# The AI Arena: A Framework for Distributed Multi-Agent Reinforcement Learning

Edward W. Staley11footnotemark: 1 , Corban G.Rivera11footnotemark: 1 , Ashley
J. Llorens111Equal contribution
Intelligent Systems Center
Johns Hopkins Applied Physics Laboratory

###### Abstract

Advances in reinforcement learning (RL) have resulted in recent breakthroughs
in the application of artificial intelligence (AI) across many different
domains. An emerging landscape of development environments is making powerful
RL techniques more accessible for a growing community of researchers. However,
most existing frameworks do not directly address the problem of learning in
complex operating environments, such as dense urban settings or defense-
related scenarios, that incorporate distributed, heterogeneous teams of
agents. To help enable AI research for this important class of applications,
we introduce the AI Arena: a scalable framework with flexible abstractions for
distributed multi-agent reinforcement learning. The AI Arena extends the
OpenAI Gym interface to allow greater flexibility in learning control policies
across multiple agents with heterogeneous learning strategies and localized
views of the environment. To illustrate the utility of our framework, we
present experimental results that demonstrate performance gains due to a
distributed multi-agent learning approach over commonly-used RL techniques in
several different learning environments.

##  1 INTRODUCTION

Reinforcement learning (RL) offers a powerful approach to generating complex
behaviors for intelligent systems that could not be explicitly derived or
programmed. In the RL setting, the problem of learning an effective control
policy is posed as a sequential decision-making problem for an agent
interacting with a learning environment [27]. Given that learning the
environment dynamics is an essential aspect of the RL problem, the ultimate
effectiveness of a learned policy is dependent on the extent to which the
learning environment reflects the essential aspects of the intended operating
environment for the target system. Hence, many RL breakthroughs to date have
focused on gaming and other applications with structured and predictable
environments [29, 25, 4].

Translating progress in RL to increasingly complex applications of artificial
intelligence (AI) will require the design of representative learning
environments with corresponding complexity. Ensuring that future progress is
reproducible and accessible for a broad community of researchers will require
tools and frameworks that enable RL solutions to gracefully scale to address
the problem of learning effectively in these increasingly complex settings. In
general, RL frameworks must balance multiple tradeoffs, including ease of
prototyping versus training at scale, high-level abstractions versus fine-
grained control, and richness of features versus ease of use.

In this work, we introduce the AI Arena: a scalable framework with flexible
abstractions for distributed multi-agent reinforcement learning. Our aim is to
help enable RL research for the class of applications that involve multiple
teams of agents where each team may have unique learning strategies and where
agents within a given team may have localized views of the environment.
Distributed multi-agent applications may be thought of as analogous to "system
of systems" applications from a systems engineering perspective where
collections (teams) of goal-oriented systems (agents) collaborate to achieve
shared objectives. These attributes may arise, for example, in smart city
applications where automated traffic control systems interact with fleets of
automated vehicles or in defense applications where heterogeneous autonomous
systems interact across time and space to achieve high-level mission
objectives. Applications such as these often include cooperation or
competition among heterogeneous teams of agents as defining features.

The AI Arena extends the commonly-used OpenAI Gym [3] interface to provide a
flexible and scalable RL framework for the class of distributed multi-agent
applications described above. Specifically, the AI Arena offers:

  * •

Multi-policy learning: Each team of agents in the environment may employ a
unique and potentially decentralized learning strategy.

  * •

Distributed learning: Each agent in a team can be instantiated as a worker
deployed to an arbitrary computing resource contributing learning experiences
to a common control policy.

  * •

Flexible curriculum design: Successive learning episodes can be easily
structured as sequences of self-play, tournament play or population-based
policy selection.

  * •

Broad compatibility: Support for a wide range of different simulation
environments and commonly-used base learning strategies.

The primary contribution of this work is the introduction of the AI Arena
framework along with a set of motivating experimental results. To illustrate
the efficacy of the our framework, we apply known algorithms to multi-agent
problems and demonstrate how our framework can easily be used to train against
these problems at scale. We present experimental results of single-agent
algorithmic approaches to a multi-team competitive environment called
TanksWorld [22], for which each environment instances houses 10 agents. We
further demonstrate how multi-agent algorithms are supported by our framework
by presenting results from the Cooperative Navigation environment [19].

##  2 RELATED WORK

The OpenAI Gym [3] interface have been widely adopted across RL research
frameworks, especially those primarily focused on single-agent environments
[8, 2, 15]. The AI Arena builds upon the standard Gym interface to provide
robust support for multi-agent environments with decentralized control. Google
Dopamine [6] is a highly configurable RL framework via for supports dynamic
function declaration. KerasRL [21] includes a good diversity of algorithms
including DDPG, DQN, and SARSA focused on single agent reinforcement learning.
Deepmind TRFL includes reinforcement learning primitives that can be direly
integrated with tensorflow, while other neural network libraries are not well
supported. Tensorforce [23] also integrates strongly with tensorflow and
includes a good abstraction for defining a round of training. The strong
integration with tensorflow is also a limitation when considering
reinforcement learning algorithms that are not based on deep networks.
Facebook’s Horizon [12] framework has extensive support for distributed and
scaled training through Spark. Horizon is focused on the scaled use case and
is relatively more verbose when prototyping. The framework has a large amount
of external dependencies and the focus is primarily on off-policy
reinforcement learning.

Coach [5] has a large number of reinforcement learning algorithms built-in. It
has native support for scaling through Kubernetes. TF-agents [13] aims for
production level training. The framework focuses on ease of prototyping and it
is tightly coupled to tensorflow. SLM-Lab [18] focuses on designing
reproducible experiments for the comparison of approaches. The framework uses
Ray for scalability and focuses on single agent training via pytorch-based
neural networks. DeeR is another deep reinforcement learning framework with
solid abstractions and algorithm support. Garage is derived from rllib and
implements a large number of reinforcement learning, meta-RL, and imitation
learning algorithms. Surreal [9] is a reinforcement learning framework with
built-in support for robotics simulation via Mujoco. RLgraph [24] focuses on
scalability also using Ray [20]. It includes abstractions that avoid coupling
the RL algorithm with the framework. ChainerRL [10] also includes a broad set
of reinforcement learning algorithms. The framework is specifically tailored
for the use of the Chainer library for deep networks. Some libraries [1, 7,
26] focus on reproducibility of experiments and ease of prototyping. These
frameworks contain a tremendous amount of capability, but only specialize in
single agent learning tasks. They lack abstractions suitable for multi-agent
multi-policy learning, which is the focus of this work.

Related to our framework, MAgent [30] has support for large numbers of multi-
agent training including multiple policy types. The abstraction for
registering agents and agent types is useful. A weakness is the built-in focus
on gridworld like environments. It might be challenging to integrate other
environments into the framework. RLlib [17] is part of the Ray project. It
includes relatively easy access to distributed training through Ray. The
library includes solid abstractions for multi-agent multi-policy training, and
the configuration in python code is relatively easy to use. RLlib includes
Tune which provides convenient access to built-in hyperparameter tuning.
However, the framework has relatively high complexity, and integrating new
algorithms can be challenging as native PyTorch or Tensorflow models must be
wrapped in the RLlib Model abstraction. ML-agents [16] includes good support
for multi-agent multi-policy training. It promotes flexible coupling between
agents and policies. However, the framework is tightly coupled to the Unity
simulation engine, and it currently lacks multi-node distributed training.

##  3 AI ARENA INTERFACE

###  3.1 INTERFACE OVERVIEW

Figure 1: _The AI Arena Interface._ The AI Arena Interface is an adjustment to
the OpenAI Gym interface such that many data objects are replaced with lists.
This is easily added to an existing environment’s reset() and step() methods
such that a list of actions is accepted and lists of response data are
returned.

The AI Arena Interface is an extension of the gym interface [3] that allows
for environments to house multiple agents, without limitation on how those
agents are attached to learning algorithms. This revision to the gym interface
causes a distinct change in thinking about how environments are constructed:
the environment is no longer completely representative of the problem a given
agent is trying to solve. Rather, the environment is a shared space among one
or more entities, each potentially having their own objectives. We use the
term “entity” to avoid confusion between a policy and the portion of the
environment under its control.

The above interface changes are manifested by simply expecting lists of values
where the gym interface expects singular values. The only exception is the
“done” flag, which is kept as a single boolean to indicate if the shared
episode has ended. This data flow is described in Figure 1.

###  3.2 MULTIAGENT ENVIRONMENTS

One primary goal of the AI Arena interface is to encourage environments in
which a variety of agents may coexist and learn together. This should
encompass everything from collaboration among identical entities to
competition among several dissimilar groups. To that end, other properties of
the environment are also converted to lists, such as action spaces or
observation spaces. This allows for a variety of agent types to coexist in a
single environment. For example, one learning entity may be making discrete
decisions about image state data, while another entity in the same environment
may expect continuous actions based on a vectorized state space.

An implication of this multi-entity setup is that all entities, as well as
their actions, observations, and rewards, are occurring in lock-step at the
same rate. Each step, the environment expects decisions corresponding to all
entities, and will return information to all of them about the consequences of
those decisions. While this may seem limiting at first, it is better to think
of this as supporting the most extreme case of multi-agent interaction: all
entities can be involved in a single frame of the environment. It is fairly
straightforward to embed special cases within this framework: an entity which
has exited an episode early can send and receive null values, or an entity
with a lower interaction frequency can easily on every $N$th observation and
repeat actions until that observation occurs. The global “done” signal is
especially useful for simulations and games in which there is a common or
mutually exclusive objective, as is often the case.

###  3.3 MULTIPLE LEARNING POLICIES

A further goal of the AI Arena interface is to enable complex distributed
training architectures in which many policies may be training simultaneously
in shared environments. The policies may be several instances of the same
algorithm or be entirely separate approaches to learning. The inclusion of
many entities in a single environment breaks from a typical training paradigm
of one policy-worker thread corresponding to one agent in one environment.
Rather, it is up to the user of this interface to distribute the many entities
in an environment (or across many environments in the distributed case) to as
many agents as desired. For example, an environment with $N$ agents may
function as $N$ workers to a single distributed algorithm, or on the other
extreme, single workers to $N$ distinct policies. They may also be grouped
such that $M$ agents are in fact controlled by a single instance of a multi-
agent policy (Figure 2). In other words, the agency of a given entity is at
the discretion of the user. While this is a potentially powerful paradigm, it
can be complex to implement. In the next section, we will describe the AI
Arena software framework, which provides a straightforward implementation.

Figure 2: _Possible worker configurations._ Policy workers may be attached in
environment entities in any desired combination. A Three entities are assigned
to three independent policies that are learning separately and may have
workers attached to other environments. B Entities are each attached to
workers of the same policy, such that some of the agents contributing to the
policy are coexisting in the same environment. C Entities are all attached to
the same policy worker, which takes all of their data into account in a
multiagent manner, possibly one of many workers for a distributed multiagent
algorithm.

##  4 AI ARENA SOFTWARE FRAMEWORK

The AI Arena software framework is an implementation of the previously
described AI Arena Interface with several goals in mind: 1) To allow usage of
any environment or RL algorithm that the user may desire, with minimal effort,
2) To provide high-level functionality such that a user can quickly describe a
very sophisticated training or testing scheme in python, and 3) to seamlessly
orchestrate the desired training at scale.

###  4.1 PROCESS-BASED ARCHITECTURE

The AI Arena software framework is designed to run in a highly distributed and
compartmentalized way, making heavy use of processes via the Message-Passing
Interface (MPI) standard [11, 28]. Each environment instance is maintained in
its own process, as well as each policy worker. The MPI paradigm allows many
processes to spin up at once (across one or more machines), and allows various
subsets of processes to communicate. This is used by the AI Arena to designate
messaging groups between an environment and any policy workers interfacing
with that environment, as well as between collections of worker processes
which all contribute to a single policy. These groupings are described in
Figure 3, and details on environment and algorithm integration in the
following subsections.

Figure 3: _Communication groupings among AI Arena processes._ MPI
communication occurs between each environment and any policy workers which are
commanding its entities. Separately, workers with a common policy communicate
to share network weights, experience, or gradient information.

