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

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.


Introduction
Order theory can formally be seen as a subject between lattice theory [23-25, 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, Halas, Larmerová, Rachånek, Niederle [8,[26][27][28][29]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 D W fx 2 A.B/jx Ä ag) 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 Doperator) 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.

Preliminaries
By a partial order on set P we mean a binary relation Ä 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 Ä on P . Less familiar is the symbol k used to denote non-comparability: we write xky if x Š y and y ‹ x. We say P has a bottom element if there exist 0 2 P (called bottom) with the property that 0 Ä x for all x 2 P . An element x 2 P is an upper bound of S if s Ä x for all s 2 L. A lower bound is defined dually. The set of all upper bounds of L is denoted by S u (read as "L upper") and the set of all lower bounds by L l (read as "L lower").
Throughout this article, 0 denotes the least element in a poset. 11]). Let P be an ordered set and x; y 2 P . We say x is covered by y (y covers x), and write x y or y x, if x < y and x Ä z < y implies z D 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 D x 0 x 1 ::: x n D y. Thus, in the finite case, the order relation determines, and is determined by the covering relation.  24]). Let L and K be lattices. A map f W L ! K is said to be a lattice homomorphism if f is join-preserving and meet-preserving, that is, for all a; b 2 L, A bijective lattice homomorphism is a lattice isomorphism.
Proposition 2.6 ( [11]). Let L and K be lattices and f W L ! K 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 and L be a family of subsets of X , ordered by inclusion, such that (i) T i2I 2 L for every non-empty family fA i g i 2I Â L, and That is to say that L is a topped intersection structure on X . Then L is a complete lattice in which A i Â Bg: 11]). Let P and Q be ordered sets and ': P ! Q be an order-isomorphism map. Then ' preserves all existing joins and meets.
Definition 2.9 ( [11]). Let L be a complete lattice and let k 2 L.
The set of finite elements of L is denoted F .L/ (ii) k is said to be compact if, for every subset S of L, T f or some f i ni t e subset T of S: The set of compact elements of L is denoted K.L/.  ). Let P be a poset. If P has a least element 0, then x 2 P is called an atom if 0 x. If P has no least element, then x 2 P 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 A 0 .P / D A.P / [ f0g and P 0 D P nA 0 .P /. The poset P is called atomic if, given a.¤ 0/ in P , there exists x 2 A.P / such that x Ä a.
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.
C -operator and D-operator can be viewed as operators between A.P / and P 0 . Naturally, we can define C A D S a2A C a for any A Â P 0 , and D B D S b2B D b for any B Â P 0 . Then we can define two kinds of operators between A.P / and P 0 via C a and D b . Based on C -operators and D-operators, we generate several new relational operators as follows: Definition 2.15. Let P be an atomic poset.
In Definition 2.14 and 2.15, these operators are very reasonable.
Then, we study some properties of several operators.

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 (C A , C A , D B and D o B ) 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.

