Scott convergence and fuzzy Scott topology on L-posets

Abstract We firstly generalize the fuzzy way-below relation on an L-poset, and consider its continuity by means of this relation. After that, we introduce a kind of stratified L-generalized convergence structure on an L-poset. In terms of that, L-fuzzy Scott topology and fuzzy Scott topology are considered, and the properties of fuzzy Scott topology are discussed in detail. At last, we investigate the Scott convergence of stratified L-filters on an L-poset, and show that an L-poset is continuous if and only if the Scott convergence on it coincides with the convergence with respect to the corresponding topological space.


Introduction
Ordered structure and topological structure are two basic and crucial structures in Mathematics, closely related to each other. Many works have been done to compare and combine the two structures [1][2][3][4][5][6]. At the beginning, classical Scott convergence and Scott topology are, in view of the theory of continuous lattices, only defined for complete lattices. Not very soon, these definitions have been found to be very fruitful for dcpos [2]. Unfortunately, they are not fit for arbitrary partially ordered sets (posets), since the join of a directed subset is involved in the definition of Scott convergence, which may not exist in a poset. Regarding this, several alternative choices have been proposed to generalize the definition of Scott convergence in posets [1,[6][7][8][9][10][11], and the Scott topology related to Scott convergence has also been studied.
In recent years, quantitative domain theory has attracted many people because it provides a model of concurrent systems. Wagner's -categories [12], Rutten's generalized metric spaces [13] and Flagg's continuity spaces [14] are examples of quantitative domain theory. Fan and Zhang [15,16] studied quantitative domain via fuzzy set theory, where fuzzy partial order was clearly proposed. After analysis, it is easily seen that -categories could be regarded as a fuzzy preordered set in [15,16], and a fuzzy partial ordered set (an L-poset) is equivalent to an L-ordered set introduced by Bělohlávek [17,18]. Later on, Lai and Zhang [19,20] studied complete and directed-complete -categories, and their continuity was also discussed. Following [15,16,19], Yao [21,22] studied the continuity of fuzzy dcpos, and further extended the Scott convergence and Scott topology on classical dcpos to fuzzy setting. But the results in [22] do not adapt to fuzzy partially ordered sets as well as in the classical case, even the continuity needs to be modified. This provides sufficient motivations for this paper. We firstly redefine the fuzzy way-below relation on L-posets and reconsider the continuity. Then we introduce a kind of stratified L-generalized convergence structure on L-posets, and restudy fuzzy Scott topology associating with it. Finally, we establish the Scott convergence theory on L-posets, and prove that it is an effective tool to characterize the continuity.
The paper is organized as follows. In Section 2, we recall some necessary definitions and results needed later on. In Section 3, we give a fuzzy way-below relation on L-posets and based on that the continuity for L-posets is considered. In Section 4, we introduce a new stratified L-generalized convergence structure on L-posets, then study and characterize fuzzy Scott topology. In Section 5, the properties of Scott convergence are given, the description for continuous L-posets via Scott convergence is constructed. In the final section, we summarize the results and draw a conclusion.
Let X be a nonempty set, L X denote the set of all L-subsets of X . 8A; B 2 L X , define: x/. Then .L X ; ; !; _;^; N 0; N 1/ is also a complete residuated lattice, and we never discriminate the constant value function N a with a, e.g., .a A/.x/ D a A.x/ and .a ! A/.x/ D a ! A.x/ for every x 2 X . A complete residuated lattice L with D^is just a complete Heyting algebra (or a frame). Throughout this paper, L always denotes a complete Heyting algebra.
Remark 2.2. In [17,18], an L-preordered set is defined to be a triple .X; R; /, where is an L-equality on X and R is an L-preorder on X which is compatible with . It is verified in [18,22] that if R is compatible with , it must hold that D R^R op . Thus, the L-equality is completely determined by R, so it can be omitted in the definition. Definition 2.4 ( [16,21,22]). Let .X; e/ be an L-poset and z 2 X , A 2 L X . Then A.y/^e.x; y//, and "A 2 L X is defined dually. (4) #z 2 L X is defined by 8x 2 X; #z.x/ D e.x; z/, and "y 2 L X is defined by 8x 2 X; "z.x/ D e.z; x/.
Moreover, A is called a lower L-set or fuzzy lower set if A.x/^e.y; x/ Ä A.y/ for all x; y 2 X . A is called an upper L-set or fuzzy upper set if A.x/^e.x; y/ Ä A.y/ for all x; y 2 X .
Let .X; e/ be a complete L-lattice, and respectively, denote join and meet in the underlying poset e X D .X; Ä e /.
Definition 2.12 ( [21,22]). Let .X; e X /; .Y; e Y / be L-posets, f W X ! Y , g W Y ! X be L-order-preserving maps. Then .f; g/ is called a fuzzy Galois connection between X and Y if e Y .f .x/; y/ D e X .x; g.y// for all x 2 X; y 2 Y , where f is called the left adjoint of g and dually g the right adjoint of f .