The AI Arena uses this grouping structure to orchestrate training runs with
multiple environments, each having multiple entities, and contributing to
multiple policies. This constitutes one training “rounds”. While many
reinforcement learning approaches train for an indeterminate amount of time,
the AI Arena also handles the bringing down of distributed processes at
specific times. This allows many rounds to be conducted seamlessly in
sequence, enabling training schemes that progress through tournament
structures, evolution of populations of agents, or schedules of environment
parameters.

###  4.2 ENVIRONMENT INTEGRATION

As described previously, each environment is contained in a dedicated CPU
process, and MPI is used to communicate between an environment and external
agents. However, these details are fully obscured from a user looking to
attach an environment. Each environment is wrapped in a python class which
manages the communication to and from the environment, as well as MPI
communication to and from policy workers. Actions from every relevant process
are aggregated and sent to the worker as a single, local list communication,
and replies are received as a list and then disseminated back to the
respective parties (Figure 3 left). From the environment design perspective,
the environment only needs to implement the previously described AI Arena
Interface.

It is not uncommon for a python environment instance to itself manage an
external environment, and the AI Arena places no restrictions on this design.
For example, once an environment process is created by the AI Arena, an
environment instance will be created. At this point, the python environment
may bring up, for example, a Unity simulation which has its own lines of
communication. This communication is completely separated from the AI Arena
framework, just as the MPI calls are completely separated from the python
environment instance.

Figure 4: _Communication between an environment and single-agent policy
workers._ Left: An environment instance is contained in a process which
manages MPI communication. The core environment only needs to manage lists of
actions and lists of replies. Right: Single-agent policy workers communicate
locally with an environment proxy that handles MPI communication to a specific
entity, expressed as a single-agent gym interface. Existing single-agent
algorithms can be easily integrated using this paradigm.

###  4.3 ALGORITHM INTEGRATION

Just as the environment instances are contained in unique CPU processes, so
are workers contributing to the training of a distributed reinforcement
learning policy. MPI is again used here to maintain communication among a set
of workers, to share gradients, data, or network parameters. Distributed
algorithms looking to train with the AI Arena framework are encouraged to be
set up in this manner, using MPI for updates as in [2]. However, other forms
of communication could be used among processes at the user’s discretion.

While the AI Arena Interface encourages highly multi-entity environments, many
DRL algorithms are constructed to be single agent, and to communicate with the
standard gym interface. To bridge this gap, the AI Arena software framework
provides to each worker a proxy environment which implements the typical
single-agent OpenAI gym interface, and communicates with the true environment
on the backend. In this manner, a single-agent policy can be run without any
modification. From its perspective, it is operating a single agent in a single
environment, and may be unaware that other entities or other policies exist
alongside it (Figure 3 right).

While this provides simple integration for many problems, it is not always
sufficient to apply a single-agent algorithm to a multi-agent environment, and
multi-agent algorithms have been developed for that use case. The AI Arena
fully supports the assignment of multiple entities to a single policy worker
rather than one, and in this case the worker will receive lists of data about
several entities and be expected to reply with lists of actions. This enables,
for example, a single worker process to manage all of the entities within an
environment (or some subset). Scaling up to many environments, this would
simply result in distributed training of a multi-agent algorithm. Furthermore,
it is perfectly compatible to mix multi-entity and single-entity assignments,
such that some entities may be attached to single-agent algorithms while
others are pooled into multiagent algorithms.

Figure 5: _Python code and process diagram for a four-policy round robin
training scheme._ A hypothetical training scheme in which four policies train
simultaneously in a round-robin style with a 2v2 environment. Left: Given an
environment that conforms to the AI Arena Interface, the core content of the
script to run this training scheme is only three lines. Four policies are
defined, arranged into six environments, and the Arena is asked to run the
provided configuration. Right: The resulting processes and their basic
organization. With scaling turned on, this brings up 30$N$ processes to run
the configuration $N$ times over simultaneously.

###  4.4 TRAINING DESCRIPTIONS AND SCALABILITY

Finally, the AI Arena software framework is designed to orchestrate much of
the above implementation without any input from the user of the framework.
Training schemes that previously would be quite cumbersome to maintain can be
described in only a few lines of code (Figure 5 left). Furthermore, the AI
Arena software framework includes many built-in utilities and conveniences,
including example environments and common algorithms. Notably, a complex
training scheme can be automatically duplicated across available compute. For
example, the round-robin setup described in Figure 5 can be automatically
repeated $N$ times over such that $6N$ environments exist, connected to $24N$
workers representing 4 policies (Figure 5 right). Policy hooks allow for
automatic logging, saving, and restoration to ease the maintenance of such
complex setups. Additional utilities allow for policies to be duplicated so
that they may be developed over several rounds in population-based approaches.

##  5 RESULTS AND DISCUSSION

In this section we highlight results that illustrate some of the key features
of the AI Arena. To that end, we demonstrate the use of multi-agent, multi-
policy, distributed curriculum training in the AI Safety Tanksworld domain
[22], and demonstrate options for multi-agent algorithms in a cooperative
navigation domain [19]. Additional examples that demonstrate advanced training
schemes made possible by the AI Arena are included in the appendix.

###  5.1 AI SAFETY TANKSWORLD DOMAIN

AI Safety TanksWorld is an environment for AI safety research with three
essential aspects: competing performance objectives, human-machine teaming,
and multi-agent competition. The AI Safety TanksWorld aims to accelerate the
advancement of safe multi-agent decision-making algorithms by providing a
software framework to support competitions with both system performance and
safety objectives [22].

Figure 6: TanksWorld Multi-Agent Environment for AI Safety Research. These
images illustrate different views of the Tanksworld environment (Left) is a
birds-eye rendering of the environment, (Center) is an agent’s-view rendering,
and (Right) is the state representation actually provided as observations to
the RL algorithm

More specifically, the AI Safety TanksWorld [22] is a competitive 5 vs. 5
environment that challenges teams of agents to simultaneously win against the
opposing team, cooperate with diverse teammates including humans, and cope
with uncertainty in the environment.

To illustrate the expressiveness of the AI Arena for complex reinforcement
learning experimental design, we organized an experiment that compares
reinforcement learning training against multiple opponents simultaneously with
and without curriculum training. As shown in Figure 9, we train against 4
different opponent policies (i.e. static, random, aggressive, and passive
policies). The policy weights for the aggressive and passive policies were
pretrained via PPO and frozen. Curriculum training was used to slowly
introducing penalties for safety violations. The curriculum was composed of
increasing penalties for safety violations from 0 to .3 in increments of .05
distributed evenly over 4 million steps We compared the curriculum training
approach to a baseline approach without a curriculum that sets the penalty for
safety violations at .3. The results of the comparison are shown in Figure 8.
The non-curriculum baseline reaches at plateau at just below 0, while the
curriculum-based approach achieves a higher overall combined score for both
safety and performance. The accompanying code shown in Figure 9 illustrates
the expressiveness of the AI Arena abstractions for multi-agent training.

Figure 7: The AI Safety TanksWorld is composed of 10 controllable entities
divided into two teams. The AI Arena is used to facilitate training of
separate policies for each teams, and in this example all 5 team members use
and train the same policy. Figure 8: Round-based curriculum training organized
with the AI Arena (brown). Successive rounds of training increased the
difficulty by slowing introducing safety penalties over three rounds of
training with penalty weights (0,.05,.1,.15,.20,.25,.3). The baseline (green)
starts with the penalty threshold of $.3$. The result illustrates the value of
successive rounds of curriculum training for teams of tanks in the AI Safety
Challenge domain  Figure 9: _Python code and process diagram for a TanksWorld
training scheme against four enemy strategies._ Left: Core code snippet needed
to setup and run this training scheme. "ppo" refers to a new ppo policy that
is being trained, while the other policies refer to custom frozen policies.
Right: The resulting processes and their organization. Four environments are
created, each housing 10 entities. All blue tanks are contributing to a single
policy.

###  5.2 Multi-agent Algorithms

When working with environments containing multiple agents, it is sometimes
desirable to coordinate their behavior beyond treating them as copies of
single-agent policies. A recommended approach in this case is to use a common
critic architecture, as introduced in multi-agent deep distributed policy
gradient [19], which uses an understanding of collective rewards during
training to create individual policies that work together. To support such an
approach with the AI Arena, multiple entities can be assigned to a single
policy worker, such that a single worker coordinates the behavior of multiple
entities.

Figure 10: _Cooperative Navigation Environment and Behavior._ The Cooperative
Navigation environment has three targets (black squares) and three agents
(circles). The agents must coordinate to cover all targets. Left: Illustration
of environment and potential solution. Right: Traces of the testing behavior
from the learned MASAC policies. The actions were reduced in magnitude to
create slow paths to the targets. Some targets appear as outlines, showing
that the agent happened to start on or near that target. The traces are
interesting in that they show clear coordination but occasionally sub-optimal
pairings of entities and targets. If the actions were not reduced, such that
the entities reached the targets in only a handful of steps, these sub-
optimalities would have little consequence on score.

In this example we recreate the Cooperative Navigation environment from MADDPG
[19], and train to convergence using multi-agent soft actor-critic with the AI
Arena framework. In this version of the environment, there are 3 agents that
can move in 2D space and must navigate to cover three targets. Targets provide
a reward of +1 if they are occupied, so the optimal behavior is to have each
entity travel to a unique target, such that all targets are occupied. There is
no penalty for colliding with other agents. The environment runs for 300
steps, so the maximum theoretical score is 900 (all entities starting directly
on a target and staying there for the duration, for 300x3=900 points). Our
implementation of Multi-agent Soft Actor Critic (MASAC) is a direct extension
of soft actor critic [14] to the multi-agent domain using the common critic
framework initially described by MADDPG [19]. MASAC can naturally be
distributed. We run with 8 copies of the environment with 8 workers, each
worker coordinating three entities in its respective environment and training
with a common critic, sharing gradients among all workers at each update.

As seen in Figure 11, our agents converge to a cooperative set of behaviors
that clear 800 points on average, which is nearly optimal. The agents move
quite quickly in this environment, but we have slowed them down in testing to
create visualizations of their movements in Figure 10. While they do not
always attempt to reach the nearest target, they have coordinated in such a
way that all the targets become occupied. Crucially, they do not communicate
during testing, so it is only through training with the common critic that
they have learned complementary policies that can deploy independently while
still working together.

Figure 11: _Training Curves for MASAC and Comparisons._ Left: Entity
assignments for the three approaches: Truly multiagent policy (MASAC),
treating each entity as an SAC worker, and grouping all entities into a single
agent. In all cases, the assignment was duplicated over several environments
for distributed training. Right: The corresponding training curves for each
approach. MASAC was the only successful algorithm, making slow and deliberate
progress for roughly 18 million steps before levelling off. Figure 12: _Python
code and process diagram for a multi-agent algorithm._ Left: Core code snippet
needed to setup and run this training scheme. To route multiple entities to a
single worker (as needed for a common critic algorithm), the entities are
simple grouped by square brackets. Here we indicate that all entities 1-3
correspond to one worker for policy 1. Right: The resulting processes and
their organization. In our experiments, eight environments were created, each
having three entities but only one worker.

As points of comparison, we note some other approaches to this multi-agent
problem. The three entities in the environment could alternatively be treated
as separate workers for a single-agent SAC algorithm (comparison 1), or the
entities could be grouped together into a single agent that has a single state
and accepts one large action (comparison 2, may require slight environment
modifications). These comparisons are further described in Figure 11.

Neither of these comparisons were successful in our experiments. Treating the
agents as separate workers for SAC does not properly assign rewards to the
agents, since all three agents are collectively rewarded based on target
occupancy, and therefore the distributed SAC approach is not able to solve the
credit-assignment problem among multiple workers. Treating all three entities
as a single agent may suffer from a similar problem in that any rewards that
are experienced do not reflect credit for the action taken but rather a subset
of the action taken. Also note that it is impossible to have identical data
collection among the three approaches. MASAC ran on 8 environments (8 workers,
each controlling 3 entities), SAC ran on 6 environments (18 workers, 3 per
environment) and the combined entity SAC ran on 8 environments (8 workers,
each controlling 3 entities as one agent).

This example demonstrates compatibility with “true” multi-agent algorithms and
the ability to scale them. These approaches can be used alongside single-agent
algorithms at scale; it is perfectly acceptable to have some entities
coordinated in a multi-agent fashion while others are treated as independent
workers of a single-agent policy.