Complete lattices via .C
The set of all those pairs of P is denoted by B.P /.
We can then see easily that the relation Ä is an order on B.P /. As we can see in Theorem 3.1, < B.P /I Ä> is a complete lattice.
Theorem 3.1. Let P be an atomic poset. Then < B.P /I Ä> is a complete lattice in which join and meet are given by _ i2I . (4) in Proposition 2.16, implies that By Theorem 2.7, B.P 0 / 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 (7) in Proposition 2.16, and S i2I A i Â A, by (2) in Proposition 2.16.
Hence A is an upper bound for fA i g i2I in B.P 0 /. Also, if X is an upper bound in B.P 0 / for fA i g i 2I , then 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 Proof. Suppose .L; v/ is a complete lattice. Define the atomic poset P D f0g S A.P / S P 0 , where A.P / D L and P 0 D L. Further, in P , let akb for 8a; b 2 A.P /; let a Ä b iff a v b for 8a 2 A.P /; b 2 P 0 ; let a Ä b iff akb for 8a; b 2 P 0 . As L is a lattice, it is easy to see that P is an atomic poset. We want to show that .L; v/ is order-isomorphic to B.P /.
First note that for any X Â P 0 , we have x/ means the upper set of x in L. On the other hand, for any Y Â A P , D Y Dfa 2 P 0 jY Â C a g Dfa 2 P 0 j8y 2 Y; y Ä ag In other words, B.P 0 / are precisely the up-closed subsets of L generated by a single element. Hence, a subset of P 0 belongs to B.P 0 / if and only if it is a principal filter.
Example 3.4. Let L D fa; b; c; d g 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, a 1 and a 2 in P is a in L, b 1 and b 2 in P is b in L, c 1 and c 2 in P is c in L, d 1 and d 2 in P is d in L. In B.P /, A D .fd 2 g; fa 1 ; b 1 ; c 1 ; d 1 g/, B D .fb 2 ; d 2 g; fa 1 ; b 1 g/, C D .fc 2 ; d 2 g; fa 1 ; c 1 g/, D D .fa 2 ; b 2 ; c 2 ; d 2 g; fa 1 ; b 1 ; c 1 ; d 1 g/.
We can then see easily that the relation Ä is an order on B o .P /. As we see in Theorem 3.5, < B o .P /I Ä> is a complete lattice.
Proof. We shall prove that B o .P / is a topped intersection structure. Let A i 2 B o .P / for i 2 I . Then D o , which shows that B o .P / is topped. By Theorem 2.7, B o .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 Therefore A is indeed the required join. We may now appeal to Theorem 2.8 to deduce that B o .P / is a complete lattice in which joins and meets are given by Theorem 3.6. For every complete lattice L, there is an atomic poset P such that L is order-isomorphic to B o .P /.
Proof. Suppose .L; Ä/ is a complete lattice. Define the atomic poset P D A.P / S P 0 , where A.P / D L and P 0 D L. Further, in P , .1/ let akb for 8a; b 2 A.P /; .2/ let a Ä b for 8a 2 A.P /; b 2 P 0 iff a D b in case that a is the least element in L, a Ä b in case that b is the largest element in L, in other cases a ‹ b 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; Ä/ is order-isomorphic to B.P /.
First note that for any X Â P 0 which does not contain the least and largest elements, we have Among them, " L .x/ means the upper set of x in L. On the other hand, If X.Â P 0 / which contains the least or largest elements, we can easily check that D o C X D #. W X /. Therefore, we have fD o C X jX Â P 0 g D f#xjx 2 Lg. Hence, X Â B o .P / iff X D # L . W X /. In other words, B o .P / are precisely the lower sets of L generated by a single element. It is obvious that B o .P / is isomorphic to L. Example 3.7. Let L D fa; b; c; d g 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 lattice B o .P / which is isomorphic to L by Theorem 3.5. In Figure 2, a 1 and a 2 in P is a in L, b 1 and b 2 in P is b in L, c 1 and c 2 in P is c in L, d 1 and d 2 in P is d in L. In B o .P /, A D fa 2 g, B D fa 2 ; b 2 g, C D fa 2 ; c 2 g, D D fa 2 ; b 2 ; c 2 ; d 2 g.

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(C A , C A , D B and D o B ) 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.

Algebraic lattices via .C A ; D B ; Â/
Let P be an atomic poset. A set A satisfies A Â P 0 and for every finite subset X Â A, D C X Â A. The set of all those sets of P is denoted by F.P /. For We can see easily that the relation Ä is an order on F.P /. As we can see in Theorem 4.1, < F.P /I Ä> is an algebraic lattice. Proof. We first show that < F.P /I Ä> is a complete lattice. To show that < F.P /I Ä> is a complete lattice it suffices to show that < F.P /I Ä> is a topped intersection structure by Lemma 2.7. Given any subset T Â F.P /, it suffices to show that T T 2 F.P /. Suppose X is a finite subset of T T . Then X Â t for each t 2 T . Since each t 2 F.P / , we have D C X Â t for each t Â T . This implies D C X Â T T and so T T 2 F.P /. It is easy to see L 0 2 F.P / and so < F.P /I Ä> is a topped intersection structure.
To show that < F.P /I Ä> is algebraic, note that D C X is a compact element for each finite X in L 0 . To see this, we first show D C X 2 F.P /. Let X 1 be a finite subset of D C X , then D C X 1 Â D C .D C X / D D C X by Proposition 2.16, which implies D C X 2 F.P /. Then let fA i ji 2 I g be a directed subset of F.P / such that By Proposition 2.16, X Â D C X . Therefore X Â S i2I A i . Since X is finite and fA i ji 2 I g is directed, X Â A k for some k 2 I . But A k 2 F.P /, therefore D C X Â A k . 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 2 F.P /, T D S fD C X jX Â f i n T g. For any X Â f i n T , as T 2 F.P /, we have D C X Â T . Then S fD C X jX Â f i n T g Â T . As X Â D C X , So T D S X Â S fD C X jX Â f i n T g. Therefore T D S fD C X jX Â f i n T g. Therefore < F.P /I Ä> forms an algebraic lattice by Definition 2.11.
Corollary 4.2. Let P be a finite atomic poset. Then F.P / D fD C X jX Â Lg.
Proof. First we will show F.P / Â fD C X jX Â P g. 8A 2 F.P /, we have A Â P 0 and for every finite subset X Â A, D C X Â A. As P is finite, we have that A is finite and D C A Â A. Since A Â D C A by Proposition 2.16, therefore A D D C A . So F.P / Â fD C X jX Â P g. Then we will show fD C X jX Â P g Â F.P /. Since we show in Theorem 4.1, D C X is a compact element in < F.P /I Ä> for each finite X in P 0 . Since P is finite, so fD C X jX Â Lg Â F.P /. Therefore F.P / D fD C X jX Â P g.
As D is an algebraic lattice, it is easy to see that P is an atomic poset. We want to show that .D; v/ is order-isomorphic to F.P /.
First note that for any X Â P 0 , we have 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 P 0 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 7 ! fa 2 K.D/ja v d g that is, F.P / is isomorphic to D.  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, a 1 and a 2 in P is a in D, b 1 and b 2 in