The continuity of L-posets via fuzzy way-below relation
Way-below relation was first imported for investigating the continuity of complete lattices. It was also an effective tool to describe the continuity of dcpos. This observation had inspired several authors to study continuous posets [30][31][32]. Unfortunately, these works on posets were rather restrictive since the definition of the way-below relation only considers certain directed subsets, of which join exists. In view of this deficiency, Erné [33] introduced another way-below relation on posets, and studied the continuous posets via it. However, in the quantitative domain theory, the fuzzy way-below relation introduced on fuzzy dcpos [21,22] has the same deficiency and is not fit for fuzzy posets either. Thus for a fuzzy poset it is necessary to define a reasonable fuzzy way-below relation and reconsider its continuity. Then .X; e/ is said to be continuous if +y is directed and t+y D y.
Note that each of the sets +y is a lower L-set (but in general not directed), and +W X X ! L can be regarded as the fuzzy way-below relation on .X; e/.
When .X; e/ is a fuzzy dcpo, then for every I 2 I L .X /, Thus, the fuzzy way-below relation as above is compatible with the usual one for fuzzy dcpos [21,22]. Some fundamental properties of the fuzzy way-below relation are listed in the following proposition.

Fuzzy Scott topology on L-posets
Classical Scott topology on complete lattices and dcpos is studied in [2]. After that, many works have been done to generalize that theory on posets [1,6]. Recently, fuzzy Scott topology has been investigated on fuzzy ordered sets with the necessary condition that fuzzy joins of all directed fuzzy set exist (i.e. dcpos) [22]. In the absence of any sort of join, the previous result is invalid, so an additional consideration for fuzzy Scott topology on L-posets is needed. This is our motivation for this section. The set of all stratified L-filters on X will be denoted by F s L .X /.  (2) Let .X; / be an L-fuzzy topological space and x 2 X . Define U x W L X ! L by

B.x/^ .B/:
Then U x is an L-filter, and it is stratified if is enriched.
(3) Let .X; ı/ be an L-topological space and x 2 X . Define U x ı W L X ! L by 8A 2 L X ; U x ı .A/ D A ı .x/, where ı is the L-interior operator of .X; ı/. Then U x ı is an L-filter, and if ı is stratified then so is U x ı .
Definition 4.4 ( [34,35,38]). A stratified L-generalized convergence structure on X is a map R W F s L .X / X ! L satisfying that (1) 8x 2 X; R.OEx; x/ D 1;   34]). Each stratified L-generalized convergence structure R on X induces an enriched L-fuzzy topology R on X given by and a stratified L-topology ı R D fA 2 L X W R .A/ D 1g.
Elicited by the well-known results, we aim to study topologies on an L-poset, then the consideration of a kind of convergence structures on it will be effective. To reach that goal we begin with the discussion of the lower bound of a stratified L-filter.
Let .X; e/ be an L-poset and F 2 F s L .X /. Define F l 2 L X by Proposition 4.6 ( [22]). Let .X; e/ be an L-poset and F; G 2 F L .X /. Then For a fuzzy ideal on .X; e/, define F I W L X ! L by .I.x/^sub."x; A// : Then F I is a stratified L-filter on X , and F l I D I (refer to [22] for detail). Let .X; e/ be an L-poset. Define a map S W F L .X / X ! L by It is easily seen that S is a stratified L-generalized convergence structure on X , and S.F; x/ can be interpreted as the degree of F Scott converges to x. Moreover, we define U x S W L X ! L by Then U x S is a stratified L-filter. By Theorem 4.5, there is an enriched L-fuzzy topology associated with S . We denote it as LF .X; e/ ( LF .X/ for short), that is,  Proof. First, *x is an upper L set obviously for 8x 2 X . For 8y 2 X; 8I 2 I L .X /,we have *x.y/ D +y.x/ D W z2X .+z.x/^+y.z//. Since +y.z/ D V J 2I L .X/ .J ul .y/ ! J.z// Ä I ul .y/ ! I.z/, so .*x.z/^I.z// :  It is worth noting that * x 2 L .X / in a continuous L-poset .X; e/ since L .X / is stratified.
Theorem 4.11. Let .X; e/ be a continuous L-poset, then f* x W 2 L; x 2 X g is a basis of L .X /.

Scott convergence on L-posets
Usually, convergence theory can not be ignored when considering topology. As shown before, on an L-poset, Lfuzzy Scott topology and fuzzy Scott topology naturally exist. A deeper problem arises in order to be compatible with the convergence under the related topology: how to define a fruitful convergence on an L-poset? This section will give the answer.
Definition 5.1. Let .X; e/ be an L-poset, x 2 X and F a stratified L-filter on X . Then we say F is Scott convergent to x if there exists I 2 I L .X / such that I Ä F l and I ul .x/ D 1. We denote this by F ! s x. Recall that for an L-fuzzy topology on X , we call a stratified L-filter F is convergent to x 2 X , denoting F ! x, if U x Ä F. Under an L-topology ı on X , a stratified L-filter F convergent to x 2 X, denoting So U x L .X/ Ä U x LF .X / , as needed.
Theorem 5.8. Let .X; e/ be a continuous L-poset, x 2 X and F a stratified L-filter on X . Then F is Scott convergent to x iff +x Ä F l .
Proof. The sufficiency is obvious. To show the necessity, assume that F is Scott convergent to x. There exists I 2 I L .X / such that I Ä F l and I ul .x/ D 1. For all y 2 X , +x.y/ Ä I ul .x/ ! I.y/ Ä F l .y/. Thus +x Ä F l .

Conclusion
In this paper, we first extend the fuzzy way-below relation on fuzzy dcpos to fuzzy ordered sets without any additional conditions, and based on that, the continuity for L-posets is studied. Later on, we propose a kind of stratified Lgeneralized convergence structure, and then study fuzzy Scott topology. The Scott convergence theory on L-posets is established finally, and the continuity is well described by Scott convergence. That is, an L-poset is continuous if and only if Scott convergence coincides with convergence under either L-fuzzy Scott topology or fuzzy Scott topology. All the works will promote the development of quantitative domain theory.