# CaveSeg: Deep Semantic Segmentation and Scene Parsing for Autonomous
Underwater Cave Exploration

Adnan Abdullah1, Titon Barua2, Reagan Tibbetts2, Zijie Chen1, Md Jahidul
Islam1, and Ioannis Rekleitis2
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
1RoboPI Laboratory, Department of ECE, University of Florida, FL 32611, USA.
2AFRL Laboratory, Department of CSE, University of South Carolina, SC 29208,
USA.  This pre-print is in review at the ICRA 2024. [as of 09/23]

###### Abstract

In this paper, we present CaveSeg - the first visual learning pipeline for
semantic segmentation and scene parsing for AUV navigation inside underwater
caves. We address the problem of scarce annotated training data by preparing a
comprehensive dataset for semantic segmentation of underwater cave scenes. It
contains pixel annotations for important navigation markers (_e.g_. caveline,
arrows), obstacles (_e.g_. ground plain and overhead layers), scuba divers,
and open areas for servoing. Through comprehensive benchmark analyses on cave
systems in USA, Mexico, and Spain locations, we demonstrate that robust deep
visual models can be developed based on CaveSeg for fast semantic scene
parsing of underwater cave environments. In particular, we formulate a novel
transformer-based model that is computationally light and offers near real-
time execution in addition to achieving state-of-the-art performance. Finally,
we explore the design choices and implications of semantic segmentation for
visual servoing by AUVs inside underwater caves. The proposed model and
benchmark dataset open up promising opportunities for future research in
autonomous underwater cave exploration and mapping.

##  1 Introduction and Background

Figure 1: (a) A tethered BlueROV2 is operating inside an underwater cave
system in FL, USA; it is teleoperated by a surface operator following the
_caveline_ as a navigation guide; (b) the corresponding POV from the robot’s
camera; (c) the proposed semantic parsing concept is shown; the envisioned
capabilities are: first-layer & second-layer obstacle avoidance, ground plain
estimation, and caveline detection, following, and 3D estimation – to enable
autonomous robot navigation inside underwater caves.

Underwater cave formations, sediment properties, and water chemistry provide
insights into the world’s past climate conditions and geological processes [1,
2]. Underwater caves also play a crucial role in monitoring and tracking
groundwater flows in Karst topographies, while almost $25\%$ of the world’s
population relies on Karst freshwater resources [3]. Despite the importance,
underwater cave exploration and mapping by humans is a tedious, labor-
intensive, and extremely dangerous operation, even for highly skilled divers
[4]. When a new section of a cave is discovered, a single and continuous line
termed _caveline_ [5] is laid out identifying the skeleton of the main
passages. The caveline is attached to other navigation guides such as _arrows_
and _cookies_ [6], marking the orientation of the cave, distance to the
entrance and presence of other divers. Such surveys by the explorers produce a
one-dimensional retraction of the 3D environment. Recording all this
information together with additional observations [7] is a challenging, time-
consuming, and error-prone process.

Enabling Autonomous Underwater Vehicles (AUVs) and Remotely Operated Vehicles
(ROVs) to navigate, explore, and map underwater caves safely and effectively
is of significant importance [8, 2]; Fig. 1(a) shows an ROV deployment
scenario inside the Orange Grove Cave System in Florida. Our earlier work
[C93] developed a high-precision camera trajectory estimation method by a
Visual-Inertial Odometry (VIO) algorithm [J17], which can generate 3D caveline
trajectory estimates that are comparable to the manually surveyed
measurements. More recently in [6], we addressed the lack of annotated samples
in visual learning for caveline detection and tracking across different scenes
for AUV navigation.

In this work, we focus on developing a deep visual learning pipeline to
extract dense semantic information for autonomous underwater cave exploration
by mobile robots. Considering the limited onboard resources available on
embedded platforms, our objective is to design a computationally light model
that can (learn to) identify the navigation markers of underwater caves
(_e.g_. caveline, arrows), obstacles to avoid (_e.g_. ground plain and
overhead layers), scuba divers (for cooperative missions), and safe open areas
for visual servoing in real-time. We identify two major difficulties to
achieve these: (i) no large-scale datasets are available for underwater cave
environments; (ii) the state-of-the-art (SOTA) models for semantic scene
parsing are computationally too demanding for robotic platforms.

We address these challenges by proposing CaveSeg, the first large-scale
semantic segmentation dataset and learning pipeline for underwater cave
exploration. We collect comprehensive training data by ROVs and scuba divers
through robotics trials in three major locations [6]: the Devil’s system in
Florida, USA; Dos Ojos Cenote in QR, Mexico; and Cueva del Agua in Murcia,
Spain. Our processed data contain $2300$ pixel-annotated samples with $10$
object categories that include first and second layer obstacles, human scuba
divers, and navigation aids (see Sec 3; Fig. 2). We also compile a CaveSeg-
Challenge test set that contains $250$ samples from unseen waterbody and cave
systems such as the Blue Grotto and Orange Grove cave systems in FL, USA. We
conduct extensive benchmark evaluation of SOTA models across the Convolutional
Neural Network (CNN) [9, 10], Conditional Random Fields (CRF) [11, 12], and
Vision Transformer (ViT) [13, 14, 15] literature, which validate that robust
semantic learning is feasible on CaveSeg.

Moreover, we develop a novel CaveSeg model by rigorous design choices to
balance the robustness-efficiency trade-off. The proposed model consists of a
transformer-based backbone, a multi-scale pyramid pooling head, and a
hierarchical feature aggregation module for robust semantic learning (see
Sec.4). Experimental evaluation and comparison with SOTA models suggest that
the CaveSeg model is over $3\times$ more memory efficiency and offers
$1.8\times$ faster inference than SOTA models, while providing comparable
benchmark performances. A series of experiments on unseen challenging scenes
prove the robustness of our model across different waterbody conditions and
optical artifacts; see more details in Sec.5.

Furthermore, we highlight several challenging scenarios and practical use
cases of CaveSeg for real-time AUV navigation inside underwater caves. Those
scenarios include safe cave exploration by caveline following and obstacle
avoidance, planning towards caveline rediscovery, finding safe open passages
and exit directions inside caves, giving uninterrupted right-of-way to scuba
divers exiting the cave, and 3D semantic mapping and state estimation. We
demonstrate that CaveSeg-generated semantic labels can be utilized effectively
for vision-based cave exploration and semantic mapping by AUVs (see more in
Sec. 6).

##  2 Related Work

Figure 2: A few sample images from the proposed CaveSeg dataset, corresponding
ground truth labels, and their overlayed visualizations are shown; color codes
for each object category are listed on the right.

###  2.1 Underwater Cave Exploration and Mapping

Exploration and mapping of underwater caves by human divers traditionally
employed photogrammetry [16] methods in order to generate informative
photorealistic representation, especially for sites with archaeological
interest [17, 1]. Autonomous underwater cave mapping by AUVs remains an open
problem due to many challenges. Mallios _et al_. [18] manually moved an AUV
collecting acoustic data for offline mapping, and Weidner _et al_. [C68, O4]
used images from a stereo camera to create a 3D reconstruction of the cave
walls, floor, and ceiling. Major challenges to vision-based state estimation
in an underwater environment include lighting variations, light absorption,
and blurriness [C86]. Rahman _et al_. [C77, C85, J17] proposed a framework
where visual, acoustic, inertial, and water depth data are fused together to
estimate the trajectory of an AUV or a sensor while in parallel generates a
sparse representation of the underwater cave. Denser models of the cave
boundaries can be obtained by mapping the moving shadows [C88], the contours
[19], or dense online stereo reconstruction [C98].

More recently, Richmond  _et al_. [2] describe a man-portable AUV named
_Sunfish_ , which can work safely inside underwater caves and bring back
chemical profiles, detailed imagery, and sonar maps of the cave. Our recent
work [6] develops a ViT-based caveline detection model for autonomous caveline
following by underwater robots. However, beyond detecting cavelines, a full-
form scene parsing is essential for safe and effective AUV navigation inside
underwater caves.

###  2.2 Semantic Segmentation of Underwater Scenes

Deep visual learning algorithms have revolutionized semantic segmentation and
scene parsing benchmarks on standard datasets, which are mostly developed for
terrestrial applications. The existing learning pipelines trained on
terrestrial imagery are not directly applicable because the object categories
and image statistics are entirely different. A unique set of underwater image
distortion artifacts and the unavailability of large-scale annotated datasets
have influenced a significant lack of research attempts on semantic
segmentation of underwater imagery. The contemporary research literature
offers application-specific image datasets for coral reef segmentation and
coverage estimation [20, 21], visual attention modeling [22], human-robot
cooperative missions [23], image restoration and foreground enhancement [24],
and fish detection [25, 26].

Figure 3: Frequencies and distributions of important object categories are
shown in the train, validation, and test sets.

More recent work by Modasshir  _et al_. combined a deep learning-based
classifier model to identify and track the locations of different types of
corals to generate semantic maps [C81] as well as volumetric models [C83].
Islam  _et al_. formulated the SUIM dataset [23] for semantic segmentation of
underwater imagery with eight object categories: fish, coral reefs, aquatic
plants, wrecks/ruins, human divers, robots/instruments, and sea-floor. Other
datasets consider even fewer object categories such as marine debris or ship
hull defects [27]. With limited training samples per object category over only
a few waterbody types, it is extremely challenging to achieve good
generalization performance by SOTA deep learning-based models for image
segmentation and scene parsing. More importantly, these object categories are
not useful for underwater cave exploration and mapping applications – which we
address in this paper.

##  3 CaveSeg Dataset: Data Preparation and Problem Formulation

Figure 4: The network architecture of our proposed CaveSeg model is shown.
Input images are partitioned into $4\times 4$ patches and fed into a four-
stage transformer backbone for coarse-to-fine feature extraction. The
extracted multi-scale features are then pooled and combined by the PPM head
for bottom-up and top-down feature aggregation. A hierarchical feature map is
then compiled by merging several multi-level feature representations. On this
feature space, a classifier performs pixel-level semantic segmentation to
generate the final outputs.

We prepared learning pipelines with data collected from three cave systems in
different geographical locations [6]: the Devil’s system in Florida, USA; Dos
Ojos Cenote in QR, Mexico; and Cueva del Agua in Murcia, Spain. For the
semantic labels, we considered the following object categories: caveline,
first layer (immediate avoidance areas), second layer (areas to be avoided
subsequently), open area (obstacle-free regions), scuba divers, navigation
aids (_e.g_., arrows, reels, and cookies), and caveline-attached rocks.
Moreover, stalactites, stalagmites, and columns are also considered if they
are present in the scene, which is the case for the Mexico caves. With these
$10$ object categories in consideration, a total of $2300$ images are labeled
in the CaveSeg dataset. A few annotated samples are shown in Fig. 2; the
dataset and relevant information are available online at
https://robopi.ece.ufl.edu/caveseg.html.