Algebraic lattices via .C
Let P be an atomic poset. A set A satisfies A Â P 0 and for every finite subset X Â A, D o C X Â A. The set of all those sets of P is denoted by F o .P /. For We can see easily that the relation Ä is an order on F o .P /. As we can see in Theorem 4.5, < F o .P /I Ä> is an algebraic lattice. Proof. We first show that < F o .P /I Ä> is a complete lattice. To show that < F o .P /I Ä> is a complete lattice it suffices to show that < F o .P /I Ä> is a topped intersection structure by Lemma 2.7. Given any subset T Â F o .P /, it suffices to show that It is easy to see P 0 2 F o .P / and so < F o .P /I Ä> is a topped intersection structure.
To show that < F o .P /I Ä> is algebraic, note that D o C X is a compact element for each finite X in P 0 . To see this, we first show Then let fA i ji 2 I g be a directed subset of F o .P / such that By Proposition 2.16, X Â D o C X . Therefore X Â S i2I A i . Since X is finite and fA i ji 2 I g is directed, X Â A k for some k 2 I . But A k 2 F o .P /, therefore D o C X Â A k . By Definition 2.9 and Lemma 2.10, D o C X is a compact element for each finite X .
Next we will show that for any T 2 F o .P /, T D S fD o C X jX Â f i n T g. For any X Â f i n T , as T 2 F o .P /, Therefore < F o .P /I Ä> forms an algebraic lattice by Definition 2.11.
Corollary 4.6. Let P be a finite atomic poset. Then F o .P / D fD o C X jX Â Lg.
Proof. First we will show F o .P / Â fD o C X jX Â P g. 8A 2 F o .P /, we have A Â P 0 and for every finite subset Then we will show fD o C X jX Â P g Â F o .P /. As we show in Theorem 4.5, D o C X is a compact element in < F o .P /I Ä> for each finite X in P 0 . As P is finite, so .
As D is an algebraic lattice, it is easy to see that P is an atomic poset. We want to show that .D; Ä/ is orderisomorphic to F o .P /.
First note that for any X.Â P 0 / which does not contain the least and largest elements, we have C X Dfb 2 A.P /j9x 2 X; b Ä xg D o C X Dfa 2 P 0 jC a Â C X g Dfa 2 K.D/jDn"a Â Dn". _ X /g Dfa 2 K.D/j"a Ã ". _ X /g If X.Â P 0 / which contains the least or largest element, we can easily check that D o C X D #. W X /. Therefore, I 2 F o .P / iff D o C X Â I for any finite subset X Â f i n I , or # K.D/ . _ X / Â 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 P 0 belongs to F o .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 7 ! fa 2 K.D/ja Ä d g that is, F o .P / is isomorphic to D.  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 lattice F o .P / which is isomorphic to L by Theorem 4.5. In Figure 4, a 1 and a 2 in P is a in D, b 1 and b 2 in P is b in D, c 1 and c 2 in P is c in D, d 1 and d 2 in P is d in D. In F o .P /, A D fa 2 g, B D fa 2 ; b 2 g, C D fa 2 ; c 2 g, D D fa 2 ; d 2 g, E D fa 2 ; b 2 ; c 2 ; d 2 ; e 2 g.

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.