##  6 CONCLUSIONS

In this work, we introduce the AI Arena: a scalable framework with flexible
abstractions for distributed multi-agent reinforcement learning. Our aim is to
help enable RL research for the class of applications that involve multiple
teams of agents where each team may have unique learning strategies and where
agents within a given team may have localized views of the environment. Our
unique abstractions allow the single agent focused OpenAI Gym interface to be
used without modification for multi-agent environments from the perspective of
learning algorithms. We demonstrate the expressiveness of the abstractions for
training with multiple environmental configurations simultaneously in the
TanksWorld domain. We compared multiple strategies for multi-agent training in
the cooperative navigation domain. The AI Arena framework software is
available as open source software on github at
_https://github.com/cgrivera/ai-arena_. The Tanksworld environment for AI
safety research is available at _https://github.com/cgrivera/ai-safety-
challenge_.

## References

  * [1] David Abel.  simple_rl: Reproducible reinforcement learning in python.  In RML@ ICLR, 2019.
  * [2] Joshua Achiam.  Spinning up in deep reinforcement learning.(2018).  URL https://spinningup. openai. com, 2018.
  * [3] Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba.  Openai gym.  arXiv preprint arXiv:1606.01540, 2016.
  * [4] Noam Brown and Tuomas Sandholm.  Superhuman ai for multiplayer poker.  Science, 365(6456):885–890, 2019.
  * [5] Itai Caspi, Gal Leibovich, Gal Novik, and Shadi Endrawis.  Reinforcement learning coach, December 2017.
  * [6] Pablo Samuel Castro, Subhodeep Moitra, Carles Gelada, Saurabh Kumar, and Marc G Bellemare.  Dopamine: A research framework for deep reinforcement learning.  arXiv preprint arXiv:1812.06110, 2018.
  * [7] Carlo D’Eramo, Davide Tateo, Andrea Bonarini, Marcello Restelli, and Jan Peters.  Mushroomrl: Simplifying reinforcement learning research.  arXiv preprint arXiv:2001.01102, 2020.
  * [8] Prafulla Dhariwal, Christopher Hesse, Oleg Klimov, Alex Nichol, Matthias Plappert, Alec Radford, John Schulman, Szymon Sidor, Yuhuai Wu, and Peter Zhokhov.  Openai baselines, 2017.
  * [9] Linxi Fan, Yuke Zhu, Jiren Zhu, Zihua Liu, Orien Zeng, Anchit Gupta, Joan Creus-Costa, Silvio Savarese, and Li Fei-Fei.  Surreal: Open-source reinforcement learning framework and robot manipulation benchmark.  In Conference on Robot Learning, pages 767–782, 2018.
  * [10] Yasuhiro Fujita, Toshiki Kataoka, Prabhat Nagarajan, and Takahiro Ishikawa.  Chainerrl: A deep reinforcement learning library.  In Workshop on Deep Reinforcement Learning at the 33rd Conference on Neural Information Processing Systems, December 2019.
  * [11] Edgar Gabriel, Graham E Fagg, George Bosilca, Thara Angskun, Jack J Dongarra, Jeffrey M Squyres, Vishal Sahay, Prabhanjan Kambadur, Brian Barrett, Andrew Lumsdaine, et al.  Open mpi: Goals, concept, and design of a next generation mpi implementation.  In European Parallel Virtual Machine/Message Passing Interface Users’ Group Meeting, pages 97–104. Springer, 2004.
  * [12] Jason Gauci, Edoardo Conti, Yitao Liang, Kittipat Virochsiri, Yuchen He, Zachary Kaden, Vivek Narayanan, Xiaohui Ye, Zhengxing Chen, and Scott Fujimoto.  Horizon: Facebook’s open source applied reinforcement learning platform.  arXiv preprint arXiv:1811.00260, 2018.
  * [13] Sergio Guadarrama, Anoop Korattikara, Oscar Ramirez, Pablo Castro, Ethan Holly, Sam Fishman, Ke Wang, Ekaterina Gonina, Neal Wu, Chris Harris, et al.  Tf-agents: A library for reinforcement learning in tensorflow, 2018.
  * [14] Tuomas Haarnoja, Aurick Zhou, Kristian Hartikainen, George Tucker, Sehoon Ha, Jie Tan, Vikash Kumar, Henry Zhu, Abhishek Gupta, Pieter Abbeel, et al.  Soft actor-critic algorithms and applications.  arXiv preprint arXiv:1812.05905, 2018.
  * [15] Ashley Hill, Antonin Raffin, Maximilian Ernestus, Adam Gleave, Rene Traore, Prafulla Dhariwal, Christopher Hesse, Oleg Klimov, Alex Nichol, Matthias Plappert, et al.  Stable baselines.  GitHub repository, 2018.
  * [16] Arthur Juliani, Vincent-Pierre Berges, Esh Vckay, Yuan Gao, Hunter Henry, Marwan Mattar, and Danny Lange.  Unity: A general platform for intelligent agents.  arXiv preprint arXiv:1809.02627, 2018.
  * [17] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox, Ken Goldberg, Joseph Gonzalez, Michael Jordan, and Ion Stoica.  Rllib: Abstractions for distributed reinforcement learning.  In International Conference on Machine Learning, pages 3053–3062, 2018.
  * [18] Keng Wah Loon, Laura Graesser, and Milan Cvitkovic.  Slm lab: A comprehensive benchmark and modular software framework for reproducible deep reinforcement learning.  arXiv preprint arXiv:1912.12482, 2019.
  * [19] Ryan Lowe, Yi Wu, Aviv Tamar, Jean Harb, Pieter Abbeel, and Igor Mordatch.  Multi-agent actor-critic for mixed cooperative-competitive environments.  CoRR, abs/1706.02275, 2017.
  * [20] Philipp Moritz, Robert Nishihara, Stephanie Wang, Alexey Tumanov, Richard Liaw, Eric Liang, Melih Elibol, Zongheng Yang, William Paul, Michael I Jordan, et al.  Ray: A distributed framework for emerging $\\{$AI$\\}$ applications.  In 13th $\\{$USENIX$\\}$ Symposium on Operating Systems Design and Implementation ($\\{$OSDI$\\}$ 18), pages 561–577, 2018.
  * [21] Abhishek Nandy and Manisha Biswas.  Reinforcement learning with keras, tensorflow, and chainerrl.  In Reinforcement Learning, pages 129–153. Springer, 2018.
  * [22] Corban G Rivera, Olivia Lyons, Arielle Summitt, Ayman Fatima, Ji Pak, William Shao, Robert Chalmers, Aryeh Englander, Edward W Staley, I Wang, et al.  Tanksworld: A multi-agent environment for ai safety research.  arXiv preprint arXiv:2002.11174, 2020.
  * [23] Michael Schaarschmidt, Alexander Kuhnle, and Kai Fricke.  Tensorforce: A tensorflow library for applied reinforcement learning.  Web page, 2017.
  * [24] Michael Schaarschmidt, Sven Mika, Kai Fricke, and Eiko Yoneki.  Rlgraph: Modular computation graphs for deep reinforcement learning.  arXiv preprint arXiv:1810.09028, 2018.
  * [25] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al.  Mastering the game of go with deep neural networks and tree search.  nature, 529(7587):484–489, 2016.
  * [26] Adam Stooke and Pieter Abbeel.  rlpyt: A research code base for deep reinforcement learning in pytorch.  arXiv preprint arXiv:1909.01500, 2019.
  * [27] Richard S. Sutton and Andrew G. Barto.  Reinforcement Learning: An Introduction.  A Bradford Book, Cambridge, MA, USA, 2018.
  * [28] Federico Tesser.  Distributed message passing with mpi4py.  In Euroscipy 2016, 2016.
  * [29] Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michaël Mathieu, Andrew Dudzik, Junyoung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al.  Grandmaster level in starcraft ii using multi-agent reinforcement learning.  Nature, 575(7782):350–354, 2019.
  * [30] Lianmin Zheng, Jiacheng Yang, Han Cai, Weinan Zhang, Jun Wang, and Yong Yu.  Magent: A many-agent reinforcement learning platform for artificial collective intelligence.  arXiv preprint arXiv:1712.00600, 2017.

##  Appendix A Appendix

###  A.1 Advanced Examples

The simplicity of the AI Arena interface allows for orchestration of high-
level training schemes for which the inner loop is an entire DRL training run.
Here we demonstrate the usefulness of this approach towards a population-based
training example for Atari Pong.

####  A.1.1 Population-based Methods

The AI Arena paradigm of allowing arbitrary connections between entities and
policies can provide for some interesting and potentially powerful training
schemes. Here we demonstrate a toy variation on PPO in which three policies
are maintained instead of one, and this population of three is evolved over
many rounds of training on the Atari game Pong. We use nine environment
instances such that each PPO instance is a small distributed learner using
three of the nine environments. Every 250,000 training steps, the policies are
paused and only the most performant is allowed to continue by copying its
network weights over to the two lesser policies. For comparison, we also
examine the standard approach of applying all computational resources to a
single policy (all nine environments to one policy). Results from both
approaches are shown in Figure 13.

Figure 13: _Evolution over PPO on Atari Pong._ Nine environments are broken
into groups of three to train three independent PPO policies. Periodically,
the best policy of the three is copied over to replace the other two, forming
a simple round-based evolution scheme. This is compared with using all nine
environments to train a single policy instance. Left: Two different
assignments of environments to policies, showing the evolution case (red) and
the typical use case (purple) Right: Three training runs of each approach,
showing evolution in red and typical PPO in purple. Orange "hairs" visualize
policies that were not selected to continue at the end of a generation (every
250k training steps). Point X shows an especially poor performing policy that
was eliminated via evolution.

While a simple example, this demonstrates the ability to set up complex
training schemes that may involve populations of agents, multiple rounds of
training, and many environments. A population-based approach may not result in
performance gains on Pong, but it provides another option for training which
may be difficult to set up in other software frameworks. In particular, this
training scheme shows some interesting properties that can be seen in Figure
13. Namely, two of the three runs converge with less data than their standard
counterparts (although not necessarily less network updates, as they have less
data per update). The population-based approach also seems to protect against
irregularities in training, such as poor initialization or catastrophic
updates, by weeding out poor performing policies. One population member had an
especially poor training generation in one of the runs (point X in Figure 13),
which was abandoned in the next generation. If this had been a single PPO run,
this catastrophic performance drop could have long-term effects. In a bit of
irony, the purpose of the PPO algorithm is to clip network updates such that
these performance hits are avoided in the first place, but it is not
infallible.


# On ideals in quantales, II

Amartya Goswami  [1] Department of Mathematics and Applied Mathematics,
University of Johannesburg, South Africa; [2] National Institute for
Theoretical and Computational Sciences (NITheCS), South Africa.
<EMAIL_ADDRESS>
###### Abstract.

As a natural extension of the ongoing development of a theory of ideals in
commutative quantales with an identity element, this article aims to study
into the analysis of certain topological properties exhibited by distinguished
classes of ideals. These ideals are equipped with quantale topologies. The
primary objectives encompass characterizing quantale spaces exhibiting
sobriety, examining several conditions pertaining to quasi-compactness, and
demonstrating that quantale spaces comprised of proper ideals adhere to the
spectral properties as defined by Hochster. We introduce the notion of a
strongly disconnected spaces and show that for a quantale with zero Jacobson
radical, strongly disconnected quantale spaces containing all maximal ideals
of the quantale imply existence of non-trivial idempotent elements in the
quantale. Additionally, a sufficient criterion for establishing the
connectedness of a quantale space is presented. Finally, we discuss on
continuous maps between quantale spaces.

###### Key words and phrases:

Quantale; quasi-compactness, sobriety, spectral space

###### 1991 Mathematics Subject Classification:

06F07; 54B35

##  1\. Introduction

The concept of introducing topological structures to different classes of
ideals traces its origins back to [Sto37]. In that work, a specific topology
referred to as the Stone topology or hull-kernel topology was established for
the set of prime ideals in a Boolean ring.

Subsequently, in [Jac45], a similar hull-kernel topology, recognized as the
Jacobson topology, was imposed on the collection of primitive ideals in a
ring. Notably, [McC48] noted the applicability of this same topology to the
set of generalized prime ideals as defined therein.