As mentioned earlier, the role of the caveline is crucial for any underwater
cave operations. It represents the direction of exploration, the main area
where the cave extends, and equally important, the path to safely exit the
cave. Caveline pixels are marked as yellow in order to generate maximum
contrast. The rock formations where the caveline is attached are often called
_placements_ or _attachment points_. These rocks most of the time signal a
change in the direction of the caveline and they are marked in blue.
Navigational markers such as _arrows_ (dark red) and _cookies_ (red) provide
important information about the direction to the nearest cave exit and the
presence of other divers in the cave, respectively.

A special category in our semantic mapping scheme is the scuba divers
(magenta). An AUV should always give the right of way to divers, especially
those exiting the cave. In case of an emergency, there should be nothing
impeding the divers from reaching the surface, which is a norm practiced by
cave divers. As such we have established a human diver category to incorporate
emergency responses when a diver is detected, for example, lowering the light
intensity, moving away from the main passage, avoiding abrupt motions, etc.

Fig. 3 shows the frequency and distribution of various object categories in
CaveSeg dataset. Human diver is present in $15\%$ of the images. Over $90\%$
of the samples contain caveline, obstacle-free open areas, as well as the
first and second layer obstacles. Other navigation markers (_e.g_., cookies,
arrows, reels) typically occupy smaller pixel areas compared to other objects,
and they are found in $10\%$ of the data. Overall, these $10$ object
categories in the scene embed useful information for vision-based planning and
navigation for AUVs inside underwater caves.

##  4 CaveSeg Model: Semantic Scene Segmentation Of Underwater Caves

###  4.1 Network Design and Learning Pipeline

We search for a computationally light architecture that provides real-time
underwater cave scene segmentation in addition to achieving state-of-the-art
(SOTA) performance. To this end, we explore both CNN-based and transformer-
based backbone architectures [28, 29, 13]. The windowed multi-head self-
attention (W-MSA) module proposed in Swin Transformer [15] is a powerful tool
to maintain efficient computation. Additionally, the window shifting technique
connects features across spatially overlapped windows. We find this connection
to be particularly useful since cave scenes hold some spatial relation among
categories. For instance, navigation markers and attachment rocks are almost
always found on the caveline, while the second layer obstacles are usually
found around the first layer or ground plain. Although computationally heavy,
we found such cross-category attention extraction to be effective in cave
scene segmentation tasks. Inspired by this, we design a novel architecture
that incorporates these capabilities with a light Swin Transformer backbone
for feature extraction. With an efficient Pyramid Pooling Module (PPM) and
hierarchical feature aggregation, our proposed CaveSeg model is over
$3.3\times$ memory efficient and offers $60\%$ faster inference rates than the
Swin Transformer base model.

####  4.1.1 CaveSeg Model Architecture

The detailed network architecture is shown in Fig 4. First, input RGB images
are partitioned into $4\times 4$ non-overlapping patches or tokens; a linear
embedding layer of dimension $48$ is then applied to each token. These feature
tokens are passed through a four-stage backbone, each containing a windowed
and shifted windowed module of multi-head self-attention [30]. In each stage,
patches are merged with $2\times 2$ neighboring regions to reduce the number
of tokens while at the same time, the linear embedded dimension is doubled.
Subsequently, bottom-up and top-down feature aggregation [31] is performed in
two separate branches. A pyramid pooling module (PPM) [32] is attached to the
backbone that further improves global feature extraction at deep layers of the
network [33]. Features from each stage of the backbone as well as from the PPM
head are then fused and compiled into a hierarchical feature map. Finally, a
fully connected convolution layer performs pixel-wise category estimation on
that feature space.

####  4.1.2 Supervised Learning Pipeline

The end-to-end training is driven by the standard cross-entropy loss [34] that
quantifies the dissimilarity in the generated and predicted pixel labels for
each category. For multi-class segmentation of an image with $N$ pixels and
$M$ classes, it is calculated as

${\mathcal{L}}_{CE}=-\frac{1}{N}\sum_{i=1}^{N}\sum_{c=1}^{M}y_{i,c}\log(p_{i,c}),$
(1)

where $p_{i,c}$ denotes the probability of pixel $i$ belonging to class $c$,
and $y$ is $0$ (or $1$) for correct (or incorrect) predictions, respectively.
The training is optimized by the Stochastic Gradient Descent (SGD) algorithm
[35] with an initial learning rate of $1e^{-4}$ and a momentum of $0.9$.

###  4.2 Training Setups for CaveSeg & Baseline Models

We configured a unified training pipeline for CaveSeg and several SOTA
benchmark models for training and evaluation. Specifically, we used the
MMSegmentation libraries [36] in PyTorch for the large-scale training on
CaveSeg dataset with a $75$:$15$:$10$ split ratio for train, validation, and
test, respectively. For baseline comparison, we considered the following SOTA
models across the CNN, CRF, and ViT literature: FastFCN [29], DeppLabV3+ [28],
Segmenter [14], Segformer [13], and Swin Transformer [15]. The proposed
CaveSeg model is pre-trained on ADE20k [37] followed by the unified training.
The input-output resolution is set to $960\times 540$ for all models; other
SOTA model-specific parameters are chosen based on their recommended
configurations.

Table 1: Semantic segmentation performances of all models are compared on $250$ test images from the CaveSeg-Challenge set; here, higher scores ($\uparrow$) are better for all metrics in consideration.  Method |  mIoU $\uparrow$ |  mAcc $\uparrow$ |  aAcc $\uparrow$
---|---|---|---
FastFCN [29] | $27.82$ | $39.30$ | $68.28$
DeepLabV3+ [28] | $30.22$ | $42.02$ | $68.61$
Segmenter [14] | $17.92$ | $24.14$ | $66.42$
Segformer [13] | $25.62$ | $34.09$ | $64.47$
Swin Transformer [15] | $32.06$ | $32.55$ | $66.53$
CaveSeg (proposed) | $32.64$ | $34.82$ | $68.01$
|  |  |

##  5 Performance Analyses of CaveSeg

Figure 5: A few qualitative performance comparisons of all models on CaveSeg-
Challenge test set are shown (results for only four top-performing models are
shown for clarity). Note that the object detection and localization accuracy
for categories such as caveline, open area, and navigation markers are
particularly important for AUV navigation.

###  5.1 Quantitative Evaluation

We use three standard metrics [38, 39] for quantitative assessments: mean
Intersection Over Union mIOU, mean class-wise Accuracy (mAcc), and Average
pixel Accuracy (aAcc). The IoU (Intersection Over Union) measures caveline
localization performance using the area of overlapping regions of the
predicted and ground truth labels. it is defined as $IoU=\frac{Area\text{
}of\text{ }overlap}{Area\text{ }of\text{ }union}$. On the other hand, mAcc and
aAcc represent the mean accuracy of each class category and over all pixels,
respectively. The quantitative results are presented in Table 1; all
evaluations are performed on the CaveSeg-Challenge set, which we curated with
$250$ test samples. These samples include low-light noisy scenarios compiled
in the CL-Challenge set (released earlier in [6]), as well as data from cave
explorations in other such as the Blue Grotto and Orange Grove cave systems in
FL, USA.

As Table 1 shows, our proposed dataset and learning pipelines can achieve up
to $68\%$ accuracy for pixel-wise segmentation. The proposed CaveSeg model
offers comparable performance margins despite having a significantly lighter
architecture. The comparisons for computational efficiency are provided in
Table 2. CaveSeg model has less than $50\%$ parameters than other competitive
models and offers up to $1.8\times$ faster inference rates. While we analyze
the class-wise performance, we find that small and rare object categories such
as arrows, cookies, and reels are challenging in general.

Table 2: Computational complexities for all models are compared based on number of parameters, memory requirements in Mega-Bytes (MB), and inference speeds in FPS. All experiments are performed on an Nvidia™ A$100$ GPU server with $16$ GB RAM. Method |  # Params $\downarrow$ |  Memory $\downarrow$ |  Speed $\uparrow$
---|---|---|---
FastFCN |  $64$ M |  $548.32$ MB |  $16.89$ FPS
DeepLabV3+ |  $58$ M |  $489.38$ MB |  $15.04$ FPS
Segmenter |  $86$ M |  $798.04$ MB |  $13.73$ FPS
Segformer |  $84$ M |  $956.62$ MB |  $10.92$ FPS
Swin Tx |  $121$ M |  $1380.16$ MB |  $12.39$ FPS
CaveSeg |  $38$ M |  $406.40$ MB |  $19.78$ FPS
|  |  |

###  5.2 Qualitative Evaluation

Figure 5 compares the qualitative performance of CaveSeg with other SOTA
models. A unique feature of CaveSeg and other transformer-based models is that
they perform better in finding large and connected areas, _e.g_., categories
such as first/second layer obstacles and open areas. The continuous regions
segmented by the CaveSeg model facilitate a better understanding of the
surroundings compared to DeepLabV3+ and FastFCN. This validates our intuition
that window-shifting technique can indeed extract global features more
accurately. While the shifted window method focuses on global feature
extraction, the deeper layers and multi-scale pooling of PPM help to preserve
the details of local features. The precise detection of caveline, which is
only a few pixels wide, further supports this design choice.

Figure 6: A BlueROV2 is being teleoperated through a cave passage where the
caveline is almost invisible; note that the yellow wire on the top image is
the ROV’s tether, not a caveline. In such scenarios, an AUV can leverage the
segmented obstacles and open areas in the scene (overlayed on the right
column) toward planning a trajectory to rediscover and follow the caveline.

##  6 Use Cases: Vision-based Cave Exploration and Semantic Mapping by AUVs

###  6.1 Safe AUV Navigation Inside Underwater Caves

The confined nature of the underwater cave environment makes obstacle
avoidance a particularly important task. Safe navigation approaches, such as
AquaNav proposed by Xanthidis _et al_. [C90] and especially AquaVis [C92] can
generate smooth paths by avoiding obstacles. To this end, having a holistic
semantic map of the scene can ensure that the AUV can safely follow the
caveline and keep it in the FOV during cave exploration. Our previous work [6]
demonstrates the utility of caveline detection and following for autonomous
underwater cave exploration. While caveline detection is paramount, having
semantic knowledge about the surrounding objects in the scene is essential to
ensure safe AUV navigation. As shown in Fig. 6, cavelines can be obscure in
particular areas due to occlusions and blends inside the cave passages. Hence,
dense semantic information provided by CaveSeg is important to ensure
continuous tracking and re-discovery of the caveline and other navigation
markers. A few more challenging scenarios are shown Fig. 7, where simply
following the caveline is not sufficient due to the cluttered scene geometry.
With the semantic knowledge of first layer and second layer obstacles,
caveline, and open areas – an AUV can plan its trajectory safely and
efficiently.

