an alegraic lattice, the Hasse diagram of D is illustrated by Figure 3. We can get an atomic poset P by Theorem 4.3, whose Hasse diagram is illustrated by Figure 3, and can also get an alegraic lattice ℱ(P) which is isomorphic to D by Theorem 4.1. In Figure 3, a1 and a2 in P is a in D, b1 and b2 in P is b in D, c1 and c2 in P is C in D, d1 and d2 in P is d in D. In ℱ(P), A = {a2}, B = {a2, b2}, C = {a2, c2}, D = {a2, d2}, E = {a2, b2, c2, d2, e2}.

The figure in Example 4.4
Fig. 3

The figure in Example 4.4

4.2 Algebraic lattices via (CA,DBo,)

Let P be an atomic poset. A set A satisfies AP0 and for every finite subset XA, DCXoA. The set of all those sets of P is denoted by ℱo(P).

For A1, A2 in ℱo(P). We write A1A2 if A1A2. We can see easily that the relation ≤ is an order on ℱo(P). As we can see in Theorem 4.5, < ℱo(P);≤> is an algebraic lattice.

Theorem 4.5

Let P be an atomic poset. Then < ℱo(P);≤> forms an algebraic lattice.

Proof

We first show that > ℱo(P);≤> is a complete lattice. To show that ℱo(P);≤> is a complete lattice it suffices to show that <ℱo(P);≤> is a topped intersection structure by Lemma 2.7. Given any subset T ⊆ ℱo(P), it suffices to show that ⋂ T ∈ ℱo(P). Suppose X is a finite subset of ⋂T. Then Xt for each tT. Since each t ∈ ℱo(P), we have DCXot for each tT. This implies DCXoT and so ⋂T ∈ ℱo(P). It is easy to see P0∈ ℱo(P) and so <ℱo(P);≤> is a topped intersection structure.

To show that < ℱo(P); ≤> is algebraic, note that DCXo is a compact element for each finite X in P0. To see this, we first show DCXoFo(P). Let X1 be a finite subset of DCXo. Then DCX1oDc(DCX)o=DCXo by Proposition 2.16, which implies DCXoFo(P). Then let {Ai|iI} be a directed subset of ℱo(P) such that DCXoiIAi.

By Proposition 2.16, XDCXo. Therefore XiIAi. Since X is finite and {Ai|iI} is directed, XAk for some kI. But Ak ∈ ℱo(P), therefore DCXoAk. By Definition 2.9 and Lemma 2.10, DCXo is a compact element for each finite X.

Next we will show that for any TFo(P),T={DCXo|XfinT}. For any XfinT, as T ∈ ℱo(P), we have DCXoT.Then{DCXo|XfinT}T.AsXDCXo,SoT=X{DCXo|XfinT}. Therefore T={DCXo|XfinT}. Therefore < ℱo(P);≤> forms an algebraic lattice by Definition 2.11. □

Corollary 4.6

Let P be a finite atomic poset. Then Fo(P)={DCXo|XL}.

Proof

First we will show Fo(P){DCXo|XP}.AFo(P), we have AP0 and for every finite subset XA,DCXoA. As P is finite, we have that A is finite and DCAoA.AsADCAo by Proposition 2.16. Therefore A=DCAo.SoFo(P){DCXo|XP}. Then we will show {DCXo|XP}Fo(P). As we show in Theorem 4.5, DCXo is a compact element in < ℱo(P);≤> for each finite X in P0. As P is finite, so {DCXo|XL}Fo(P).ThereforeFo(P)={DCXo|XP}. □

Theorem 4.7

For every algebraic lattice D, there is an atomic poset P such that D is order-isomorphic too(P).

Proof

Suppose (D, ≤) is an algebraic lattice. Define a poset (P, ≤) = 𝒜(P) ⋃ P0 with 𝒜(P) = D, P0 = K(D), where K(D) stands for the set of compact elements of D. Further, in P,

  1. let ab for ∀a, b ∈ 𝒜(P);

  2. let ab for ∀ a ∈ 𝒜(P), bP0 iff a = b in the case that a is the least in D, ab in the case that b is the largest element in L, in other cases ab in L;

  3. let ab for ∀a, bP0 iff ab in D.

As D is an algebraic lattice, it is easy to see that P is an atomic poset. We want to show that (D, ≤) is order-isomorphic to ℱo(P).

First note that for any X(⊆ P0) which does not contain the least and largest elements, we have CX={bA(P)|xX,bx}={bD|xX,bx}=D{bD|xX,bx}=DxXD(x)=DD(X)

Among them, ↑D(x) means the upper set of x in D. On the other hand, DCXo={aP0|CaCX}={aK(D)|DaD(X)}={aK(D)|a⊇↑(X)}={aK(D)|aX}=K(D)(X)

If X(⊆ P0) which contains the least or largest element, we can easily check that DCXo=↓(X). Therefore, IFo(P)iffDCXoI for any finite subset XfinI, or K(D)(X)I

this is equivalent to say that I is a downward closed, directed subset of compact elements of D. A downward closed, directed subset is called an ideal. Hence, a subset of P0 belongs to ℱo(P) if and only if it is an ideal.

At last, by the classical result about algebraic domains [3]: an algebraic domain is isomorphic to the ideal completion of the poset of its compact elements through the isomorphism d{aK(D)|ad}

that is, ℱo(P) is isomorphic to D. □

Example 4.8

Let D= {a, b, c, d, e} be an alegraic lattice, the Hasse diagram of D is illustrated by Figure 4. We can get an atomic poset P by Theorem 4.7, whose Hasse diagram is illustrated by Figure 4, and can also get an alegraic latticeo(P) which is isomorphic to L by Theorem 4.5. In Figure 4, a1 and a2 in P is a in D, b1 and b2 in P is b in D, c1 and c2 in P is C in D, d1 and d2 in P is d in D. Ino(P), A = {a2}, B = {a2, b2}, C = {a2, c2}, D = {a2, d2}, E = {a2, b2, c2, d2, e2}.

The figure in Example 4.8
Fig. 4

The figure in Example 4.8

5 Conclusions

In this paper, to promote the research and development of completion of poset, we thoroughly study C-operators and D-operators. It is aiming at illustrating fresh methodological achievement in lattice which will also be of soaring importance in the future. We have defined C-operators and D-operators. Next, we investigate some related properties. A distinctive completion of lattice via C-operators and D-operators is followed. Our future work on this topic will focus on studying of completion and algebraization using C-operators and D-operators in poset.

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About the article

Received: 2016-02-26

Accepted: 2016-08-25

Published Online: 2017-03-19


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 238–251, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0027.

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© 2017 Yu et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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