In the context of affine schemes, as explored in [Gro60], the hull-kernel
topology (known as the Zariski topology) was considered for the set of prime
ideals in a commutative ring. Building upon this, [Olb16] demonstrated the
utilization of the Zariski topology in establishing that every ideal can be
uniquely represented in terms of irreducible ideals, eliminating redundancy.

Further investigations ensued in the field. [HJ65] studied into the hull-
kernel topology’s behaviour concerning minimal prime ideals within a
commutative ring. Meanwhile, [Azi08] embarked on a study of strongly
irreducible ideals in a commutative ring, specifically within the context of
the Zariski topology. Exploration of the topological properties exhibited by
distinct classes of ideals in a commutative ring extended to [DG22] and
[FGS23].

Beyond the realm of rings, investigations into the imposition of topologies on
diverse classes of ideals or substructures extend to various algebraic
structures. For instance, within the domain of $R$-modules, [FFS16] examines
topologies applied to ideals and proper ideals, while [FS20] delves into
primary ideals and principal ideals with constructible topologies.

Turning to semirings, topologies applied to prime, maximal, and strongly
irreducible ideals are explored in [Isé56, Isé56’, IM56, Gol99]. In the
specialized context of $(m,n)$-semirings, [H al.18] investigates the
structural aspects of $n$-ary prime $k$-ideals, $n$-ary prime full $k$-ideals,
$n$-ary prime ideals, maximal ideals, and strongly irreducible ideals.

In the domain of semigroups, [Gos23”] delves into the application of
topologies to primitive ideals, while [Gos22] undertakes a similar exploration
for various types of subgroups within a group.

A comprehensive exploration of different topologies related to prime spectra,
including variations like minimal prime, coprime, and fully prime spectra, is
available in [A11, A15, A10, AF14, AH19, AKF16, AP17, BH08, MMS97, MS06,
Tek09, UTS15]. Moreover, the study of prime spectra within multiplicative
lattices is addressed in [Fac23, FFJ22].

The objective of this paper is to extend these investigations into the realm
of lattice theory, specifically within the context of commutative unital
quantales. This expansion is not only reasonable but also well-founded, as the
collection of ideals in a commutative ring inherently forms a quantale.
Furthermore, by considering intersection as the multiplication operation, we
uncover additional instances of quantales that emerge as substructures of the
aforementioned algebraic entities.

For a comprehensive exploration of the broader theory of quantales (also known
as a special type of multiplicative lattices), we direct our readers to
references such as [Ros90, Mul86, And74, Dil36, Dil62, KP08, War37, WD39]. The
concept of an ideal within a quantale was introduced in [Gos23’], where
algebraic properties of specific distinguished classes of quantale ideals were
examined.

Within the scope of this paper, we introduce the adaptation of a specific
topology, termed the “quantale topology,” which can be applied to any class of
ideals in a quantale. Our primary goal is to study various topological aspects
intrinsic to these corresponding spaces.

Let us briefly outline the paper’s structure. In Section 2, we revisit crucial
definitions and pertinent facts concerning quantale ideals. The groundwork for
imposing quantale topologies on various classes of ideals is laid out in
Section 3. We elucidate our choice of subbasic closed sets as the fundamental
constituents for constructing our topology. Section 4 deals with the
examination of quasi-compactness of quantale spaces, offering a range of
characterizations. The discussion then shifts to lower separation axioms for
quantale spaces, encompassing $T_{0}$, $T_{1}$, and sobriety in Section 5.
Demonstrating the spectrality of the quantale space of proper ideals is also
the focus of study in this section. Finally, in Section 6, we talk about the
concept of continuity among quantale spaces and introduce a distinctive form
of connectedness that correlates with the presence of idempotent elements in a
quantale.

##  2\. Preliminaries

Let us now revisit fundamental definitions pertaining to ideals of quantales.
For a detailed study of these concepts and related results, we refer our
readers to [Gos23’]. Recall that an _unital commutative quantale_ is a
complete lattice $(\mathcal{Q},\preccurlyeq,\bot,\top)$ endowed with a binary
operation $\&$ on $\mathcal{Q}$, satisfying the following axioms:

  1. (1)

$x\mbox{{\small{$\&$}}}(y\mbox{{\small{$\&$}}}z)=(x\mbox{{\small{$\&$}}}y)\mbox{{\small{$\&$}}}z$,

  2. (2)

$x\mbox{{\small{$\&$}}}y=y\mbox{{\small{$\&$}}}x$,

  3. (3)

$x\mbox{{\small{$\&$}}}\left(\bigvee\limits_{\lambda\in\Lambda}y_{\lambda}\right)=\bigvee\limits_{\lambda\in\Lambda}\left(x\mbox{{\small{$\&$}}}y_{\lambda}\right)$,

  4. (4)

$x\mbox{{\small{$\&$}}}\top=x$,

for all $x,$ $y,$ $y_{\lambda}$, $z\in\mathcal{Q}$, and for all
$\lambda\in\Lambda$, where $\Lambda$ is an index set. Throughout this paper,
unless otherwise stated, by a “quantale,” we shall always refer to a
commutative quantale with the top element $\top$ as the identity with respect
to $\&$. We shall use the notation $x^{n}$ to denote
$x\mbox{{\small{$\&$}}}\cdots\mbox{{\small{$\&$}}}x$ (repeated $n$ times). A
nonempty subset $I$ in $\mathcal{Q}$ is called an _ideal_ , if for all $x,$
$y,$ $l\in\mathcal{Q}$, the following two axioms hold:

  1. (1)

$x,$ $y\in I$ implies that $x\vee y\in I$, and

  2. (2)

$x\in I$ and $l\preccurlyeq x$ implies that $l\in I$.

We shall denote the set of all ideals in $\mathcal{Q}$ by
$\mathcal{I}_{\mathcal{Q}}$, and by $\theta$, the zero ideal in $\mathcal{Q}$.
Let $\mathcal{Q}$ be a quantale. The _meet_
$\bigwedge_{\lambda\in\Lambda}I_{\lambda}$ of ideals
$\\{I\\}_{\lambda\in\Lambda}$ in $\mathcal{Q}$ is given by their intersection.
The _join_ of ideals $\\{I\\}_{\lambda\in\Lambda}$ of $\mathcal{Q}$ is defined
by

$\bigvee_{\lambda\in\Lambda}I_{\lambda}=\left\\{l\in\mathcal{Q}\mid
l\preccurlyeq\bigvee_{\text{finite}}x_{\lambda}\;\text{for}\;x_{\lambda}\in
I_{\lambda}\right\\}.$

The _product_ of two ideals $I$ and $J$ in $\mathcal{Q}$ is defined as

$I\mbox{{\small{$\&$}}}J=\left\\{l\in\mathcal{Q}\mid
l\preccurlyeq\bigvee_{\text{finite}}\left\\{x\mbox{{\small{$\&$}}}y\mid x\in
I,y\in J\right\\}\right\\}.$

If $S$ is a nonempty subset in a $\mathcal{Q}$, then the _ideal generated by_
$S$ is defined by

$\langle S\rangle=\left\\{x\in\mathcal{Q}\mid
x\preccurlyeq\bigvee_{i=1}^{n}a_{i},\;\text{for
some}\;n\in\mathds{N}\;\text{and}\;a_{i}\in\mathcal{Q}\mbox{{\small{$\&$}}}S\right\\}.$

Let $\mathcal{Q}$ be a quantale. An ideal $I$ of $\mathcal{Q}$ is called
_proper_ if $I\neq\mathcal{Q}$, and we shall denote the set of all proper
ideals in $\mathcal{Q}$ by $\mathcal{I}^{+}_{\mathcal{Q}}$. A proper ideal $M$
of $\mathcal{Q}$ is called _maximal_ if there is no other proper ideal $I$ of
$\mathcal{Q}$ properly containing $M$, and we denote the set of all maximal
ideals in $\mathcal{Q}$ by $\mathrm{Max}_{\mathcal{Q}}$. A quantale
$\mathcal{Q}$ with exactly one maximal ideal is called _local_. An ideal $I$
of $\mathcal{Q}$ is called _minimal_ if $I\neq 0$ and $I$ properly contains no
other nonzero ideals in $\mathcal{Q}$.

A proper ideal $P$ of $\mathcal{Q}$ is called _prime_ if
$x\mbox{{\small{$\&$}}}y\in P$ implies $x\in P$ or $y\in P$ for all $x,$
$y\in\mathcal{Q}$. By $\mathrm{Spec}_{\mathcal{Q}}$, we denote the set of all
prime ideals in $\mathcal{Q}$. An ideal $P$ of $\mathcal{Q}$ is called
_minimal prime_ if $P$ is both a minimal ideal and a prime ideal. A quantale
$\mathcal{Q}$ is called _Noetherian_ if every ascending chain of ideals in
$\mathcal{Q}$ is eventually stationary. An element $e$ in a quantale
$\mathcal{Q}$ is called _idempotent_ if $e^{2}=e$. The _Jacobson radical_
$\mathcal{J}(\mathcal{Q})$ of a quantale $\mathcal{Q}$ is defined as the
intersection of all maximal ideals in $\mathcal{Q}$.

A proper ideal $P$ in a quantale $\mathcal{Q}$ is called _primary_ if for $x,$
$y\in\mathcal{Q}$ and $x\mbox{{\small{$\&$}}}y\in I$ imply that $x\in I$ or
$y^{n}\in I$ for some $n\in\mathds{N}$. An ideal $I$ of $\mathcal{Q}$ is
called _irreducible_ if $J\wedge J^{\prime}=I$ implies that either $J=I$ or
$J^{\prime}=I$ for all $J,$ $J^{\prime}\in\mathcal{I}_{\mathcal{Q}}$. An ideal
$I$ of $\mathcal{Q}$ is called _strongly irreducible_ if $J\wedge
J^{\prime}\preccurlyeq I$ implies that either $J\preccurlyeq I$ or
$J^{\prime}\preccurlyeq I$ for all $J,$
$J^{\prime}\in\mathcal{I}_{\mathcal{Q}}$. By $\Sigma_{\mathcal{Q}}$, we shall
mean either $\mathcal{I}^{+}_{\mathcal{Q}}$ or any subclass of ideals of it.
All the classes of ideals mentioned above are examples of
$\Sigma_{\mathcal{Q}}$.

##  3\. Quantale topology

Consider a quantale $\mathcal{Q}$ and let $\Sigma_{\mathcal{Q}}$ be a class of
ideals in $\mathcal{Q}$. Our objective is to define a topology on
$\Sigma_{\mathcal{Q}}$. We aim to achieve a topology that coincides with the
Stone topology when $\Sigma_{\mathcal{Q}}$ consists of maximal ideals or with
the Zariski topology when $\Sigma_{\mathcal{Q}}$ consists of prime ideals in
$\mathcal{Q}$, and so on. In case of rings, however, it is well-known that
neither the Zariski topology, the Stone topology, nor any hull-kernel topology
can be imposed on an arbitrary class of ideals.