Figure 7: A few examples of semantic scene parsing by CaveSeg for important
navigation information are shown. The first scene shows a traceable caveline
towards the open area and surrounding obstacles. When no open areas are
visible (second image), arrows on the caveline are helpful to know the exit
direction of the cave. Lastly, the accurate detection of human divers is a
crucial step for giving the right-of-way as well as diver-robot cooperation.

###  6.2 Working With and/or Alongside Human Scuba Divers

The underwater cave environment is extremely hostile due to the lack of direct
access to the surface. Cave divers enforce strict protocols on light
configurations (always a primary and two backup lights), use of breathing gas
(use only $30\%$ on the way in, leaving $30\%$ for the return and $30\%$ for
emergencies), and always keep near the caveline, ensuring there is an
uninterrupted line to open water [5]. Of particular importance is the right of
way for an exiting team. As such, CaveSeg maintains a label for divers (see
Fig. 7) to ensure appropriate actions by the AUV. Specifically, in the
presence of a diver, the lights of the vehicle will be dimmed so that the
approaching diver can see the caveline. The AUV will reduce its speed, and
refrain from using the downward-facing propellers in order to avoid stirring
up the sediment; see Fig. 8 where the teleoperated ROV disturbed the sediment
on the floor. Finally, the vehicle will move away from the caveline towards
the walls of the cave.

Figure 8: A BluROV2 is being teleoperated inside the Blue Grotto cave, FL:
(left) entering the cave; (right) stirring the sediments.

###  6.3 3D Semantic Mapping and State Estimation

Another prominent use case of CaveSeg is the 3D estimation and reconstruction
of caveline for AUV navigation. Specifically, 2D caveline estimation can be
combined with camera pose estimation from Visual-Inertial-Odometry (VIO) to
generate 3D estimates. This could potentially be used to compare and improve
manual surveys of existing caves. Moreover, 3D caveline estimations can also
be utilized to reduce uncertainty in VIO and SLAM systems since the line
directions in 3D point clouds can provide additional spatial constraints [40].
We demonstrate a sample result in Fig. 9, where _ray-plane triangulation_ of
cavelines is achieved using Visual-Inertial pose estimation for enabling 3D
perception capabilities for underwater cave exploration. This experiment is
performed on field data collected by ROVs in the Orange Grove cave system, FL,
USA; we are currently exploring these capabilities for more comprehensive 3D
semantic mapping of underwater caves.

Figure 9: Ray-plane triangulation of cavelines using VIO pose estimation and
2D line segments from deep segmentation pipeline are shown. The reconstructed
3D Lines are colored based on average re-projection error within its
neighborhood in a local connectivity graph.

##  7 Conclusion and Future Work

This paper presents CaveSeg – the first comprehensive dataset and deep visual
learning pipeline for underwater cave scene segmentation. With the primary
focus on vision-guided cave exploration and mapping by AUVs, we include
semantic labels for object categories such as caveline, layered obstacles,
arrows, cookies, attachment points, reels, as well as human scuba divers. We
perform extensive benchmark evaluations of several SOTA models that
demonstrate the utility of such dataset for robust semantic segmentation of
underwater cave scenes. We further propose a computationally light model that
offers up to $1.8\times$ faster inference in addition to providing SOTA
performance. More importantly, we demonstrate that the predicted scene parsing
labels can be utilized for safe AUV navigation inside underwater caves. We are
currently investigating the scope of augmenting semantic maps with geometric
information from a Visual-Inertial SLAM system such as SVIn2 [J17]. This will
potentially reveal the 3D position of the caveline as well as the relative
location of obstacles and helpful markers. The synergy between geometry and
semantics will provide additional visual features to further disambiguate
weakly labeled areas during cave exploration. For future CaveSeg releases, we
will augment semantic labels for objects such as vertical columns, stalagmite,
and stalactite that are found in decorated caves.

##  8 Acknowledgements

This research has been supported in part by the NSF grants $1943205$ and
$2024741$. The authors would also like to acknowledge the help of the
Woodville Karst Plain Project (WKPP), El Centro Investigador del Sistema
Acuífero de Quintana Roo A.C. (CINDAQ), Global Underwater Explorers (GUE), and
Ricardo Constantino, Project Baseline in collecting data, providing access to
challenging underwater caves, and mentoring us in underwater cave exploration.
We also appreciate the help from Evan Kornacki for coordinating our field
experimental setups. Lastly, we thank the RoboPI lab members and former
students (Ailani Morales and Boxiao Yu) who helped us with image labeling and
annotation tasks for CL-ViT and in the early stages of this project.

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# Unbounded order convergence on infinitely distributive lattices

Kevin Abela  Kevin Abela
Department of Mathematics
Faculty of Science
University of Malta
Msida MSD 2080, Malta<EMAIL_ADDRESS>and  Emmanuel Chetcuti
Emmanuel Chetcuti, Department of Mathematics
Faculty of Science
University of Malta
Msida MSD 2080 Malta<EMAIL_ADDRESS>
###### Abstract.

In this article we study unbounded order convergence
($\mathfrak{u}O$-convergence) on infinitely distributive lattices. For a
sublattice $Y$ of an infinitely distributive lattice $\mathscr{L}$, we show
that the order closure and unbounded order closure are sublattices. We also
extend several results that hold for Riesz spaces, in particular [7, Theorem
3.15] and [9, Theorem 2.1, Theorem 2.13].

##  1\. Introduction

In the literature, order convergence has been thoroughly studied on Riesz
spaces, lattices and partially ordered sets [9, 13, 17, 20]. Thus, over the
years, one can find several different definitions of O-convergence. The
interested reader can look at [1] to see different definitions and under which
conditions these definitions agree or differ. A concept closely related to
order convergence is unbounded order convergence. Unbounded order convergence
($\mathfrak{u}O$-convergence) was first introduced by Hidegorô Nakano under
the name individual convergence [14, 15]. Later, Ralph DeMarr coined the
commonly used term unbounded order convergence [4]. For sequences
$\mathfrak{u}O$-convergence is generally studied on Riesz spaces due to its
natural relation to pointwise convergence on $\ell_{p}$ for $(1\leq
p\leq\infty)$. Furthermore, for sequences in $L_{p}(\mu)$ for $1\leq p<\infty$
and finite measure $\mu$, $\mathfrak{u}O$-convergence is also equivalent to
convergence almost everywhere. This relationship between almost everywhere
convergence and $\mathfrak{u}O$-convergence was further investigated by
Wickstead in [18]. He studied $\mathfrak{u}O$-convergence and weak convergence
in Banach lattices, and showed that for norm bounded nets, weak and
$\mathfrak{u}O$-convergence are equivalent. Samuel Kaplan studied
$\mathfrak{u}O$-convergence on Riesz spaces with a weak order unit [11]. He
showed that in a Riesz space with a weak order unit,
$\mathfrak{u}O$-convergence has a simpler form. This form was used to give a
new proof of a result by Hakano.

Gao and Xanthos showed that every weakly compact $\mathfrak{u}O$-convergent
net is norm convergent in Banach lattices with the positive Schur property.
The notion of $\mathfrak{u}O$-Cauchy nets was used to show that every relative
weakly compact $\mathfrak{u}O$-Cauchy net is $\mathfrak{u}O$-convergent in an
order continuous Banach lattice [10]. Niushan Gao studied
$\mathfrak{u}O$-convergence in the dual of Banach spaces. He showed that every
norm bounded $\mathfrak{u}O$-convergence net in $X^{*}$ is $w^{*}$-convergent
if and only if $X$ has order continuous continuous norm. Furthermore, every
$w^{*}$-convergent net in $X^{*}$ was shown to be $\mathfrak{u}O$-convergent
if and only if $X$ is atomic with order continuous norm [8].

A pivotal study on $\mathfrak{u}O$-convergence was done by Gao, Xanthos and
Troitsky [7]. They proved that $\mathfrak{u}O$-convergence passes freely to
and from regular Riesz sublattices. This was eventually used to improve
several results in [8, 10]. They also proved that a Riesz sublattice $Y$ in a
Riesz space $X$, is $O$-closed if and only if it is $\mathfrak{u}O$-closed.
The relationships between $\mathfrak{u}O$-closure and $O$-closure was further
investigated in [9]. Bilokopytov and Troitsky studied
$\mathfrak{u}O$-convergence in spaces of continuous functions, in particular
$C(X),C_{b}(X),C_{0}(X)$ and $C^{\infty}(X)$ where $X$ is a completely regular
Hausdorff topological space. They characterized $\mathfrak{u}O$-convergence in
$C(X)$. Furthermore, they proved that a sequence $\mathfrak{u}O$-converges if
and only if it converges pointwise on a co-meagre set [3].

Unbounded order convergence is generally studied on Riesz spaces, in this
article we consider $\mathfrak{u}O$-convergence on infinitely distributive
lattices. We study several properties that are true in Riesz spaces and extend
them to this setting. In particular we extend several results found in [7, 9].

##  2\. $\mathfrak{u}$O-convergence on Infinitely distributive lattices

Recall that a subset $D$ of an Archimedean Riesz space $X$ 111Unless otherwise
stated $X$ is assumed to be an Archimedean Riesz space. is directed (filtered)
provided it is non-empty and every finite subset of $D$ has an upper bound
(lower bound) in $D$. Also, $Y$ is said to be regular if for every subset
$A\subseteq Y$, $\inf A$ is the same in $X$ and in $Y$ whenever $\inf A$
exists in $Y$. A net in a set $X$ is a function $g$ from a directed set
$\Gamma$ into $X$. For convenience a net will be written as
$(x_{\gamma})_{\gamma\in\Gamma}$. For the definition of a subnet, we adopt the
definition of Willard [19]. For a thorough exposition on nets and subnets, one
can look at [16]. We start this section by defining order convergence.

######  Definition 1.

Let $(x_{\gamma})_{\gamma\in\Gamma}$ be a net and $x$ a point in a poset
$\mathscr{P}$. Then $(x_{\gamma})_{\gamma\in\Gamma}$ is said to _$O$
-converge_ to $x$ in $\mathscr{P}$ if there exists a directed subset
$\mathscr{M}\subset\mathscr{P}$, and a filtered subset
$\mathscr{N}\subset\mathscr{P}$, such that
$\sup\mathscr{M}=\inf\mathscr{N}=x$, and for every
$(m,n)\in\mathscr{M}\times\mathscr{N}$ the net is eventually contained in
$[m,n]$.

If we are working on lattices instead of partially ordered sets, the fact that
$\mathscr{M}$ is directed and $\mathscr{N}$ is filtered is redundant.