We will now demonstrate that a similar limitation applies to quantales. The
following theorem characterizes the “largest” class of ideals in a quantale on
which a hull-kernel topology can be imposed. This result, which was initially
established in [McK53, §2.2, p.11] for rings, is adapted to the context of
quantales. For a class $\Sigma_{\mathcal{Q}}$ of ideals of $\mathcal{Q}$, we
consider the collection

$\mathcal{B}_{\Sigma_{\mathcal{Q}}}:=\left\\{X^{\uparrow}\mid
X\preccurlyeq\mathcal{I}_{\mathcal{Q}}\right\\}$

of subsets of $\Sigma_{\mathcal{Q}}$ given by:

(3.1)
$X^{\uparrow}=\begin{cases}\left\\{J\in\Sigma_{\mathcal{Q}}\mid\mathcal{D}_{X}:=\wedge
X\preccurlyeq J\right\\},&X\neq\emptyset;\\\
\emptyset,&X=\emptyset.\end{cases}$

Whenever $X=\\{I\\},$ we shall write $I^{\uparrow}$ for $\\{I\\}^{\uparrow}.$

###### Theorem 3.1. —

The collection $\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ induces a closed-set
topology on $\Sigma_{\mathcal{Q}}$ if and only if the class
$\Sigma_{\mathcal{Q}}$ of ideals satisfies the following property:

(3.2) $I\wedge J\preccurlyeq S\;\;\text{implies}\;\;I\preccurlyeq
S\;\;\text{or}\;\;J\preccurlyeq S,$

for all $I,$ $J\in\mathcal{I}_{\mathcal{Q}}$ and for all
$S\in\Sigma_{\mathcal{Q}}.$

###### Proof.

We shall show that $(-)^{\uparrow}$ is a Kuratowski closure operator on
$\Sigma_{\mathcal{Q}}$ if and only if the class $\Sigma_{\mathcal{Q}}$ of
ideals satisfies the condition (3.2). The facts that
$\emptyset^{\uparrow}=\emptyset$ and $X\preccurlyeq X^{\uparrow}$ follow from
(3.2), whereas the condition $(X^{\uparrow})^{\uparrow}=X^{\uparrow}$ follows
from the monotone property of $(-)^{\uparrow}$. Finally, what remains is to
show that $(X\vee Y)^{\uparrow}=X^{\uparrow}\vee Y^{\uparrow}$ holds for any
$X,$ $Y\preccurlyeq\mathcal{I}_{\mathcal{Q}}$ if and only if
$\Sigma_{\mathcal{Q}}$ satisfies (3.2). The fact that $X^{\uparrow}\vee
Y^{\uparrow}\preccurlyeq(X\vee Y)^{\uparrow}$ follows from monotonicity of
$(-)^{\uparrow}$ and from the properties proven above. Suppose $J\in(X\vee
Y)^{\uparrow}.$ Then $(\mathcal{D}_{X})\vee(\mathcal{D}_{Y})\preccurlyeq J,$
and hence, $J\in X^{\uparrow}\vee Y^{\uparrow}$ if and only if
$\Sigma_{\mathcal{Q}}$ satisfies (3.2). ∎

###### Remark 3.2. —

It is worth noting that the class of ideals in a quantale that satisfies
property (3.2) defined earlier corresponds exactly to the class of strongly
irreducible ideals. Moreover, it can be easily observed that the classes of
maximal, prime, and minimal prime ideals in a quantale also satisfy property
(3.2). Therefore, these classes serve as suitable candidates upon which a
hull-kernel topology can be imposed.

Whenever $(-)^{\uparrow}$ is a Kuratowski closure operator on a class
$\Sigma_{\mathcal{Q}}$ of ideals in $\mathcal{Q}$, the collection
$\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ of subsets of $\Sigma_{\mathcal{Q}}$ are
exactly the closed subsets of the hull-kernel topology induced on
$\Sigma_{\mathcal{Q}}$. Therefore, $\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ is a
closed basis of that topology. In fact, we can obtain even more, as
demonstrated by the following theorem.

###### Theorem 3.3. —

If $\mathcal{Q}$ is a quantale, then $(-)^{\uparrow}$ is a Kuratowski closure
operator on a class of ideals $\Sigma_{\mathcal{Q}}$ if and only if
$\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ is a closed basis of the corresponding
hull-kernel topology.

###### Proof.

If $(-)^{\uparrow}$ is a Kuratowski closure operator on
$\Sigma_{\mathcal{Q}}$, then obviously $\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ is
a closed basis of the corresponding hull-kernel topology. To prove the
converse, we shall use the following fact: if $\mathcal{B}$ is a closed basis
on a topological space $X$, then the closure operator induced by the topology
is expressible in terms of $\mathcal{B}$, in the sense that

$\mathcal{C}(T)={\wedge}\\{B\in\mathcal{B}\mid T\preccurlyeq B\\},$

for any $T\preccurlyeq X$. Suppose that $\mathcal{B}_{\Sigma_{\mathcal{Q}}}$
is a closed basis for $\Sigma_{\mathcal{Q}}$. Let $\mathcal{C}(\cdot)$ be the
desired Kuratowski closure operator. It suffices to show that
$(\cdot)^{\uparrow}=\mathcal{C}(\cdot)$. For any
$T\preccurlyeq\Sigma_{\mathcal{Q}}$, we obtain

$\mathcal{C}(T)=\bigwedge\left\\{S^{\uparrow}\in\mathcal{B}_{\Sigma_{\mathcal{Q}}}\mid
T\preccurlyeq S^{\uparrow}\right\\}.$

Since $T^{\uparrow}\in\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ and $T\preccurlyeq
T^{\uparrow}$, we deduce from the equality
$(T^{\uparrow})^{\uparrow}=T^{\uparrow}$ that $\mathcal{C}(T)\preccurlyeq
T^{\uparrow}$. To have the reverse inclusion, let $J\in T^{\uparrow}$.
Consider any $S\preccurlyeq\Sigma_{\mathcal{Q}}$ with $T\preccurlyeq
S^{\uparrow}$. Then, we have

$\mathcal{D}_{S}=\wedge S^{\uparrow}\preccurlyeq\mathcal{D}_{T}.$

Since $\mathcal{D}_{T}\preccurlyeq J$, as $J\in T^{\uparrow}$, it follows that
$\mathcal{D}_{S}\preccurlyeq J$. Thus, $J\in S^{\uparrow}$. In all then, we
have shown that $J$ belongs to every set included in the intersection that
describes $\mathcal{C}(T)$, and so $J\in\mathcal{C}(T)$, and so
$T^{\uparrow}\preccurlyeq\mathcal{C}(T)$. Consequently,
$T^{\uparrow}=\mathcal{C}(T)$, and hence, $(\cdot)^{\uparrow}$ is a Kuratowski
closure operator. ∎

###### Remark 3.4. —

From Theorem 3.3, we can conclude that if $(-)^{\uparrow}$ is not a Kuratowski
closure operator, then $\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ cannot serve as a
closed basis for a topology on a class $\Sigma_{\mathcal{Q}}$ of ideals in
$\mathcal{Q}$. Therefore, our only option is to utilize
$\mathcal{B}_{\Sigma_{\mathcal{Q}}}$ as a closed subbasis to generate a
topology on $\Sigma_{\mathcal{Q}}$.

Let $\mathcal{Q}$ be a quantale and $\Sigma_{\mathcal{Q}}$ be a class of
ideals in $\mathcal{Q}$. A _quantale topology_ on $\Sigma_{\mathcal{Q}}$,
denoted by $\tau_{\mathcal{Q}}$, is defined as the topology generated by the
sets of the form (3.1) as a subbasis of closed sets. When the class
$\Sigma_{\mathcal{Q}}$ of ideals is equipped with this quantale topology, it
is referred to as a _quantale space_. For the sake of brevity, a quantale
space $(\Sigma_{\mathcal{Q}},\tau_{\mathcal{Q}})$ will be denoted simply as
$\Sigma_{\mathcal{Q}}$, employing the same notation as the underlying set of
the topological space.

###### Remark 3.5. —

(1) The concept we have referred to as quantale topology is also known by
other names in different contexts. In the context of general lattices, it is
referred to as coarse lower topology (see [DST19, A.8, p. 589]) or lower
topology (see [G.al.03, Definition O-5.4, p. 43]). In the context of rings, it
is called as ideal topology (see [DG22]).

(2) From Theorem 3.3, it follows that a quantale topology coincides with a
Zariski topology or a Stone topology when the underlying class of ideals is
respectively prime or maximal. Furthermore, in a more general setting, a
quantale topology can be viewed as a hull-kernel topology whenever the
underlying set is the class of strongly irreducible ideals or any subclass
thereof.

##  4\. Quasi-compactness

Recall that a topological space is said to be _quasi-compact_ if it satisfies
the finite intersection property (or, equivalently, every open cover has a
finite subcover), without assuming the space to be $T_{2}$. It is well-known
that spectrum of prime ideals of a ring endowed with Zariski topology is
quasi-compact. We first provide a similar result encompassing more classes of
ideals of a quantale. However, the proof of this result relies on the
Alexander’s subbasis theorem.

###### Theorem 4.1. —

Let $\mathcal{Q}$ be a quantale. If $\Sigma_{\mathcal{Q}}$ is a class of
ideals in $\mathcal{Q}$ with the property that
$\mathrm{Max}_{\mathcal{Q}}\preccurlyeq\Sigma_{\mathcal{Q}},$ then the
quantale space $\Sigma_{\mathcal{Q}}$ quasi-compact.

###### Proof.

Suppose $\\{\mathcal{K}_{\lambda}\\}_{\lambda\in\Lambda}$ is a family of
subbasic closed sets of $\Sigma_{\mathcal{Q}}$ with
$\bigwedge_{\lambda\in\Lambda}\mathcal{K}_{\lambda}=\emptyset.$ This implies
that $\mathcal{K}_{\lambda}=X_{\lambda}^{\uparrow}$, for all
$\lambda\in\Lambda$ and $X_{\lambda}\preccurlyeq\Sigma_{\mathcal{Q}}$ such
that

$\emptyset=\bigwedge_{\lambda\in\Lambda}X_{\lambda}^{\uparrow}=\left(\bigvee_{\lambda\in\Lambda}X_{\lambda}\right)^{\uparrow}.$

The second equality indeed holds. For

$J\in\left(\bigvee_{\lambda\in\Lambda}X_{\lambda}\right)^{\uparrow}\Leftrightarrow\mathcal{D}_{\bigvee_{\lambda\in\Lambda}X_{\lambda}}\preccurlyeq
J\Leftrightarrow\forall\lambda\in\Lambda,\mathcal{D}_{X_{\lambda}}\preccurlyeq
J\Leftrightarrow\forall\lambda\in\Lambda,J\in
X_{\lambda}^{\uparrow}\Leftrightarrow
J\in\bigwedge_{\lambda\in\Lambda}X_{\lambda}^{\uparrow}.$

If $\bigvee_{\lambda\in\Lambda}X_{\lambda}\neq\mathcal{Q},$ then there exists
$M\in\mathrm{Max}_{\mathcal{Q}}$ such that
$\bigvee_{\lambda\in\Lambda}X_{\lambda}\preccurlyeq M.$ Moreover,

$\mathcal{D}_{X_{\lambda}}\preccurlyeq
X_{\lambda}\preccurlyeq\bigvee_{\lambda\in\Lambda}X_{\lambda}\preccurlyeq M,$

for all $\lambda\in\Lambda.$ Therefore, by hypothesis, $M\in
X_{\lambda}^{\uparrow}=\mathcal{K}_{\lambda},$ for all $\lambda\in\Lambda$, a
contradiction. Hence $\bigvee_{\lambda\in\Lambda}X_{\lambda}=\mathcal{Q},$ and
thus $\top\in\bigvee_{\lambda\in\Lambda}X_{\lambda}.$ This implies the
existence of a finite subset $\\{\lambda_{\scriptscriptstyle
1},\ldots,\lambda_{\scriptscriptstyle n}\\}$ of $\Lambda$ such that
$\top=\bigvee_{i=1}^{n}x_{\lambda_{i}},$ for some $x_{\lambda_{i}}\in
X_{\lambda_{i}}$. Hence $\mathcal{Q}=\bigvee_{i=1}^{n}X_{\lambda_{i}},$ which
establishes the finite intersection property. Therefore,
$\Sigma_{\mathcal{Q}}$ is quasi-compact by Alexander’s subbasis theorem. ∎

###### Corollary 4.2. —

The quantale spaces of proper, maximal, prime, radical, primary, strongly
irreducible, and irreducible ideals are all quasi-compact.

While Theorem 4.1 presents a sufficient condition for a quantale space to
possess quasi-compactness, the subsequent proposition delivers a
characterization of these spaces. However, before proceeding with the
proposition, we need to establish a prerequisite result from the domain of
topological spaces.

###### Lemma 4.3. —

If $X$ is a quasi-compact, T0-space, then every chain in $X$ has an upper
bound.

###### Proof.

Let $K\preccurlyeq X$ be a chain and let $\Omega:=\left\\{\mathcal{C}(k)\mid
k\in K\right\\}$. Then $\mathcal{G}$ is clearly a chain of nonempty closed
sets of $X$ and thus it has the finite intersection property. Since $X$ is
quasi-compact, there is a point $z\in\bigwedge\Omega$ and, by definition,
$k\preccurlyeq z$, for every $k\in K$, and this proves the result. ∎

###### Proposition 4.4. —

Let $\Sigma_{\mathcal{Q}}$ be a class of ideals of a quantale $\mathcal{Q}$.
Suppose $\mathcal{M}(\Sigma_{\mathcal{Q}})$ denotes the set of maximal
elements of $\Sigma_{\mathcal{Q}}$. Then the following are equivalent:

  1. (1)

$\Sigma_{\mathcal{Q}}$ is quasi-compact;

  2. (2)

for every $I\in\Sigma_{\mathcal{Q}},$ there exists a
$J\in\mathcal{M}(\Sigma_{\mathcal{Q}})$ such that $I\preccurlyeq J$, and
$\mathcal{M}(\Sigma_{\mathcal{Q}})$ is quasi-compact.