######  Lemma 1.

Let $\mathscr{P}$ be a poset and $(x_{\gamma})_{\gamma\in\Gamma}$ be an
increasing (decreasing) net in $\mathscr{P}$ with
$\sup_{\gamma\in\Gamma}x_{\gamma}=x$ ($\inf_{\gamma\in\Gamma}x_{\gamma}=x)$.
Then $x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$.

If $\mathscr{L}$ is a distributive lattice, we have the following proposition.

######  Proposition 2 ([2], Proposition 3.10).

For a lattice $\mathscr{L}$ the following statements are equivalent:

  1. (i)

$\mathscr{L}$ is distributive.

  2. (ii)

Let $f_{s,t}:\mathscr{L}\mapsto\mathscr{L}$ such that $f_{s,t}(x)=(x\wedge
t)\vee s$, then $f_{s,t}$ is a lattice homomorphism for all
$s,t\in\mathscr{L}$.

  3. (iii)

Let $g_{s,t}:\mathscr{L}\mapsto\mathscr{L}$ such that $g_{s,t}(x)=(x\vee
s)\wedge t$, then $g_{s,t}$ is a lattice homomorphism for all
$s,t\in\mathscr{L}$.

(b) If $\mathscr{L}$ is distributive, then $f_{s,t}=f_{s,s\vee t}=g_{s,s\vee
t}$ and $g_{s,t}=g_{s\wedge t,t}=f_{s\wedge t,t}$ for every
$s,t\in\mathscr{L}$.

######  Definition 2.

A net $(x_{\gamma})_{\gamma\in\Gamma}$ in a lattice $\mathscr{L}$ is said to
unbounded order converge ($\mathfrak{u}O$-converge) to $x\in\mathscr{L}$, if
$(x_{\gamma}\wedge t)\vee
s\mathop{\overset{\text{O}}{\longrightarrow}}(x\wedge t)\vee s$ for every
$s,t\in\mathscr{L}$ and $s\leq t$.

We note that due to Proposition 2 (b), if the lattice $\mathscr{L}$ is
distributive, the fact that $s\leq t$ is redundant in the definition of
$\mathfrak{u}O$-convergence.

######  Proposition 3 ([5]).

Let $X$ be a Riesz space. Then, $(x_{\gamma})_{\gamma\in\Gamma}$
$\mathfrak{u}O$-converges to $x$ if and only if $|x_{\gamma}-x|\wedge
u\mathop{\overset{\text{O}}{\longrightarrow}}0$ for every $u\in X_{+}$.

By Proposition 3, it can be seen that Definition 2 extends the notion of
$\mathfrak{u}O$-convergence that has originally been considered for Riesz
spaces.

######  Proposition 4.

Let $\mathscr{L}$ be a distributive lattice and
$(x_{\gamma})_{\gamma\in\Gamma}$ be an increasing (decreasing) net in
$\mathscr{L}$ such that $x_{\gamma}\mathop{\overset{\text{$\mathfrak{u}$O
}}{\longrightarrow}}x$. Then,
$\sup_{\gamma\in\Gamma}x_{\gamma}=x(\inf_{\gamma\in\Gamma}x_{\gamma}=x)$.

###### Proof.

We start by showing that $x$ is an upper bound for the net
$(x_{\gamma})_{\gamma\in\Gamma}$. Fix $\gamma_{0}\in\Gamma$ and take
$t=x_{\gamma_{0}}$ and $s=x$. Then, $(x_{\gamma}\wedge x_{\gamma_{0}})\vee
x\mathop{\overset{\text{O}}{\longrightarrow}}x$. Thus, there exist subsets
$\mathscr{M}$ and $\mathscr{N}$ with $\sup\mathscr{M}=x=\inf\mathscr{N}$, and
for every $(m,n)\in\mathscr{M}\times\mathscr{N}$ there exists
$\gamma(m,n)\in\Gamma$ such that $(x_{\gamma}\wedge x_{\gamma_{0}})\vee
x\in[m,n]$ for $\gamma\geq\gamma(m,n)$. Moreover, there exists $\gamma_{1}$
such that $\gamma_{1}\geq\gamma(m,n)$ and $\gamma_{1}\geq\gamma_{0}$. As the
net is increasing, this means that $x_{\gamma_{0}}\vee x\in[m,n]$. Thus,
$x_{\gamma_{0}}\vee x=x$, or equivalently that $x_{\gamma_{0}}\leq x$. As
$\gamma_{0}$ was arbitrary, it follows that $x_{\gamma}\leq x$ for every
$\gamma\in\Gamma$.

To see that $x$ is the supremum, let $y$ be another upper bound for
$(x_{\gamma})_{\gamma\in\Gamma}$ and fix $\gamma_{0}\in\Gamma$. Now take $t=x$
and $s=x_{\gamma_{0}}$ in the definition of $\mathfrak{u}O$-convergence, such
that $x_{\gamma}\vee x_{\gamma_{0}}=(x_{\gamma}\wedge x)\vee
x_{\gamma_{0}}\mathop{\overset{\text{O}}{\longrightarrow}}x$. There exist
subsets $\mathscr{M}$ and $\mathscr{N}$ with
$\sup\mathscr{M}=x=\inf\mathscr{N}$ and for every
$(m,n)\in\mathscr{M}\times\mathscr{N}$ there exists $\gamma(m,n)\in\Gamma$
such that $(x_{\gamma}\vee x_{\gamma_{0}})\in[m,n]$ for
$\gamma\geq\gamma(m,n)$. As $y$ is an upper bound of
$(x_{\gamma})_{\gamma\in\Gamma}$, it follows that $m\leq y$ for every
$m\in\mathscr{M}$, concluding that $x\leq y$. ∎

Although on Riesz spaces, $\mathfrak{u}O$-convergence is order continuous, the
following example shows that this is not necessarily true for distributive
lattices.

######  Example 5.

Let $\mathscr{L}$ be the collection of closed subsets of $[0,1]$, ordered by
inclusion. Clearly this is a distributive lattice. However, from the
increasing sequence $(\big{[}\frac{1}{2^{n}},1\big{]})_{n\in\mathds{N}}$ we
have that
$\big{[}\frac{1}{2^{n}},1\big{]}\mathop{\overset{\text{O}}{\longrightarrow}}[0,1]$
but
$\big{[}\frac{1}{2^{n}},1\big{]}\wedge\\{0\\}\nrightarrow^{\text{O}}\\{0\\}$.

######  Proposition 6.

Let $\mathscr{L}$ be an infinitely distributive lattice. If
$(x_{\gamma})_{\gamma\in\Gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$
and
$(y_{\omega})_{\omega\in\Omega}\mathop{\overset{\text{O}}{\longrightarrow}}y$,
then $(x_{\gamma}\vee
y_{\omega})_{\gamma\times\omega\in\Gamma\times\Omega}\mathop{\overset{\text{O}}{\longrightarrow}}x\vee
y$ and dually.

###### Proof.

There exists directed sets $\mathscr{M}_{x},\mathscr{M}_{y}$ and filtered sets
$\mathscr{N}_{x},\mathscr{N}_{y}$ such that for
$(a_{x},b_{x})\in\mathscr{M}_{x}\times\mathscr{N}_{x}$ and
$(a_{y},b_{y})\in\mathscr{M}_{y}\times\mathscr{N}_{y}$, there exists
$\gamma(a_{x},b_{x}),\omega(a_{y},b_{y})$ such that
$x_{\gamma}\in[a_{x},b_{x}]$ for $\gamma\geq\gamma(a_{x},b_{x})$ and
$y_{\omega}\in[a_{y},b_{y}]$ for $\omega\geq\omega(a_{y},b_{y})$. Take
$\mathscr{M}=\mathscr{M}_{x}\vee\mathscr{M}_{y}$ and
$\mathscr{N}=\mathscr{N}_{x}\vee\mathscr{N}_{y}$. Then, $\mathscr{M}$ is
directed, $\mathscr{N}$ is filtered and by infinite distributivity,
$\sup\mathscr{M}=x\vee y=\inf\mathscr{N}$. Furthermore, for $a_{x}\vee
a_{y}\in\mathscr{M}_{x}\vee\mathscr{M}_{y}$ and $b_{x}\vee
b_{y}\in\mathscr{N}_{x}\vee\mathscr{N}_{y}$, it follows that $x_{\gamma}\vee
y_{\omega}\in[a_{x}\vee a_{y},b_{x}\vee b_{y}]$ for
$\gamma\times\omega\geq\gamma(a_{x},b_{x})\times\omega(a_{y},b_{y})$. ∎

######  Proposition 7.

Let $\mathscr{L}$ be an infinitely distributive lattice. If
$(x_{\gamma})_{\gamma\in\Gamma}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x$
and
$(y_{\omega})_{\omega\in\Omega}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}y$,
then $(x_{\gamma}\vee
y_{\omega})_{\gamma\times\omega\in\Gamma\times\Omega}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x\vee
y$ and dually.

###### Proof.

The proof is identical to that of Proposition 6. ∎

Now we show that for distributive lattices $\mathscr{L}$,
$\mathfrak{u}O$-convergence being order continuous is equivalent to
$\mathscr{L}$ being infinitely distributive.

######  Theorem 8.

Let $\mathscr{L}$ be a distributive lattice and
$(x_{\gamma})_{\gamma\in\Gamma}$ a net in $\mathscr{L}$. Then,
$\mathfrak{u}O$-convergence is order continuous if and only if $\mathscr{L}$
is infinitely distributive.

###### Proof.

If $\mathscr{L}$ is infinitely distributive, then $\mathfrak{u}O$-convergence
is order continuous by Proposition 6. Conversely, assume that
$\mathfrak{u}O$-convergence is order continuous. Let
$\mathscr{A}\subseteq\mathscr{L}$ with $\sup\mathscr{A}=x$, $b\in\mathscr{A}$
and $y\in\mathscr{L}$. We show that $\sup\\{a\wedge
y:a\in\mathscr{A}\\}=x\wedge y$. Let
$\mathscr{B}=\\{B\subseteq\mathscr{A}:|B|<\omega\ \text{ and }b\in B\\}$ and
let $x_{B}=\bigvee_{\gamma\in B}x_{\gamma}$ to get an increasing net
$\\{x_{B}:B\in\mathscr{B}\\}$ with supremum $x$. Then by order continuity of
$\mathfrak{u}O$-convergence,
$x_{B}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x$. Take $t=y$
and $s=b\wedge y$ in the definition of $\mathfrak{u}O$-convergence to get that
$(x_{B}\wedge y)\vee(y\wedge
b)\mathop{\overset{\text{O}}{\longrightarrow}}x\wedge y$. Again by order
continuity of $\mathfrak{u}O$-convergence we have that $(x_{B}\wedge
y)\vee(y\wedge
b)\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x\wedge y$. Then
by Proposition 4,

$\displaystyle\sup_{B\in\mathscr{B}}((x_{B}\wedge y)\vee(y\wedge b))$
$\displaystyle=x\wedge y$
$\displaystyle\sup_{B\in\mathscr{B}}(y\wedge(x_{B}\vee b))$
$\displaystyle=x\wedge y$ $\displaystyle\sup_{B\in\mathscr{B}}(y\wedge x_{B})$
$\displaystyle=x\wedge y$ $\displaystyle\sup_{a\in\mathscr{A}}(y\wedge a)$
$\displaystyle=x\wedge y.$

A similar argument can be made to prove the dual. ∎

######  Proposition 9.

Let $\mathscr{L}$ be an infinitely distributive lattice. If
$(x_{\gamma})_{\gamma\in\Gamma}$ is an order bounded net, then
$x_{\gamma}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x$ if and
only if $x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$.

###### Proof.

That $x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ implies
$x_{\gamma}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x$
follows by Theorem 8. On the other-hand assume that there exists
$a,b\in\mathscr{L}$ such that $x_{\gamma}\in[a,b]$ for every $\gamma\in\Gamma$
and $x_{\gamma}\mathop{\overset{\text{$\mathfrak{u}$O}}{\longrightarrow}}x$.
Then $(x_{\alpha}\wedge x_{\gamma})\vee x_{\beta}\in[a,b]$ and thus $(x\wedge
x_{\gamma})\vee x_{\beta}\in[a,b]$. Repeating this argument it follows that
$x\in[a,b]$. Finally, take $s=a$ and $t=b$ such that
$x_{\gamma}=(x_{\gamma}\wedge b)\vee
a\mathop{\overset{\text{O}}{\longrightarrow}}(x\wedge b)\vee a=x$. ∎

Let $\mathscr{P}$ be a poset. By the adherence of a subset $\mathscr{X}$, we
mean all those points which have a net in $\mathscr{X}$ O-converging to them.
We denote the first adherence $\overline{\mathscr{X}}^{O}$ by
$\mathscr{X}_{1}$ and for an ordinal $\lambda$, we denote the
$\lambda$-adherence
$\overline{\bigcup_{\beta<\lambda}\mathscr{X}_{\beta}}^{O}$ by
$\mathscr{X}_{\lambda}$. A subset $\mathscr{X}$ of $\mathscr{P}$ is said to be
_$O$ -closed_ if there is no net in $\mathscr{X}$ that is O-converging to a
point outside of $\mathscr{X}$. Thus, $\mathscr{X}$ is O-closed iff
$\mathscr{X}=\mathscr{X}_{1}$. The collection of all O-closed subsets of
$\mathscr{P}$ form the _order topology $\tau_{O}(\mathscr{P})$_. It is easily
seen that $[a,b]$ is O-closed for every $a\leq b\in\mathscr{P}$. Similarly, a
subset $\mathscr{X}$ is $\mathfrak{u}O$-closed if there is no net in
$\mathscr{X}$ that is $\mathfrak{u}$O-converging to a point outside of
$\mathscr{X}$. Thus, the first $\mathfrak{u}$O-adherence
$\overline{\mathscr{X}}^{\mathfrak{u}O}$ is denoted by
$\mathscr{X}_{1}^{\mathfrak{u}O}$ and for an ordinal $\lambda$ we denote the
$\lambda$-$\mathfrak{u}$O-adherence by
$\mathscr{X}_{\lambda}^{\mathfrak{u}O}$.

######  Lemma 10.

Let $Y$ be a sublattice of an infinitely distributive lattice $\mathscr{L}$.
Then both $Y_{1}$ and $Y_{1}^{\mathfrak{u}O}$ are sublattices.

###### Proof.

$Y_{1}$ : Let $x,y\in Y_{1}$ then there exist two nets
$(x_{\gamma})_{\gamma\in\Gamma}\subseteq Y$ and
$(y_{\omega})_{\omega\in\Omega}\subseteq Y$ such that
$x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ and
$y_{\omega}\mathop{\overset{\text{O}}{\longrightarrow}}y$. As $Y$ is a
sublattice, $(x_{\gamma}\vee
y_{\omega})_{\gamma\times\omega\in\Gamma\times\Omega}\subseteq Y$ and by Lemma
6 $x_{\gamma}\vee y_{\omega}\mathop{\overset{\text{O}}{\longrightarrow}}x\vee
y$. Thus, $x\vee y\in Y_{1}$. The same argument can be used to show that
$x\wedge y\in Y_{1}$.

$Y_{1}^{\mathfrak{u}O_{2}}$ : Let $x,y\in Y_{1}^{\mathfrak{u}O}$ and
$s,t\in\mathscr{L}$. Then there exist two nets
$(x_{\gamma})_{\gamma\in\Gamma}$ and $(y_{\omega})_{\omega\in\Omega}$ in $Y$
such that $(x_{\gamma}\wedge t)\vee
s\mathop{\overset{\text{O}}{\longrightarrow}}(x\wedge t)\vee s$ and
$(y_{\omega}\wedge t)\vee
s\mathop{\overset{\text{O}}{\longrightarrow}}(y\wedge t)\vee s$. By a similar
argument to the above we deduce that $((x_{\gamma}\vee y_{\omega})\wedge
t)\vee s\mathop{\overset{\text{O}}{\longrightarrow}}((x\vee y)\wedge t)\vee
s$. Thus, $x\vee y\in Y_{1}^{\mathfrak{u}O}$. By a dual argument it can be
shown that $x\wedge y\in Y_{1}^{\mathfrak{u}O}$. ∎

For some cardinal $\kappa$, let $\kappa^{+}$ denoted the successor cardinal.

######  Proposition 11.

Let $\mathscr{L}$ be a lattice and $Y\subseteq\mathscr{L}$ be a sublattice.
Assume that $|\mathscr{L}\rvert=\kappa$ for some cardinal $\kappa$. Then,
$Y_{\kappa^{+}}$ is $O$-closed.

###### Proof.

Assume that $Y_{\kappa^{+}}$ is not $O$-closed. Then for every
$\gamma\leq\kappa^{+}$, $Y_{\gamma}\neq Y_{\beta}$ for every $\beta<\gamma$.
Take $x_{\gamma}\in
Y_{\gamma}\setminus\bigg{(}\bigcup_{\beta<\gamma}Y_{\beta}\bigg{)}$. Then,
$|\\{x_{\gamma}:\gamma\leq\kappa^{+}\\}\rvert=\kappa^{+}>\kappa$, a
contradiction. ∎

######  Theorem 12.

Let $\mathscr{L}$ be an infinitely distributive lattice and
$Y\subseteq\mathscr{L}$ be a sublattice. Then $Y_{\lambda}$ is a sublattice
for every $\lambda\leq|\mathscr{L}|^{+}$. In particular the order closure of
$Y$ is a sublattice.

###### Proof.

We do this by transfinite induction. By Lemma 10, $Y_{1}$ is a sublattice.
Assume, $Y_{\beta}$ is a sublattice for $\beta<\lambda$. If $\lambda$ is a
successor ordinal, $Y_{\lambda}$ is a sublattice by the same argument as in
Lemma 10. Assume $\lambda$ is a limit ordinal. For $x,y\in Y_{\lambda}$ there
exists
$(x_{\gamma})_{\gamma\in\Gamma}\subseteq\bigcup_{\beta<\lambda}Y_{\beta}$ and
$(y_{\omega})_{\omega\in\Omega}\subseteq\bigcup_{\beta<\lambda}Y_{\beta}$ such
that $x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ and
$y_{\omega}\mathop{\overset{\text{O}}{\longrightarrow}}y$. By the inductive
step and Proposition 6, the net $(x_{\gamma}\vee
y_{\omega})_{{\gamma\times\omega}\in\Gamma\times\Omega}\subseteq\bigcup_{\beta<\lambda}Y_{\beta}$
and $x_{\gamma}\vee y_{\beta}\mathop{\overset{\text{O}}{\longrightarrow}}x\vee
y$. It follows that $x\vee y\in Y_{\lambda}$. Similarly, it can be shown that
$x\wedge y\in Y_{\lambda}$. The last assertion follows by Proposition 11
because $Y_{\beta}$ is $O$-closed for some $\beta\leq|\mathscr{L}|$. ∎

######  Example 13.

The following is an example of a non-distributive lattice where the O-closure
of a sublattice is not a sublattice [6].

0(1,1)(1,2)(1,3)(2,1)(2,2)(2,3)(3,1)(3,2)(3,3)1(1,0)(2,0)(3,0)$\vdots$
$\vdots$ (2,0)$\vdots$ $\vdots$ (3,0)$\vdots$ $\vdots$ $(\infty,0)$
23$\cdots$$\cdots$$\cdots$$\cdots$(1,$\infty$)(2,$\infty$)(3,$\infty$)$\cdots$
($\infty$,$\infty$)

Let $Y=\\{(n,m):n,m\in\mathds{N}\\}\cup\\{n:n\in\mathds{N}\\}\cup\\{0\\}$.
Clearly, this is a sublattice with
$Y_{1}=Y\cup\\{(n,0):n\in\mathds{N}\\}\cup\\{(\infty,\infty)\\}$. Clearly,
this is not a sublattice because in $\mathscr{L}$,
$(1,0)\wedge(2,0)=(\infty,0)$ but in $Y_{1}$, $(1,0)\wedge(2,0)=0$.

Let us also note that this lattice is not distributive. Indeed,
$(1,\infty)\vee((2,\infty)\wedge(3,\infty))=(1,\infty)\vee(\infty,\infty)=(1,\infty)$.
However, $((1,\infty)\vee(2,\infty))\wedge((1,\infty)\vee(3,\infty))=1$.

######  Question 14.

Can infinite distributivity be weakened to distributivity in Theorem 12?

Now we extend some results that exist on Riesz spaces to infinitely
distributive lattices. In Proposition 16 we give a generalization of
Proposition 15. Note that in our context, we have $Y_{3}$ not $Y_{2}$.

######  Proposition 15 ([9], Proposition 2.1).

Let $Y$ be a sublattice of a Riesz space $X$. Then, $Y_{1}\subseteq
Y_{1}^{\mathfrak{u}O}\subseteq Y_{2}$. Moreover,

  1. (i)

if $Y_{1}$ is order closed, then it is the smallest order closed sublattice of
$X$ containing $Y$, and $Y_{1}=Y_{1}^{\mathfrak{u}O}$;

  2. (ii)

if $Y_{1}^{\mathfrak{u}O}$ is order closed, then it is the smallest order
closed sublattice of $X$ containing $Y$, and $Y_{1}^{\mathfrak{u}O}=Y_{2}$.