###### Proof.

To show (2)$\Rightarrow$(1), let $\mathcal{O}$ be an open cover of
$\Sigma_{\mathcal{Q}}$. Then, $\mathcal{O}$ also covers
$\mathcal{M}(\Sigma_{\mathcal{Q}})$, and thus there is a finite subcover
$\mathcal{O}^{\prime}$ of $\mathcal{M}(\Sigma_{\mathcal{Q}})$. Consider
$I\in\Sigma_{\mathcal{Q}}$. By hypothesis, there is
$J\in\mathcal{M}(\Sigma_{\mathcal{Q}})$ such that $I\preccurlyeq J$, and
$O\in\mathcal{O}^{\prime}$ such that $J\in O$. Then, $I\in O$, and thus
$\mathcal{O}^{\prime}$ is also a finite subcover for $\Sigma_{\mathcal{Q}}$.
Hence, $\Sigma_{\mathcal{Q}}$ is quasi-compact. Next we prove that
(1)$\Rightarrow$(2). Suppose that $I\in\Sigma_{\mathcal{Q}}$, and consider

$\Sigma_{\mathcal{Q}}^{\prime}:=I^{\uparrow}\wedge\Sigma_{\mathcal{Q}}.$

Then, $\Sigma_{\mathcal{Q}}^{\prime}$ is a closed subset of
$\Sigma_{\mathcal{Q}}$, and thus it is quasi-compact; by Lemma 4.3, every
ascending chain in $\Sigma_{\mathcal{Q}}^{\prime}$ is bounded, and hence by
Zorn’s Lemma $\Sigma_{\mathcal{Q}}^{\prime}$ has maximal elements, i.e.,
$I\preccurlyeq J,$ for some $J\in\mathcal{M}(\Sigma_{\mathcal{Q}})$. Now,
suppose $\mathcal{O}$ is an open cover of $\mathcal{M}(\Sigma_{\mathcal{Q}})$.
Then, $\mathcal{O}$ is also an open cover of $\Sigma_{\mathcal{Q}}$, and thus
it admits a finite subcover, which will be also a finite subcover of
$\mathcal{M}(\Sigma_{\mathcal{Q}})$. Thus, $\mathcal{M}(\Sigma_{\mathcal{Q}})$
is quasi-compact. ∎

###### Remark 4.5. —

While Theorem 4.1 can now be derived from Proposition 4.4, it is worth noting
that utilizing Theorem 4.1 simplifies the verification of quasi-compactness
for a quantale space (see Corollary 4.2).

In Theorem 4.1, we have observed that the inclusion of all maximal ideals in a
class $\Sigma_{\mathcal{Q}}$ of ideals is a sufficient condition for the
quantale space $\Sigma_{\mathcal{Q}}$ to possess quasi-compactness. In the
subsequent proposition, we establish that this inclusion is also a necessary
condition for the class of finitely generated (proper) ideals.

###### Proposition 4.6. —

Let $\mathcal{Q}$ be a quantale. If the quantale space $\Sigma_{\mathcal{Q}}$
of finitely generated proper ideals is quasi-compact, then
$\mathrm{Max}_{\mathcal{Q}}\preccurlyeq\Sigma_{\mathcal{Q}}$.

###### Proof.

Suppose that $M\in\mathrm{Max}_{\mathcal{Q}}$ with the property that $M$ is
not finitely generated. We show that $\Sigma_{\mathcal{Q}}$ is not quasi-
compact. Let us consider the collection of closed subspaces:

$\mathcal{K}=\left\\{\langle x\rangle^{\uparrow}\wedge\Sigma_{\mathcal{Q}}\mid
x\in M\right\\}.$

We claim that $\bigwedge\mathcal{K}=\emptyset$. If not, let
$I\in\bigwedge\mathcal{K}$. Then $I$ is finitely generated and $M\preccurlyeq
I$. Since $M$ is not a finitely generated ideal, we must have $I\neq M,$ and
that implies $I=\mathcal{Q}$, which contradicts the fact that
$\Sigma_{\mathcal{Q}}$ is proper. But clearly $\mathcal{K}$ has the finite
intersection property, and hence, $\Sigma_{\mathcal{Q}}$ is not quasi-compact.
∎

Let us now explore the characterization of Noetherian quantales in terms of
quasi-compact quantale spaces.

###### Theorem 4.7. —

A quantale $\mathcal{Q}$ is Noetherian if and only if every quantale space
$\Sigma_{\mathcal{Q}}$ is quasi-compact.

###### Proof.

To obtain the claim, it suffices to show that $\mathcal{Q}$ is Noetherian if
and only if $\mathcal{I}_{\mathcal{Q}}$ is a Noetherian quantale space. We
show that every subclass $\Sigma_{\mathcal{Q}}$ of $\mathcal{I}_{\mathcal{Q}}$
is quasi-compact. Consider a collection

$\mathcal{L}:=\left\\{\Sigma_{\mathcal{Q}}\wedge
I_{\lambda}^{\uparrow}\right\\}_{\lambda\in\Lambda}$

of subbasic closed sets of $\Sigma_{\mathcal{Q}}$ with the finite intersection
property. Since $\mathcal{Q}$ is Noetherian, the ideal
$J:=\bigvee_{\lambda\in\Lambda}I_{\lambda}$ is finitely generated, and assume
that $J=\langle\alpha_{1},\ldots,\alpha_{n}\rangle$. For every $1\leqslant
j\leqslant n$, there exists a finite subset $\Omega_{j}$ of $\Lambda$ such
that $\alpha_{j}\in\bigvee_{\lambda\in\Omega_{j}}I_{\lambda}$. If
$\Omega:=\bigcup\limits_{j=1}^{n}\Omega_{j}$, it immediately follows that
$J=\bigvee_{\omega\in\Omega}I_{\omega}$. Hence we have

$\bigwedge\,\mathcal{L}=\Sigma_{\mathcal{Q}}\wedge
J^{\uparrow}=\Sigma_{\mathcal{Q}}\wedge\left\\{\bigvee_{\omega\in\Omega}I_{\omega}\right\\}^{\uparrow}=\bigwedge_{\omega\in\Omega}\left(\Sigma_{\mathcal{Q}}\wedge
I_{\omega}^{\uparrow}\right)\neq\emptyset,$

since $H$ is finite and $\mathcal{L}$ has the finite intersection property.
Then the claim follows by Alexander’s subbasis Theorem. Conversely, let
$\mathcal{I}_{\mathcal{Q}}$ be a Noetherian space and that there exists an
ideal $I\in\mathcal{I}_{\mathcal{Q}}$ that is not finitely generated. Then the
subspace

$\Sigma_{\mathcal{Q}}:=\\{J\in\mathcal{I}_{\mathcal{Q}}\mid J\;\text{is
finitely generated and}\;J\preccurlyeq I\\}$

is not quasi-compact. In fact, the collection of closed sets

$\mathcal{L}:=\left\\{\Sigma_{\mathcal{Q}}\wedge\langle
i\rangle^{\uparrow}\right\\}_{i\in I}$

of $\Sigma_{\mathcal{Q}}$ clearly has the finite intersection property, but
has empty intersection. ∎

In the proposition that follows, we present a sufficient condition for the
quasi-compactness of the quantale space of regular ideals in $\mathcal{Q}$.

###### Proposition 4.8. —

If $\mathcal{Q}$ is a local quantale, then the quantale space of regular
ideals is quasi-compact.

###### Proof.

Let $\mathcal{Q}$ be a local quantale with a unique maximal ideal $M$. We will
consider two cases:

  1. (1)

If $M$ is regular, the claim follows from Theorem 4.1.

  2. (2)

If $M$ consists of zero-divisors, then every regular element in $\mathcal{Q}$
is invertible. Consequently, the set of regular ideals in $\mathcal{Q}$ is
empty. By definition, an empty set is also quasi-compact. ∎

##  5\. Separation axioms

The purpose of this section is to investigate the lower separation axioms of
quantale spaces, specifically $T_{0}$, $T_{1}$, and sobriety. Notably,
sobriety will hold significant importance as it aids in characterizing
quantale spaces that are spectral. We commence by presenting an assertive yet
obvious separation property that holds for all quantale spaces.

###### Proposition 5.1. —

Every quantale space is a $T_{0}$-space.

###### Proof.

Let $I,$ $J\in\Sigma_{\mathcal{Q}}$ such that $I^{\uparrow}=J^{\uparrow}.$
Since $J\in I^{\uparrow},$ we have $I\preccurlyeq J$. Similarly, we obtain
$J\preccurlyeq I$. Hence, $I=J$, and we have the claim. ∎

To proceed further with separation axioms, we first need the following lemma.
This lemma, however, is also going to be useful in the context of
connectedness.

###### Lemma 5.2. —

Every subbasic closed set of the form $\left\\{I^{\uparrow}\mid I\in
I^{\uparrow}\right\\}$ of a quantale space $\Sigma_{\mathcal{Q}}$ is
irreducible.

###### Proof.

In fact, we show more, i.e., $I^{\uparrow}=\mathcal{C}(I)$ for every $I$ such
that $I\in I^{\uparrow}$. By definition of $\mathcal{C}(\cdot)$ and
hypothesis, we have $\mathcal{C}(I)\preccurlyeq I^{\uparrow}$. We obtain the
other inclusion by considering the following two cases:

(1) $\mathcal{C}(I)=\Sigma_{\mathcal{Q}}$. Then we have the claim from the
fact that

$\Sigma_{\mathcal{Q}}=\mathcal{C}(I)\preccurlyeq
I^{\uparrow}\preccurlyeq\Sigma_{\mathcal{Q}}.$

(2) $\mathcal{C}(I)\neq\Sigma_{\mathcal{Q}}$. Since $\mathcal{C}(I)$ is a
closed set, there exists an index set $\Omega$ such that for each
$\omega\in\Omega$, there is a positive integer $n_{\omega}$ and $I_{\omega
1},\dots,I_{\omega n_{\omega}}\in\mathcal{I}_{\mathcal{Q}}$ such that

$\mathcal{C}(I)={\bigwedge_{\omega\in\Omega}}\left({\bigvee_{i\,=1}^{n_{\omega}}}I_{\omega
i}^{\uparrow}\right).$

Since $\mathcal{C}(I)\neq\Sigma_{\mathcal{Q}}$, we may assume that
${\bigvee_{i\,=1}^{n_{\omega}}}I_{\omega i}^{\uparrow}$ is non-empty for each
$\omega$. Therefore, $I\in{\bigvee_{i\,=1}^{n_{\omega}}}I_{\omega
i}^{\uparrow}$, for each $\omega$, and from which we can deduce that
$I^{\uparrow}\preccurlyeq{\bigvee_{i=1}^{n_{\omega}}}I_{\omega i}^{\uparrow}$,
that is, $I^{\uparrow}\preccurlyeq\mathcal{C}(I)$. ∎

###### Corollary 5.3. —

Every non-empty subbasic closed set of the quantale space
$\mathcal{I}^{+}_{\mathcal{Q}}$ is irreducible.

###### Theorem 5.4. —

A quantale space $\Sigma_{\mathcal{Q}}$ is a $T_{1}$-space if and only if
$M\in\Sigma_{\mathcal{Q}}$ for all $M\in\mathrm{Max}_{\mathcal{Q}}$.

###### Proof.

Suppose that $I\in\Sigma_{\mathcal{Q}}$. Then $I\in I^{\uparrow}$, and so, by
Corollary 5.3, $\mathcal{C}(I)=I^{\uparrow}$. Let $M$ be a maximal ideal of
$\mathcal{Q}$ such that $I\preccurlyeq M$. Then

$M\in I^{\uparrow}=\mathcal{C}(I)=\\{I\\},$

where the second equality follows from the fact that $\Sigma_{\mathcal{Q}}$ is
a $T_{1}$-space. Therefore $M=I$, showing that
$\Sigma_{\mathcal{Q}}\preccurlyeq\mathrm{Max}_{\mathcal{Q}}$. Conversely,
$M^{\uparrow}=\\{M\\}$ holds for every $M\in\mathrm{Max}_{\mathcal{Q}}$, so
that $M\in M^{\uparrow}$. Hence, $\mathcal{C}(M)=\\{M\\}$, by Corollary 5.3
and this proves the desired claim. ∎

Next we consider the separation axiom of sobriety. Recall that a topological
space $X$ is called _sober_ if every non-empty irreducible closed subset
$\mathcal{K}$ of $X$ is of the form: $\mathcal{K}=\mathcal{C}(\\{x\\})$, the
closure of an unique singleton set $\\{x\\}$. The following result
characterizes sober quantale spaces.

###### Theorem 5.5. —

An quantale space $\Sigma_{\mathcal{Q}}$ is sober if and only if

(5.1) $\bigwedge_{\begin{subarray}{c}I\preccurlyeq
J,\,J\in\Sigma_{\mathcal{Q}}\end{subarray}}J^{\uparrow}\preccurlyeq
I^{\uparrow},$

for every non-empty irreducible subbasic closed subset of $I^{\uparrow}$ of
$\Sigma_{\mathcal{Q}}$.

###### Proof.

If $\Sigma_{\mathcal{Q}}$ is a sober space and $I^{\uparrow}$ is a non-empty
irreducible subbasic closed subset of $\Sigma_{\mathcal{Q}}$, then
$I^{\uparrow}=\mathcal{C}(\\{J\\})=J^{\uparrow}$ for some
$J\in\Sigma_{\mathcal{Q}}$, and we have

$\displaystyle J=\bigwedge_{\begin{subarray}{c}I\preccurlyeq
J,\,J\in\Sigma_{\mathcal{Q}}\end{subarray}}J^{\uparrow}\in\Sigma_{\mathcal{Q}}.$

Conversely, suppose the condition (5.1) holds for every non-empty irreducible
subset of $\Sigma_{\mathcal{Q}}$. Let $\mathcal{K}$ be an irreducible closed
subset of $\Sigma_{\mathcal{Q}}$. Then

$\mathcal{K}=\bigwedge_{i\in\Omega}\left(\bigvee_{j=1}^{m}J_{ji}^{\uparrow}\right)$

for some ideals $J_{ji}$ of $\mathcal{Q}$. Since $\mathcal{K}$ is irreducible,
for every $i\in\Omega$ there exists an ideal $J_{ji}$ of $\mathcal{Q}$ such
that $\mathcal{K}\preccurlyeq
J_{ji}^{\uparrow}\preccurlyeq\bigvee_{j=1}^{m}J_{ji}^{\uparrow}$ and thus, if
$Y=\bigvee_{i\in\Omega}J_{ji}$, then we have

$\mathcal{K}=\bigwedge_{i\in\Omega}J_{ji}^{\uparrow}=Y^{\uparrow}=\left(\bigwedge_{\begin{subarray}{c}Y\preccurlyeq
L,\,L\in\Sigma_{\mathcal{Q}}\end{subarray}}L\right)^{\uparrow}=\mathcal{C}\left({\bigwedge_{\begin{subarray}{c}Y\preccurlyeq
L,\,L\in\Sigma_{\mathcal{Q}}\end{subarray}}L}\right).$

The uniqueness part of the claim follows from Proposition 5.1. ∎

###### Corollary 5.6. —

The quantale spaces of proper ideals, prime ideals, and strongly irreducible
ideals are sober.

Hochster in [Hoc69] has characterized spectral spaces as spectra of prime
ideals of rings endowed with Zariski topology. A topological space is called
_spectral_ if it is quasi-compact, sober, admitting a basis of quasi-compact
open subspaces that is closed under finite intersections. Our next goal is to
prove the following result.

###### Theorem 5.7. —

Let $\mathcal{Q}$ be a quantale. Then the quantale space
$\mathcal{I}^{+}_{\mathcal{Q}}$ of proper ideals is spectral;

###### Proof.

We prove the claim by applying the following lemma111for a proof of Lemma 5.8,
we refer to [Gos23, Lemma 2.2].

###### Lemma 5.8. —

A quasi-compact, sober, open subspace of a spectral space is spectral.

The advantage of the above lemma is that we can avoid proving the existence of
a compact open basis and closedness of open compact sets under finite
intersections.

Now the quantale space $\mathcal{I}_{\mathcal{Q}}$ of all ideals is spectral
follows from the fact that an algebraic lattice endowed with coarse lower
topology222which is same as quantale topology in our context. is a spectral
space (see [Pri94, Theorem 4.2]). Since the quantale space
$\mathcal{I}^{+}_{\mathcal{Q}}$ is a subspace of $\mathcal{I}_{\mathcal{Q}}$,
according to Lemma 5.8, we need to verify that $\mathcal{I}^{+}_{\mathcal{Q}}$
is quasi-compact, sober, and is an open subspace of the quantale space
$\mathcal{I}_{\mathcal{Q}}$. Now quasi-compactness and sobriety respectively
follow from Corollary 4.2 and Corollary 5.6. Therefore, what remains is to
show that the quantale space $\mathcal{I}^{+}_{\mathcal{Q}}$ is open in
$\mathcal{I}_{\mathcal{Q}}$. Since $\mathcal{Q}\in\mathcal{I}_{\mathcal{Q}},$
by Lemma 5.2, $\mathcal{Q}=\mathcal{Q}^{\uparrow}=\mathcal{C}(\mathcal{Q}),$
and therefore
$\mathcal{I}_{\mathcal{Q}}\backslash\mathcal{I}^{+}_{\mathcal{Q}}$ is closed,
and that implies $\mathcal{I}^{+}_{\mathcal{Q}}$ is open. ∎

##  6\. Connectedness and continuity

Like Alexander subbasis theorem, there is no characterization of connectedness
in terms of subbasic closed sets. Nevertheless, we wish to present a
disconnectivity result of quantale spaces of a quantale that bears resemblance
to the fact that if spectrum of prime ideals (of a commutative ring with
identity) endowed with Zariski topology is disconnected, then the ring has a
proper idempotent element (see [Bou72, §4.3, Corollary 2]).

We say a closed subbasis $\mathcal{S}$ of a topological space $X$ _strongly
disconnects_ $X$ if there exist two non-empty subsets $A,$ $B$ of
$\mathcal{S}$ such that $X=A\vee B$ and $A\wedge B=\emptyset$. It is clear
that if some closed subbasis strongly disconnects a topological space, then
the space is disconnected. Also, if a space is disconnected, then some closed
subbasis (for instance the collection of all its closed subspaces) strongly
disconnects it.

###### Proposition 6.1. —

Suppose that $\mathcal{Q}$ is a quantale with
$\mathcal{J}(\mathcal{Q})=\theta$. Let $\Sigma_{\mathcal{Q}}$ be a spectrum of
$\mathcal{Q}$ containing all maximal ideals in $\mathcal{Q}.$ If the subbasis
$\mathcal{S}$ of the quantale space $\Sigma_{\mathcal{Q}}$ strongly
disconnects $\Sigma_{\mathcal{Q}}$, then $\mathcal{Q}$ has a non-trivial
idempotent element.

###### Proof.

Let $I$ and $J$ be ideals in $\mathcal{Q}$ such that (1) $I^{\uparrow}\wedge
J^{\uparrow}=\emptyset,$ (2) $I^{\uparrow}\vee
J^{\uparrow}=\Sigma_{\mathcal{Q}},$ and (3)
$I^{\uparrow}\neq\emptyset,J^{\uparrow}\neq\emptyset.$ Since
$I^{\uparrow}\wedge J^{\uparrow}=(I\vee J)^{\uparrow}$, we therefore have
$(I\vee J)^{\uparrow}=\emptyset$ and hence $I\vee J=\mathcal{Q}$ because
$\Sigma_{\mathcal{Q}}$ contains all maximal ideals in $\mathcal{Q}$. On the
other hand,

$\Sigma_{\mathcal{Q}}=I^{\uparrow}\vee
J^{\uparrow}\preccurlyeq(I\mbox{{\small{$\&$}}}J)^{\uparrow},$

which then implies that $I\mbox{{\small{$\&$}}}J$ is contained in every
maximal ideal in $\mathcal{Q}$, and is therefore the zero ideal since
$\mathcal{Q}$ has zero $\mathcal{J}(\mathcal{Q})$. Note that the condition (3)
implies that neither $I$ nor $J$ is the entire quantale $\mathcal{Q}$. So the
equality $I\vee J=\mathcal{Q}$ furnishes non-zero elements $x\in I$ and $y\in
J$ such that $x\vee y=\top$. Since $x\mbox{{\small{$\&$}}}y=\bot$ as
$x\mbox{{\small{$\&$}}}y\in I\mbox{{\small{$\&$}}}J=\theta$, we therefore have

$x=x\mbox{{\small{$\&$}}}(x\vee y)=x^{2}\vee(x\mbox{{\small{$\&$}}}y)=x^{2},$

showing that $x$ is a non-zero idempotent element in $\mathcal{Q}$. Since
$I\neq\mathcal{Q}$, and $x\neq\top$, and hence $x$ is a non-trivial idempotent
element in $\mathcal{Q}$. ∎

The next theorem gives a sufficient condition for a quantale space to be
connected.

###### Theorem 6.2. —

If a spectrum $\Sigma_{\mathcal{Q}}$ of a quantale $\mathcal{Q}$ contains the
zero ideal, then the quantale space $\Sigma_{\mathcal{Q}}$ is connected.

###### Proof.

Since $\Sigma_{\mathcal{Q}}=\bot^{\uparrow}$ and irreducibility implies
connectedness, the desired claim now follows from Corollary 5.3. ∎

###### Corollary 6.3. —

Quantale spaces of proper, finitely generated, principal ideals are connected.

We now discuss about continuous maps between quantale spaces of quantales.
Recall that a map $\phi\colon\mathcal{Q}\to\mathcal{Q}^{\prime}$ from
$\mathcal{Q}$ to $\mathcal{Q}^{\prime}$ is called a _quantale homomorphism_ if

  1. (1)

$x\preccurlyeq x^{\prime}$ implies that $\phi(x)\preccurlyeq\phi(x^{\prime});$

  2. (2)

$\phi(x\vee x^{\prime})=\phi(l)\vee\phi(l^{\prime});$

  3. (3)

$\phi(x\wedge x^{\prime})=\phi(x)\wedge\phi(x^{\prime});$

  4. (4)

$\phi(x\mbox{{\small{$\&$}}}x^{\prime})=\phi(x)\mbox{{\small{$\&$}}}\phi(x^{\prime}),$

for all $x,$ $x^{\prime}\in\mathcal{Q}$. Suppose that
$\phi\colon\mathcal{Q}\to\mathcal{Q}^{\prime}$ is a quantale homomorphism. If
$J$ is an ideal in $\mathcal{Q}^{\prime}$, then the _contraction of_ $J$,
denoted by $J^{c}$, is defined by
$\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(J).$ Observe that
although inverse image of an ideal under a quantale homomorphism is an ideal,
but the same may not hold for an arbitrary class $\Sigma_{\mathcal{Q}}$ of
ideals. To resolve this problem, we need to impose that property on a class of
ideal. We say a class $\Sigma_{\mathcal{Q}}$ of ideals satisfies the
_contraction_ property if for any quantale homomorphism
$\phi\colon\mathcal{Q}\to\mathcal{Q}^{\prime},$ the inverse image
$\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(I^{\prime})$ is in
$\Sigma_{\mathcal{Q}}$, whenever $I^{\prime}$ is in
$\Sigma_{\mathcal{Q}^{\prime}}.$ Since the sets $\\{I^{\uparrow}\mid
I\in\mathcal{I}_{\mathcal{Q}}\\}$ only form a (closed) subbasis, all our
arguments need to be at this level rather than just closed sets.

###### Proposition 6.4. —

Let $\Sigma_{\mathcal{Q}}$ be a spectrum satisfying the contraction property.
Let $\phi\colon\mathcal{Q}\to\mathcal{Q}^{\prime}$ be a quantale homomorphism
and $I^{\prime}\in\Sigma_{\mathcal{Q}^{\prime}}.$

  1. (1)

The induced map
$\phi_{*}\colon\Sigma_{\mathcal{Q}^{\prime}}\to\Sigma_{\mathcal{Q}}$ defined
by
$\phi_{*}(I^{\prime})=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(I^{\prime})$
is continuous.