######  Proposition 16.

Let $Y$ be a sublattice of an infinitely distributive lattice $\mathscr{L}$.
Then, $Y_{1}\subseteq Y_{1}^{\mathfrak{u}O}\subseteq Y_{3}$. In particular, it
follows that

  1. (i)

if $Y_{1}$ is order closed, then it is the smallest order closed sublattice of
$\mathscr{L}$ containing $Y$, and $Y_{1}=Y_{1}^{\mathfrak{u}O}$;

  2. (ii)

if $Y_{1}^{\mathfrak{u}O}$ is order closed, then it is the smallest order
closed sublattice of $\mathscr{L}$ containing $Y$, and
$Y_{1}^{\mathfrak{u}O}=Y_{3}$.

###### Proof.

$Y_{1}\subseteq Y_{1}^{\mathfrak{u}O}$ : From Theorem 8,
$\mathfrak{u}O$-convergence is order continuous.

$Y_{1}^{\mathfrak{u}O}\subseteq Y_{3}$ : If $x\in Y_{1}^{\mathfrak{u}O}$,
there exists $(x_{\gamma})_{\gamma\in\Gamma}\subseteq Y$ such that
$x_{\gamma}\mathop{\overset{\text{$\mathfrak{u}O$}}{\longrightarrow}}x$, i.e
$(x_{\gamma}\wedge t)\vee
s\mathop{\overset{\text{O}}{\longrightarrow}}(x\wedge t)\vee s$ for
$s,t\in\mathscr{L}$. For any $\gamma,\alpha,\beta\in\Gamma$, $x_{\gamma}\wedge
x_{\alpha}\in Y$, and

$(x_{\gamma}\wedge x_{\beta})\vee(x_{\beta}\wedge
x_{\alpha})\mathop{\overset{\text{O}}{\longrightarrow}}(x\wedge
x_{\beta})\vee(x_{\beta}\wedge x_{\alpha}),$

it then follows that $(x\wedge x_{\beta})\vee(x_{\beta}\wedge x_{\alpha})\in
Y_{1}$ for any $\alpha,\beta\in\Gamma$. Again it follows that $x\wedge
x_{\beta}\in Y_{2}$ for every $\beta\in\Gamma$. As $Y_{2}$ is a sublattice, we
note that $(x\wedge x_{\omega})\vee(x\wedge x_{\beta})\in Y_{2}$ for every
$\alpha,\omega\in\Gamma$. This concludes that $x\in Y_{3}$.

  1. (i)

We start by showing that if $Y_{1}$ is $O$-closed then it is the smallest
$O$-closed sublattice containing $Y$. From Lemma 10, $Y_{1}$ is a sublattice.
Finally, assume $\mathscr{A}\subseteq\mathscr{L}$ is an $O$-closed sublattice
and contains $Y$. Then, $Y_{1}\subseteq\mathscr{A}$, as required. It is clear
that if $Y_{1}$ is $O$-closed, $Y_{1}=Y_{3}$, concluding that
$Y_{1}=Y_{1}^{\mathfrak{u}O}=Y_{3}$.

  2. (ii)

Similarly to (i), if $Y_{1}^{\mathfrak{u}O}$ is $O$-closed. Then,
$Y_{1}^{\mathfrak{u}O}=Y_{3}$.

∎

######  Theorem 17.

Let $Y$ be a sublattice of an infinitely distributive lattice $\mathscr{L}$.
Then, $Y$ is $O$-closed if and only if $Y$ is $\mathfrak{u}O$-closed.

In [9], Lemma 26, one can find a vector sublattice such that $Y_{1}\neq
Y_{1}^{\mathfrak{u}O}$.

######  Question 18.

Is there an example of a sublattice $Y$ of an infinitely distributive lattice
$\mathscr{L}$ such that $Y^{\mathfrak{u}O}_{1}\neq Y_{2}$.

######  Definition 3.

A set $\mathscr{A}$ in a lattice $\mathscr{L}$ is a down-set if every
$a\in\mathscr{A}$, $x\in\mathscr{L}$ with $x\leq a$ implies $x\in\mathscr{A}$.
Furthermore, $\mathscr{A}$ is an ideal if it is a down-set and $a\vee
b\in\mathscr{A}$ for every $a,b\in\mathscr{A}$.

######  Proposition 19.

Let $\mathscr{A}$ be a down-set in a lattice $\mathscr{L}$. Then, every
$x\in\mathscr{A}_{1}$ is the supremum of an increasing net in $\mathscr{A}$.
Moreover, if $\mathscr{L}$ is infinitely distributive, $\mathscr{A}_{1}$ is a
down-set.

###### Proof.

Let $x\in\mathscr{A}_{1}$, then there exists a net
$(x_{\gamma})_{\gamma\in\Gamma}$ such that
$x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$. Then there exists a
directed set $\mathscr{M}$ and a filtered set $\mathscr{N}$ such that
$\sup\mathscr{M}=x=\inf\mathscr{N}$ and for every
$(m,n)\in\mathscr{M}\times\mathscr{N}$ the net
$(x_{\gamma})_{\gamma\in\Gamma}$ is eventually in $[m,n]$. As $\mathscr{A}$ is
a down-set, it follows that $\mathscr{M}\subseteq\mathscr{A}$. Result follows
from the fact that the set $\mathscr{M}$ is directed, so it can be viewed as
an increasing net indexed over itself.

Furthermore, if $\mathscr{L}$ is infinitely distributive,
$a\in\mathscr{A}_{1}$ and $x\in\mathscr{L}$ with $x\leq a$. By the argument
above, there exists an increasing net
$(a_{\gamma})_{\gamma\in\Gamma}\subseteq\mathscr{A}$ such that
$\sup_{\gamma\in\Gamma}a_{\gamma}=a$. Using the fact that $\mathscr{A}$ is a
down-set, $\\{a_{\gamma}\wedge x:\gamma\in\Gamma\\}\subseteq\mathscr{A}$ and
$\sup\\{a_{\gamma}\wedge x:\gamma\in\Gamma\\}=a\wedge x=x$ concluding that
$x\in\mathscr{A}_{1}$.∎

######  Theorem 20.

Let $\mathscr{L}$ be an infinite distributive lattice and
$\mathscr{A}\subseteq\mathscr{L}$ be an ideal. Then $\mathscr{A}_{1}$ and
$\mathscr{A}_{1}^{\mathfrak{u}O}$ are ideals. Furthermore, $\mathscr{A}_{1}$
is $O$-closed and $\mathscr{A}_{1}=\mathscr{A}_{1}^{\mathfrak{u}O}$.

###### Proof.

If $\mathscr{A}$ is an ideal, then $\mathscr{A}_{1}$ is an ideal by Lemma 10
and Proposition 19.

Next we show that $\mathscr{A}_{1}$ is $O$-closed. By Proposition 19 and the
fact that $\mathscr{A}_{1}$ is an ideal, for every $x\in\mathscr{A}_{2}$ there
exists an increasing net $(x_{\gamma})_{\gamma\in\Gamma}$ in $\mathscr{A}_{1}$
such that $\sup_{\gamma\in\Gamma}x_{\gamma}=x$. Let
$\mathscr{B}:=\\{a\in\mathscr{A}:\exists\gamma\in\Gamma\ \mbox{such that
}a\leq x_{\gamma}\\}$. Then $\mathscr{B}$ is non-empty, directed, and $x$ is
an upper-bound of $\mathscr{B}$. Finally, observe that if $k\in\mathscr{L}$
and $b\leq k$ for every $b\in\mathscr{B}$, then $x_{\gamma}\leq k$ for every
$\gamma\in\Gamma$, and therefore $x\leq k$. This shows that that $x$ is the
supremum of $\mathscr{B}$ and so $x\in\mathscr{A}_{1}$. So
$\mathscr{A}_{2}=\mathscr{A}_{1}$ and therefore $\mathscr{A}_{1}$ is
$O$-closed. Then Proposition 16(i) yields
$\mathscr{A}_{1}=\mathscr{A}_{1}^{\mathfrak{u}O}$. In particular,
$\mathscr{A}_{1}^{\mathfrak{u}O}$ is an ideal. ∎

##  3\. MacNeille Completion and Sublattices

Let $\mathscr{P}$ be a poset and $\mathscr{D}$ a subset of $\mathscr{P}$. The
set of upper-bounds and lower-bounds of $\mathscr{D}$ are denoted by
$\mathscr{D}^{+}$ and $\mathscr{D}^{-}$, respectively. A set
$\mathscr{D}\subseteq\mathscr{P}$ is said to be a cut if
$\mathscr{D}^{+-}=\mathscr{D}$. The MacNeille completion of $\mathscr{P}$
denoted by $\mathscr{P}^{\delta}$ is the set of all cuts of $\mathscr{P}$. The
set $\mathscr{P}^{\delta}$ is a complete lattice with respect to set-theoretic
inclusion. For every $x\in\mathscr{P}$, the set $(\leftarrow,x]$ is a cut and
the function $\varphi:\mathscr{P}\rightarrow\mathscr{P}^{\delta}$ defined by
$\varphi(x)=(\leftarrow,x]$ is an order isomorphism from $\mathscr{P}$ onto
the subspace $\varphi[\mathscr{P}]$ of $\mathscr{P}^{\delta}$. Moreover,
$\varphi[\mathscr{P}]$ is join and meet dense in $\mathscr{P}^{\delta}$.

We remark that the MacNeille completion $X^{\delta}$ of an Archimedean Riesz
space $X$ is obtained by removing the top (i.e. $X$) and bottom (i.e.
$\emptyset$) elements from the MacNeille completion (as described in the
previous paragraph) of the lattice associated with $X$. In this case,
therefore, $X^{\delta}$ is only Dedekind complete. We further remark that when
$\mathscr{P}$ is not a lattice, removing the top and bottom elements from
$\mathscr{P}^{\delta}$ might disrupt the lattice structure, as exhibited in
Example 21 (see from[12, p. 190]).

######  Example 21.

Let $\mathscr{P}=\\{x_{1},x_{2}\\}$ with the partial ordering $x_{1}\leq
x_{2}$ $\iff$ $x_{1}=x_{2}$. For $\mathscr{A}\subseteq\mathscr{P}$, the set
$\mathscr{A}^{+-}$ consists of either $\emptyset,\\{x_{1}\\},\\{x_{2}\\}$ or
$\mathscr{P}$. Thus,
$\mathscr{P}^{\delta}=\\{\emptyset,\\{x_{1}\\},\\{x_{2}\\},\mathscr{P}\\}$.

However, it is easy to see that if $\mathscr{P}$ is a lattice, then removing
the top or bottom elements from $\mathscr{P}^{\delta}$ does not affect the
lattice structure. We further recall that Riesz spaces are infinitely
distributive, and since the the MacNeille completion $X^{\delta}$ of a Riesz
space is again a Riesz space, it follows that the $X^{\delta}$ is also
infinitely distributive. The following example shows that this is not
necessarily true when considering the MacNeille completion of a lattice
$\mathscr{L}$.

######  Example 22.

Take $\mathscr{L}=\\{(0,b):0\leq b<1\\}\cup\\{(1,b):0\leq b<1\\}$. Then the
MacNeille completion $\mathscr{L}^{\delta}$ is $\\{(0,b):0\leq
b<1\\}\cup\\{(1,b):0\leq b\leq 1\\}$. This is not infinitely distributive.
Indeed let $x_{n}=(0,1-\frac{1}{n})$. Then
$\sup_{\mathscr{L}^{\delta}}x_{n}=(1,1)$ and
$(\sup_{\mathscr{L}^{\delta}}x_{n})\wedge(1,\frac{1}{2})=(1,\frac{1}{2})$. On
the other-hand,
$\sup_{\mathscr{L}^{\delta}}(x_{n}\wedge(1,\frac{1}{2}))=(0,\frac{1}{2})$.

######  Definition 4.

Let $\mathscr{L}$ be a lattice and let $\mathscr{L}_{0}\subseteq\mathscr{L}$
be a sublattice. $\mathscr{L}_{0}$ is said to be regular if for every
$\mathscr{A}\subseteq\mathscr{L}_{0}$ such that
$\inf_{\mathscr{L}_{0}}\mathscr{A}$ exists in $\mathscr{L}_{0}$, then it
exists in $\mathscr{L}$ and
$\inf_{\mathscr{L}_{0}}\mathscr{A}=\inf_{\mathscr{L}}\mathscr{A}$, similarly
if $\sup_{\mathscr{L}_{0}}\mathscr{A}$ exists in $\mathscr{L}_{0}$, then it
exists in $\mathscr{L}$ and
$\sup_{\mathscr{L}_{0}}\mathscr{A}=\sup_{\mathscr{L}}\mathscr{A}$.

It is known that in Riesz spaces, regularity of a Riesz sublattice is
preserved when passing to the respective MacNeille completions [7, Theorem
2.10]. Again Example 22 shows that this is not true on lattices.

######  Definition 5.

Let $\mathscr{L}$ be a lattice and $\mathscr{A}\subseteq\mathscr{L}$. Then
$\mathscr{A}$ is said to be convex if for $x,y\in\mathscr{A}$ with $x\leq y$
then $[x,y]\subseteq\mathscr{A}$.

######  Proposition 23.

Every convex sublattice of a lattice $\mathscr{L}$ is regular.

###### Proof.

Let $\mathscr{A}\subseteq\mathscr{L}_{0}$ with
$\sup_{\mathscr{L}_{0}}\mathscr{A}=x$. Assume $b\in\mathscr{L}$ with $a\leq b$
for every $a\in\mathscr{A}$. Then $x\wedge b$ is an upper-bound of
$\mathscr{A}$ and $a\leq x\wedge b\leq x$ for every $a\in\mathscr{A}$. Using
the convexity of $\mathscr{L}_{0}$, it follows that $x\wedge
b\in\mathscr{L}_{0}$. This concludes that $x=x\wedge b$, i.e
$\sup_{\mathscr{L}}\mathscr{A}=x$. The argument for the infimum follows in a
similar way. ∎

It is easy to see that if
$\mathscr{B}\subseteq\mathscr{L}_{0}\subseteq\mathscr{L}$, then
$\mathscr{B}^{+_{\mathscr{L}_{0}}}=\mathscr{B}^{+}\cap\mathscr{L}_{0}$ and
$\mathscr{B}^{-_{\mathscr{L}_{0}}}=\mathscr{B}^{-}\cap\mathscr{L}_{0}$. It is
also easy to see that if $\\{\mathscr{A}_{\alpha}:\alpha\in\mathscr{A}\\}$ is
a family of subsets, then
$\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{-}=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{-}$
and
$\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{-_{\mathscr{L}_{0}}}=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{-_{\mathscr{L}_{0}}}$.

######  Proposition 24.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}\subseteq\mathscr{L}$ be a
sublattice. Let $\mathscr{A},\mathscr{B}\subseteq\mathscr{L}_{0}$ such that
$\mathscr{A}^{+-}\subseteq\mathscr{B}^{+-}$. Then,
$\mathscr{A}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}\subseteq\mathscr{B}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}$.

###### Proof.

We start by showing that
$\mathscr{B}^{+_{\mathscr{L}_{0}}}\subseteq\mathscr{A}^{+}$. Indeed,
$\mathscr{B}^{+_{\mathscr{L}_{0}}}\subseteq\mathscr{B}^{+}=\mathscr{B}^{+-+}\subseteq\mathscr{A}^{+-+}=\mathscr{A}^{+}$.

Next note that
$\mathscr{A}^{+-_{\mathscr{L}_{0}}}\subseteq\mathscr{B}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}$.
Indeed, let $y\in\mathscr{A}^{+-_{\mathscr{L}_{0}}}$. Then
$y\in\mathscr{L}_{0}$ and $y\leq b$ for every $b\in\mathscr{A}^{+}$. From
$\mathscr{B}^{+_{\mathscr{L}_{0}}}\subseteq\mathscr{A}^{+}$ it is clear that
$y\leq c$ for every $c\in\mathscr{B}^{+_{\mathscr{L}_{0}}}$. This concludes
that
$\mathscr{A}^{+-_{\mathscr{L}_{0}}}\subseteq\mathscr{B}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}$.
Finally from
$\mathscr{A}\subseteq\mathscr{A}^{+-_{\mathscr{L}_{0}}}\subseteq\mathscr{B}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}$
it follows that
$\mathscr{A}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}\subseteq\mathscr{B}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}$.
∎

######  Proposition 25.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}\subseteq\mathscr{L}$ be a
sublattice. Then for $\mathscr{A}\subseteq\mathscr{L}_{0}$,
$(\mathscr{A}^{+-}\cap\mathscr{L}_{0})^{+-}=\mathscr{A}^{+-}$.

###### Proof.

First note that
$(\mathscr{A}^{+-}\cap\mathscr{L}_{0})\subseteq\mathscr{A}^{+-}$ implies that
$(\mathscr{A}^{+-}\cap\mathscr{L}_{0})^{+-}\subseteq\mathscr{A}^{+-}$. The
other inclusion follows by noting that $\mathscr{A}\subseteq\mathscr{A}^{+-}$
and $\mathscr{A}\subseteq\mathscr{L}_{0}$, then
$\mathscr{A}\subseteq\mathscr{A}^{+-}\cap\mathscr{L}_{0}$ concludes that
$\mathscr{A}^{+-}\subseteq(\mathscr{A}^{+-}\cap\mathscr{L}_{0})^{+-}$. ∎

######  Definition 6.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}$ be a sublattice. Then
$\mathscr{L}_{0}$ has property (A) if for
$\mathscr{A}\subseteq\mathscr{L}_{0}$ and $x\in\mathscr{A}^{-}$, there exists
$y\in\mathscr{A}^{-_{\mathscr{L}_{0}}}$ such that $x\leq y$. Equivalently, it
is said to have property (B) if $x\in\mathscr{A}^{+}$, then there exists
$y\in\mathscr{A}^{+_{\mathscr{L}_{0}}}$ satisfying $x\geq y$.

######  Lemma 26.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}$ be a sublattice
satisfying (A) and (B). Then $\mathscr{L}_{0}$ is regular.

###### Proof.

Let $\mathscr{A}\subseteq\mathscr{L}_{0}$ and
$b=\sup_{\mathscr{L}_{0}}\mathscr{A}$. Assume that there exists
$x\in\mathscr{A}^{+}$, then by property (A) there exists
$a\in\mathscr{A}^{+\mathscr{L}_{0}}$ such that $a\leq x$, thus concluding that
$b=\sup_{\mathscr{L}}\mathscr{A}$. The dual can be proven similarly. ∎

######  Proposition 27.

Let $\mathscr{L}$ be a lattice, $\mathscr{L}_{0}\subseteq\mathscr{L}$ and
$\mathscr{A}\subseteq\mathscr{L}_{0}$.

  1. (i)

Property $(A)$ implies that
$\mathscr{A}^{-+}=\mathscr{A}^{-_{\mathscr{L}_{0}}+}$;

  2. (ii)

Property $(B)$ implies that
$\mathscr{A}^{+-}=\mathscr{A}^{+_{\mathscr{L}_{0}}-}$.

###### Proof.

We shall show the proof of (i), as the proof for (ii) follows by a similar
argument. It is easy to see that
$\mathscr{A}^{-+}\subseteq\mathscr{A}^{-_{\mathscr{L}_{0}}+}$. To see the
reverse inclusion let $x\in\mathscr{A}^{-_{\mathscr{L}_{0}}+}$ and
$z\in\mathscr{A}^{-}$. Then there exists
$y\in\mathscr{A}^{-_{\mathscr{L}_{0}}}$ such that $z\leq y$. This implies that
$z\leq y\leq x$ which concludes that $x\in\mathscr{A}^{-+}$. ∎

######  Proposition 28.

Let $\mathscr{L}$ be a lattice, $\mathscr{L}_{0}\subseteq\mathscr{L}$ a
sublattice and $\\{\mathscr{A}_{\alpha}:\alpha\in\mathscr{A}\\}$ a collection
of sets in $\mathscr{L}_{0}$ such that
$\mathscr{A}_{\alpha}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}=\mathscr{A}_{\alpha}$.

  1. (i)

If $\mathscr{L}_{0}$ satisfies property (A) then
$\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+-}=(\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+-}$.

  2. (ii)

If $\mathscr{L}_{0}$ satisfies property (B) then
$(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}+-}=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+-})^{+-}$.

###### Proof.

  1. (i)

We first note that the inclusion
$(\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+-}\subseteq\bigcap_{\alpha}\mathscr{A}_{\alpha}^{+-}$
is trivial. To see the reverse inclusion, first note that
$\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+-}=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+})^{-}$.
Furthermore,

$\displaystyle(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+})^{-}$
$\displaystyle=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+})^{-+-}$
$\displaystyle\subseteq(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+_{\mathscr{L}_{0}}})^{-+-}$
$\displaystyle=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+_{\mathscr{L}_{0}}})^{-_{\mathscr{L}_{0}}+-}\text{
(By property (A) )}$
$\displaystyle=(\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}})^{+-}$
$\displaystyle=(\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+-}.$

  2. (ii)

To see this equality, we note that

$\displaystyle(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}+-}$
$\displaystyle=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}+_{\mathscr{L}_{0}}-}\text{
(By property (B) )}$
$\displaystyle=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+_{\mathscr{L}_{0}}-}$
$\displaystyle=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+-}\text{
(By property (B) )}$
$\displaystyle=(\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha}^{+-})^{+-}.$

∎

######  Proposition 29.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}\subseteq\mathscr{L}$ be a
sublattice. Then for every $a\in\mathscr{L}_{0}$,
$(\\{a\\}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}})^{+-}=((\leftarrow,a]\cap\mathscr{L}_{0})^{+-}=(\leftarrow,a]=\\{a\\}^{+-}$.