  2. (2)

If $\phi$ is surjective, then the quantale space
$\Sigma_{\mathcal{Q}^{\prime}}$ is homeomorphic to the closed subspace
$\mathrm{Ker}\phi^{\uparrow}$ of the quantale space $\Sigma_{\mathcal{Q}}.$

  3. (3)

The subset $\phi_{*}(\Sigma_{\mathcal{Q}^{\prime}})$ is dense in
$\Sigma_{\mathcal{Q}}$ if and only if
$\mathrm{Ker}\phi\preccurlyeq\bigwedge_{I\in\Sigma_{\mathcal{Q}}}I.$

###### Proof.

To show (1), let $I\in\mathcal{I}_{\mathcal{Q}}$ and $I^{\uparrow}$ be a
subbasic closed set of the quantale space $\Sigma_{\mathcal{Q}}.$ Then

$\displaystyle\phi_{*}^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(I^{\uparrow})$
$\displaystyle=\left\\{I^{\prime}\in\Sigma_{\mathcal{Q}^{\prime}}\mid\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(I^{\prime})\in
I^{\uparrow}\right\\}$
$\displaystyle=\left\\{I^{\prime}\in\Sigma_{\mathcal{Q}^{\prime}}\mid\phi(I)\preccurlyeq
I^{\prime}\right\\}=\left\langle\phi(I)\right\rangle^{\uparrow},$

and hence the map $\phi_{*}$ continuous.

(2) Observe that
$\mathrm{Ker}\phi\preccurlyeq\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(I^{\prime})$
follows from the fact that $\theta\preccurlyeq I^{\prime}$ for all
$I^{\prime}\in\Sigma_{\mathcal{Q}^{\prime}}.$ It can thus been seen that
$\phi_{*}(I^{\prime})\in\mathrm{Ker}\phi^{\uparrow},$ and hence
$\mathrm{Im}\phi_{*}=\mathrm{Ker}\phi^{\uparrow}.$ If
$I^{\prime}\in\Sigma_{\mathcal{Q}^{\prime}},$ then

$\phi\left(\phi_{*}\left(I^{\prime}\right)\right)=\phi\left(\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I^{\prime}\right)\right)=I^{\prime}.$

Thus $\phi_{*}$ is injective. To show that $\phi_{*}$ is closed, first we
observe that for any subbasic closed set $I^{\uparrow}$ of
$\Sigma_{\mathcal{Q}^{\prime}}$, we have

$\displaystyle\phi_{*}\left(I^{\uparrow}\right)$
$\displaystyle=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I^{\uparrow}\right)$
$\displaystyle=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left\\{I^{\prime}\in\Sigma_{\mathcal{Q}^{\prime}}\mid
I\preccurlyeq
I^{\prime}\right\\}=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}(I)^{\uparrow}.$

Now if $\mathcal{K}$ is a closed subset of $\Sigma_{\mathcal{Q}^{\prime}}$ and
$\mathcal{K}=\bigwedge_{\omega\in\Omega}\left(\bigvee_{i\,=1}^{n_{\omega}}I_{i\lambda}^{\uparrow}\right),$
then

$\displaystyle\phi_{*}(\mathcal{K})$
$\displaystyle=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(\bigwedge_{\omega\in\Omega}\left(\bigvee_{i=1}^{n_{\omega}}I_{i\lambda}^{\uparrow}\right)\right)$
$\displaystyle=\bigwedge_{\omega\in\Omega}\,\bigvee_{i=1}^{n_{\omega}}\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I_{i\omega}\right)^{\uparrow}$

a closed subset of $\Sigma_{\mathcal{Q}}.$ Since by (1), $\phi_{*}$ is
continuous, we have the desired claim.

(3) First we wish to show:

$\mathcal{C}\left(\phi_{*}\left(I^{\prime\uparrow}\right)\right)=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I^{\prime}\right)^{\uparrow},$

for all ideals $X^{\prime}$ of $\mathcal{Q}^{\prime}$. For that, let
$J\in\phi_{*}\left(I^{\prime\uparrow}\right).$ This implies
$\phi\left(J\right)\in I^{\prime\uparrow},$ and that means
$I^{\prime}\preccurlyeq\phi(J).$ Therefore,
$J\in\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I^{\prime}\right)^{\uparrow}.$
Since
$\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I^{\prime\uparrow}\right)=\phi^{\raisebox{0.60275pt}{$\scriptscriptstyle-1$}}\left(I^{\prime}\right)^{\uparrow}$,
the other inclusion follows. If we take $I^{\prime}$ as the trivial ideal
$\theta^{\prime}$ in $\mathcal{Q}^{\prime}$, the above identity reduces to
$\mathcal{C}\left(\phi_{*}\left(\Sigma_{\mathcal{Q}^{\prime}}\right)\right)=\mathrm{Ker}\phi^{\uparrow},$
and hence $\mathrm{Ker}\phi^{\uparrow}$ to be equal to $\Sigma_{\mathcal{Q}}$
if and only if
$\mathrm{Ker}\phi\preccurlyeq\bigwedge_{I\in\Sigma_{\mathcal{Q}}}I.$ ∎

## References

  * [1]
  * [A11] J.Y. Abuhlail, A Zariski topology for modules, Commun. Algebra, 39(11) (2011), 4163–4182.
  * [A15] ———, Zariski topologies for coprime and second submodules, Algebra Colloq., 22(47) (2015), 47–72.
  * [A10] R. Ameri, Some properties of Zariski topology of multiplication modules, Houst. J. Math., 36(2) (2010), 337–344.
  * [AF14] H. Ansari-Toroghy and F. Farshadifar, The Zariski topology on the second spectrum of a module, Algebra Colloq., 21(4) (2014), 671–688.
  * [AH19] J. Abuhlail and H. Hroub, Zariski-like topologies for lattices with applications to modules over associative rings, J. Alg. Appl., 18(7) (2019), 1950131, 34 pp.
  * [AKF16] H. Ansari-Toroghy, S. Keyvani, F. Farshadifar, The Zariski topology on the second spectrum of a module (II), Bull. Malays. Math. Sci. Soc., 39(3) (2016), 1089–1103.
  * [AP17] H. Ansari-Toroghy and S. S. Pourmortazavi, On the minimal submodules of a module, Matematiche (Catania) 72(1) (2017), 185–200.
  * [And74] D. D. Anderson, Multiplicative lattices, PhD dissertation, University of Chicago, 1974.
  * [Azi08] A. Azizi, Strongly irreducible ideals, _J. Aust. Math. Soc._ , 84 (2008), 145–154.
  * [Bou72] N. Bourbaki, _Elements of mathematics: Commutative Algebra_ , Addison-Wesley, Reading, MA, 1972.
  * [BH08] M. Behboodi and M. R. Haddadi, Classical Zariki topology of modules and spectral spaces I, Int. Electron. J. Algebra, 4 (2008), 104–130.
  * [Dil62] R. P. Dilworth, Abstract commutative ideal theory, Pacific J. Math., 12 (1962), 481–498.
  * [Dil36] by same author, Abstract residuation over lattices, Bull. Amer. Math. Soc., 44 (1938), 262–268.
  * [DG22] T. Dube and A. Goswami, Ideal spaces: An extension of structure spaces of rings, J. Algebra Appl., (online) (2022).
  * [DST19] M. Dickmann, N. Schwartz, and M. Tressel, _Spectral spaces_ , Cambridge Univ. Press, 2019.
  * [Fac23] A. Facchini, Algebraic structures from the point of view of complete multiplicative lattices, in Algebra and coding theory, 113–131, Contemp. Math., 785, Amer. Math. Soc., 2023.
  * [FFJ22] by same author, C. A. Finocchiaro and G. Janelidze, Abstractly constructed prime spectra, Algebra Universalis, 83(8) (2022), 38 pp.
  * [FFS16] by same author, M. Fontana, and D. Spirito, A topological version of Hilbert’s Nullstellensatz, J. Algebra, 461 (2016), 25–41.
  * [FGS23] by same author, A. Goswami, and D. Spirito, Distinguished classes of ideal spaces and their topological properties, Comm. Algebra, 51(4), (2023), 1752–1760.
  * [FS20] by same authorand D. Spirito, Suprema in spectral spaces and the constructible closure, New York J. Math., 26 (2020), 1064–1092.
  * [Gol99] J. S. Golan, Semirings and their applications, Springer, 1999.
  * [Gos23] A. Goswami, Proper spaces are spectral, Appl. Gen. Topol., 24(1), (2023), 95–99.
  * [Gos23’] by same author, On ideals in quantales, I, arXiv:5045264 [math.RA].
  * [Gos23”] by same author, Primitive ideals and Jacobson’s structure spaces of semigroups, arXiv:4512251 [math.GR].
  * [Gos22] by same author, Normal structure spaces of groups, Rendiconti del Circolo Matematico di Palermo Series 2 (2022) (online ready).
  * [Gro60] A. Grothendieck, _Éléments de géométrie algébrique I, Le langage des schémas_ , Inst. Hautes Études Sci. Publ. Math., No. 4., 1960.
  * [G.al.03] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, _Continuous lattices and domains_ , Cambridge Univ. Press, 2003.
  * [Hoc69] M. Hocster, Prime ideal structure in commutative rings, Trans. Am. Math. Soc., 142 (1969), 43–60.
  * [HJ65] M. Henriksen and M. Jerison, The space of minimal prime ideals in a commutative ring, Trans. Amer. Math. Soc., 115 (1965), 110–130.
  * [H al.18] K. Hila, S. Kar, Sh. Kuka, and K. Naka, Some structure spaces of generalized semirings. Filomat 32(13) (2018), 4461–4472.
  * [Isé56] K. Iséki, Ideal theory of semiring , Proc. Japan. Acad., 32 (1956), 554–559.
  * [Isé56’] by same author, Notes on topological spaces. V. On structure spaces of semiring, Proc. Japan Acad., 32 (1956), 426–429.
  * [IM56] by same authorand Y. Miyanaga, Notes on topological spaces. III. On space of maximal ideals of semiring, Proc. Japan Acad., 32 (1956), 325-328 .
  * [Jac45] N. Jacobson, A topology for the set of primitive ideals in an arbitrary rings, Proc. Nat. Acad. Sei. U.S.A., 31 (1945), 333–338.
  * [KP08] D. Kruml and J. Paseka, Algebraic and categorical aspects of quantales, in Handbook of algebra, vol. 5, Elsevier, 2008.
  * [McC48] N. H. McCoy, _Rings and ideals_ , Mathematical Association of America, 1948.
  * [McK53] J. D. McKnight, Jr., _On the characterisation of rings of functions_ , Purdu doctoral thesis, 1953.
  * [Mul86] C. J. Mulvey, &, Rend. Circ. Mat. Palermo (2) Suppl. No. 12 (1986), 99–104.
  * [MMS97] R. L. McCasland, M. E. Moore, and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra, 25 (1997), 79–103.
  * [MS06] R. L. McCasland and P. F. Smith, Zariski spaces of modules over arbitrary rings, Comm. Algebra, 34(11) (2006), 3961–3973.
  * [Olb16] B. Olberding, Topological aspects of irredundant intersections of ideals and valuation rings in _Multiplicative ideal theory and factorization theory_ , 170 (2016), 277–307.
  * [Pri94] H. A. Priestley, Intrinsic spectral topologies, in: Papers on general topology and applications (Flushing, NY, 1992), 728, 78–95, New York Acad. Sci., New York, 1994.
  * [Ros90] K. I. Rosenthal, Quantales and their applications, Pitman research notes in Math. No. 234, Longman, Scientific and Technical, 1990.
  * [Sto37] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Am. Math. Soc., 41 (1937), 375–481.
  * [Tek09] U. Tekir, The Zariski topology on the prime spectrum of a module over noncommutative rings, Algebra Colloq., 16(4) (2009), 691–698.
  * [UTS15] G. Ulucak, U. Tekir, and K. P. Shum, On ${Spec(M)}$ and separation axioms between $T_{0}$ and $T_{1}$, Southeast Asian Bull. Math., 39 (2015), 717–725.
  * [War37] M. Ward, Residuation in structures over which a multiplication is defined, Duke Math. J., 3 (1937), 627–636.
  * [WD39] by same authorand R. P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc., 45 (1939), 335–354.