###### Proof.

The result follows by Proposition 25. ∎

######  Theorem 30.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}\subseteq\mathscr{L}$ be a
sublattice. Then

$\mathscr{L}_{0}^{*}:=\\{\mathscr{A}^{+-}:\mathscr{A}\subseteq\mathscr{L}_{0}\text{
and }\mathscr{A}^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}}=\mathscr{A}\\}$

is order-isomorphic to $\mathscr{L}_{0}^{\delta}$.

###### Proof.

Consider the map $\varphi:\mathscr{L}_{0}^{\delta}\mapsto\mathscr{L}_{0}^{*}$.
First note that by Proposition 24, $\varphi$ is a bijective map. Next we show
that $\varphi$ is an order-embedding. Let
$\mathscr{A},\mathscr{B}\in\mathscr{L}_{0}^{\delta}$ such that
$\mathscr{A}\subseteq\mathscr{B}$. Then,
$\mathscr{A}^{+-}\subseteq\mathscr{B}^{+-}$. On the other-hand if
$\mathscr{A},\mathscr{B}\subseteq\mathscr{L}_{0}$ such that
$\mathscr{A}^{+-},\mathscr{B}^{+-}\in\mathscr{L}_{0}^{*}$ and
$\mathscr{A}^{+-}\subseteq\mathscr{B}^{+-}$. Then,
$\mathscr{A}\subseteq\mathscr{B}$ by Proposition 24, concluding that $\varphi$
is an order-embedding. ∎

For a collection
$\\{\mathscr{A}_{\alpha}^{+-}:\alpha\in\mathscr{A}\\}\subseteq\mathscr{L}_{0}^{*}$,
the supremum and the infimum of this collection are
$((\bigcup_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+_{\mathscr{L}_{0}}-_{\mathscr{L}_{0}}})^{+-}$
and $(\bigcap_{\alpha\in\mathscr{A}}\mathscr{A}_{\alpha})^{+-}$, respectively.
From Theorem 30, $\mathscr{L}_{0}$ can be embedded in $\mathscr{L}_{0}^{*}$ by
considering the map $\rho:\mathscr{L}_{0}\rightarrow\mathscr{L}_{0}^{*}$ where
$\rho(a)=(\leftarrow,a]$. Furthermore as $\mathscr{L}_{0}^{*}$ and
$\mathscr{L}_{0}^{\delta}$ are order-isomorphic, then
$\mathscr{L}_{0}^{\delta}$ can be seen as a subset of $\mathscr{L}^{\delta}$.

######  Theorem 31.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}\subseteq\mathscr{L}$ be a
sublattice.

  1. (i)

If $\mathscr{L}_{0}$ satisfies property (B), then arbitrary suprema in
$\mathscr{L}_{0}^{\delta}$ are preserved in $\mathscr{L}^{\delta}$.

  2. (ii)

If $\mathscr{L}_{0}$ satisfies property (A), then arbitrary infima in
$\mathscr{L}_{0}^{\delta}$ are preserved in $\mathscr{L}^{\delta}$.

###### Proof.

By Theorem 30, $\mathscr{L}_{0}^{\delta}$ and $\mathscr{L}_{0}^{*}$ are order-
isomorphic. Choose a collection,
$\\{\mathscr{A}_{\alpha}^{+-}:\alpha\in\mathscr{A}\\}\subseteq\mathscr{L}_{0}^{*}$.
Assume there exists $\mathscr{A}^{+-}\in\mathscr{L}_{0}^{*}$ satisfying
$\mathscr{A}^{+-}=\sup_{\mathscr{L}_{0}^{*}}\\{\mathscr{A}_{\alpha}^{+-}:\alpha\in\mathscr{A}\\}$.
Then result follows by Proposition 28. The infimum case follows similarly. ∎

######  Corollary 32.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}$ be a sublattice
satisfying properties (A) and (B). Then $\mathscr{L}_{0}^{\delta}$ is regular
in $\mathscr{L}^{\delta}$.

######  Proposition 33.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}$ be a convex sublattice.

  1. (i)

If $\mathscr{L}_{0}$ has a maximal element, it satisfies property (B);

  2. (ii)

If $\mathscr{L}_{0}$ has a minimal element, it satisfies property (A).

###### Proof.

We will prove (i), as (ii) can be proved dually. Let $a_{\max}$ be the maximal
element of $\mathscr{L}_{0}$. Let $\mathscr{A}\subseteq\mathscr{L}_{0}$ and
$x\in\mathscr{A}^{+}$. Then by convexity, $a_{\max}\wedge x\in\mathscr{L}_{0}$
and $a_{max}\wedge x\in\mathscr{A}^{+_{\mathscr{L}_{0}}}$. ∎

######  Proposition 34.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}$ be sublattice. If
$\mathscr{L}$ has an antitone involution $x\mapsto x^{\prime}$, and
$x^{\prime}\in\mathscr{L}_{0}$ for every $x\in\mathscr{L}_{0}$ then
$\mathscr{L}_{0}^{\delta}$ to be regular in $\mathscr{L}^{\delta}$, it
suffices to have any of properties (A) or (B).

###### Proof.

The proof is omitted as it is routine. ∎

######  Proposition 35.

Let $\mathscr{L}$ be a complete lattice and $\mathscr{L}_{0}$ a sublattice of
$\mathscr{L}$. Then $\mathscr{L}_{0}$ is $O$-closed if and only if
$\mathscr{L}_{0}$ is supremum and infimum closed.

###### Proof.

Assume that $\mathscr{L}_{0}$ is O-closed. Let
$\mathscr{A}\subseteq\mathscr{L}$ with $\sup_{\mathscr{L}}\mathscr{A}=x$. Let
$\mathscr{F}=\\{\mathscr{B}\subseteq\mathscr{A}:|\mathscr{B}|<\omega\\}$ such
that $\mathscr{F}$ is directed. For every $\mathscr{B}\in\mathscr{F}$ let
$x_{\mathscr{B}}=\bigvee\mathscr{B}$ to get an increasing net
$\\{x_{\mathscr{B}}:\mathscr{B}\in\mathscr{F}\\}$ in $\mathscr{L}_{0}$ with
$\sup\\{x_{\mathscr{B}}:\mathscr{B}\in\mathscr{F}\\}=x$, concluding that
$x\in\mathscr{L}_{0}$. The dual can be proved similarly.

Conversely, assume that for any subset $\mathscr{A}$ of $\mathscr{L}_{0}$, its
supremum and infimum in $\mathscr{L}$ belong to $\mathscr{L}_{0}$, and let
$x\in(\mathscr{L}_{0})_{1}$. Then there exists a net
$\\{x_{\gamma}:\gamma\in\Gamma\\}$ such that

$\sup_{\gamma_{0}\in\Gamma}\inf_{\beta\geq\gamma_{0}}x_{\beta}=x=\inf_{\gamma_{0}\in\Gamma}\sup_{\beta\geq\gamma_{0}}x_{\beta}$

By hypothesis it follows that $x\in\mathscr{L}_{0}$. ∎

######  Proposition 36.

Let $\mathscr{L}$ be a complete lattice and $\mathscr{L}_{0}$ a complete
regular sublattice of $\mathscr{L}$. Then $\mathscr{L}_{0}$ is $O$-closed.

###### Proof.

The result follows by Proposition 35 because $\mathscr{L}_{0}$ is supremum and
infimum closed. ∎

######  Theorem 37.

Let $\mathscr{L}$ be a lattice and $\mathscr{L}_{0}\subseteq\mathscr{L}$ a
sublattice satisfying properties (A) and (B). Then
$(\mathscr{L}_{0})_{1}^{\mathfrak{u}O}=(\mathscr{L}_{0})_{1}$, and both are
order closed.

###### Proof.

We start by showing that
$\mathscr{L}_{0}^{\delta}\cap\mathscr{L}=(\mathscr{L}_{0})_{1}$, where
$\mathscr{L}_{0}^{\delta}$ is identified with $\mathscr{L}_{0}^{*}$. For every
$x\in(\mathscr{L}_{0})_{1}$ there exists a net
$\\{x_{\gamma}:\gamma\in\Gamma\\}\subseteq\mathscr{L}_{0}$ such that
$x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ in $\mathscr{L}$.
From [1, Theorem 3] we have that
$x_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ in
$\mathscr{L}^{\delta}$ which implies that $x\in\mathscr{L}_{0}^{\delta}$
concluding that
$(\mathscr{L}_{0})_{1}\subseteq\mathscr{L}_{0}^{\delta}\cap\mathscr{L}$. To
see the converse, take $x\in\mathscr{L}_{0}^{\delta}\cap\mathscr{L}$. Let
$\mathscr{A}=\\{y\in\mathscr{L}_{0}:y\leq x\\}$. By Theorem 31,
$\mathscr{L}_{0}^{\delta}$ is regular in $\mathscr{L}^{\delta}$ and
$\sup_{\mathscr{L}_{0}^{\delta}}\mathscr{A}=x$, thus
$\sup_{\mathscr{L}^{\delta}}\mathscr{A}=x$. This implies that
$\sup_{\mathscr{L}}\mathscr{A}=x$. Generating a net using finite subsets of
$\mathscr{A}$ we get an increasing net in $\mathscr{L}_{0}$ with supremum $x$
in $\mathscr{L}$, concluding that $x\in(\mathscr{L}_{0})_{1}$.

Finally we show $(\mathscr{L}_{0})_{1}$ = $(\mathscr{L}_{0})_{2}$. Let
$x\in(\mathscr{L}_{0})_{2}$. Then there exists a net
$\\{y_{\gamma}:\gamma\in\Gamma\\}\subseteq(\mathscr{L}_{0})_{1}$ such that
$y_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ in $\mathscr{L}$. By
the previous argument we have that
$\\{y_{\gamma}:\gamma\in\Gamma\\}\subseteq\mathscr{L}_{0}^{\delta}$ and
$y_{\gamma}\mathop{\overset{\text{O}}{\longrightarrow}}x$ in
$\mathscr{L}^{\delta}$. By Theorem 31 and Proposition 36,
$\mathscr{L}_{0}^{\delta}$ is $O$-closed in $\mathscr{L}^{\delta}$, so
$x\in\mathscr{L}_{0}^{\delta}$. Then,
$x\in\mathscr{L}_{0}^{\delta}\cap\mathscr{L}=(\mathscr{L}_{0})_{1}$, as
required. ∎

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