1. Introduction
Directed graphs and similar structures, such as labelled transition systems and Kripke frames, are mathematical objects often used to represent, by means of operational semantics, the behaviour of (nondeterministic) computer programs [SOS]. For this reason a large body of research has focused on the study of logics for expressing useful properties of directed graphs. Among these, modal logic (see, .e.g, [BdRVModal, Stirling96, modallogic2008book]) and its extensions (e.g., CTL [CES83], modal calculus [Kozen83], etc.) play a fundamental role. After decades of research, the current state of knowledge regarding modal logics is substantial:

Model Theory: the class of transition (often referred to as “relational”) structures interpreting the language of modal logics and their interplay with modal formulas is well understood. This theory includes key concepts such as that of bisimulation, algorithmically relevant properties such as the finite model property, expressiveness and definability results and advanced constructions such as ultraproducts. We refer to [GorankoOtto2007] for an overview.

Algebraic Semantics: another natural approach to give semantics to modal logic is algebraic: formulas are interpreted over algebras equipped with operations corresponding to the connectives of the logic and subject to certain axioms. In the case of basic modal logic (i.e., system K [modallogic2008book]), the signature is and the algebras considered are called modal Boolean algebras and satisfy the usual axioms of Boolean algebras together with the additional axioms and for the modal connective. Starting with the seminal works of Jónsson, McKinsey and Tarski ([bennett_1946, jonssontarski1951, jonssontarski1952], a precise correspondence between algebraic and transition semantics has been established. They key tool being used is that of Stone duality: to each Boolean algebra there corresponds a certain topological “dual” space , and the modal operation of each modal Boolean algebra corresponds to a transition (topologically closed) relation :
modal Boolean algebra (topological) Kripke frame .
The correspondence is in fact a duality of categories when morphisms between Kripke frames are defined using the framework of coalgebra theory [KKV2004, Jacobs2016]. This duality provides deep mathematical insights and is considered by Johan van Benthem as “one of the three pillars of wisdom in the edifice of modal logic” [JVB1984] (the other two being completeness and correspondence theory). We refer to [SambinVaccaro1988] and [KKV2004] for an overview.

Axiomatizations and Proof Systems: axiomatizations (sound and complete with respect to the semantics) have been found for modal logic and many of its extensions, including CTL [reynolds2001] and modal calculus [WalukiewiczCompletenessofKozen]. Furthermore, structural (analytic) proof systems based on Gentzen’s sequent calculus have been designed. These constitute the purely syntactical side of the theory of modal logic. We refer to [prooftheorymodallogic] for a general introduction and to [Studer07, DHL2006, doumanephd] for a selection of some recent results.
1.1. Probabilistic logics
Despite their wide applicability, directed graph structures are not adequate for modelling all kinds of programs. Most notably probabilistic programs, such as those involving commands for generating random numbers (e.g., x=rand()
in C++
), are naturally modelled by Markov chains or similar structures (e.g., Markov decision processes). See, e.g.,
[BaierKatoenBook] for an overview. Consequently, a number of logics specifically designed to express properties of Markov chains have been investigated: e.g., Kozen’s probabilistic PDL [Kozen1981, Kozen1983], Larsen–Skou modal logic [LS91], probabilistic CTL ([HJ94, LS1982]), among others. We generally refer to such logics as probabilistic logics.The current status of the theory of probabilistic logics is, compared with that of ordinary modal logics, rather unsatisfactory. For example, for most probabilistic logics capable of expressing properties useful in model checking (probabilistic CTL is a main example), the following problems have been open for more than 35 years (since, at least, [LS1982]):

find a sound and complete axiomatization of the set of valid formulas,

find structural proof systems (e.g., sequent calculus) for deriving valid formulas,

establish if the set of valid formulas is decidable or not.
These problems are evidently intrinsically difficult but one reason that makes them harder to tackle is, possibly, the fact that most probabilistic logics (including probabilistic CTL) have been designed with special focus on model–checking (e.g., the ability to express properties useful in practice, availability of efficient algorithms for verifying finite–state systems, etc.) rather than mathematical convenience. This has led to successful results, with real–world probabilistic programs formally verified using model checking techniques. But, on the other hand, very little progress has been made on the open problems listed above.
1.2. Real–valued probabilistic logics
The seminal work of Kozen on probabilistic PDL [Kozen1983] is noteworthy as being among the first to the design probabilistic logics with main focus on convenient mathematical foundations. A key novelty of probabilistic PDL is the fact that its semantics is real–valued: formulas are not interpreted as true or false, as in most other probabilistic logics with a Boolean semantics (including probabilistic CTL), but are rather interpreted as real numbers in . So in real–valued logics the semantics of a given formula can be a number like , , and . The adequacy and mathematical convenience of a real–valued semantics in the context of probabilistic logics is discussed in detail in [Kozen1981].
However, the logic probabilistic PDL is, using the terminology introduced by Pnueli [PNUELI77], exogenous: the language of formulas is both an assertion language able to express properties of probabilistic programs and a programming language. The logic probabilistic CTL and most other logics for model checking are, on the other hand, endogenous: the language of formulas is independent from the concrete syntax of any given programming language. This distinction is important because the “programming languages” embedded in exogenous logics are usually quite abstract and restricted (e.g., consist only of the usual Kleene algebra operations) while endogenous logics express properties of models generated by arbitrary programs. One of the consequences is, for example, that the logic probabilistic PDL enjoys the finite model property [Kozen1983] while probabilistic CTL does not [LS1982, Brazdil2008]. The failure of the final model property is a fundamental characteristic of probabilistic CTL and a main source of complexity. This means that it does not seem possible to directly apply the results available for probabilistic PDL to solve the open problems regarding probabilistic CTL (and other endogenous logics) listed above.
For this reason, following Kozen, some research has subsequently explored the design of real–valued endogenous probabilistic logics based on the idea of interpreting formulas as real numbers. Early works include [MM07, HM96, prakash2000, deAlfaro2003]. A shortcoming of these attempts is, however, that these probabilistic logics are not sufficiently expressive to interpret the logic probabilistic CTL and other endogenous probabilistic logics having the usual Boolean semantics. Recent works [MioThesis, MIO2012b, MIO2014a, MioSimpsonFI2017] have shown, however, that the desired expressivity can be achieved by extending a simple real–valued probabilistic modal logic (called Łukasiewicz modal logic) with fixed–point operators, in the style of Kozen’s modal –calculus [Kozen83]. Indeed, the resulting real–valued logic, called Łukasiewicz –calculus can interpret the logic probabilistic CTL. Hence the real–valued approach to endogenous modal logics for probabilistic systems suffices to express most properties of interest:
simple real–valued modal logic  
pCTL  
(co)inductively defined operators 
This observation suggests the following research program:

Identify a simple real–valued endogenous modal logic having convenient mathematical foundations which, once extended with fixed–point operators, is sufficiently expressive to interpret probabilistic CTL and other probabilistic logics, just like the Łukasiewicz modal logic mentioned above.

Develop the theory of the probabilistic real–valued logic : model theory, algebraic theory, axiomatizations and proof systems.

Extend with the (co)inductive operators required to increase its expressive power.

Develop the theory of the extended logic using the large body of knowledge on methods for reasoning about fixed points and (co)inductive definitions.
The main contribution of this work is to set down the mathematical foundation of a logic enjoying the properties outlined above.
1.3. Contributions of this work
We introduce the Riesz modal logic, a real–valued probabilistic endogenous modal logic named in honour of the Hungarian mathematician Frigyes Riesz. The design of the syntax and the semantics of this logic is inspired by the theory of Riesz spaces, also known as vector lattices [Luxemburg, JVR1977], a branch of mathematics at the intersection of algebra and functional analysis, pioneered in the 1930’s by F. Riesz, L. Kantorovich and H. Freudenthal among others, with applications in the study of function spaces.
A Riesz space (see Section 2.3 for the details) is a real–vector space equipped with a lattice order () such that the basic vector space operations of addition and scalar multiplication satisfy:
if then , if then , for any scalar .
For example, the linearly ordered set of real numbers
is a Riesz space. Hence the concept of Riesz space is obtained by combining the notion of lattice, which is pervasive in logic, with those of addition and scalar multiplication, which are pervasive in probability theory (e.g., convex combinations, linearity of the expected value operator,
etc.)In the context of our work, it is convenient to think at Riesz spaces as a quantitative generalization of Boolean algebras, obtained by replacing the Boolean algebra with the Riesz space . This is not just a vague analogy as the theory of Riesz spaces is very rich and includes key results such as:

generates the variety of Riesz spaces, just like generates the variety of Boolean algebras,

Yosida duality, which is the equivalent of Stone duality for Boolean algebras, provides a bridge between algebra and topology,

completion theorems, just as in Boolean algebras, allow to embed Riesz spaces into other Riesz spaces whose order has certain closure properties (e.g., it is a complete lattice), etc. This is convenient, for example, when it is required to guarantee the existence of fixed–points of monotone operators as in the Knaster–Tarski fixed–point theorem.
What makes Riesz spaces particularly convenient for our applications to probabilistic logics is that, being vector spaces, the notion of linear transformation plays a fundamental role in the theory. For example the theory of Riesz spaces include results such as:

theory of linear functionals: representation theorems such as, e.g., the Riesz representation theorem for (probability) measures,

extension theorems: e.g., generalizations of the HahnBanach theorem.
For these reasons we claim that the theory of Riesz spaces is a very convenient mathematical setting to develop the theory of probabilistic logics, just as the theory of Boolean algebras is the natural setting for ordinary modal and temporal logics.
We define the transition semantics of Riesz modal logic with respect to transition systems modelled as (topological) Markov chains, which we refer to as Markov processes. Formally, these are defined as coalgebras of the Radon monad on the category of compact Hausdorff space (see Section 2.2). The theory of coalgebra then provides automatically appropriate definitions of morphisms between models, products, quotients, bisimulation, etc. Beside the choice of the category to work with, which is motivated by mathematical convenience and is at the same time very general (see discussion in Section 9), the semantics is essentially standard, it agrees with several other works in the literature and is based on the interpretation of the modality with the expected value operator. And indeed we show that Riesz modal logic can interpret other basic real–valued logics that have appeared in the literature (including the Łukasiewicz modal logic).
1.4. Technical results
Our main technical contribution is to set the mathematical foundation of Riesz modal logic by developing its duality theory, following closely the duality theory framework of ordinary modal logic. To do this, we define an algebraic semantics for Riesz modal logic.
The algebras are called modal Riesz spaces and are Riesz spaces equipped with an additional unary operation subject to the following axioms (see Figure 3 in Section 4):
Linearity:  , 

Positivity:  if then , and 
decreasing:  . 
This variety of algebras forms a category by taking the natural notion of morphisms (i.e., mappings preserving all operations). By applying the machinery of Yosida duality, and other results from the theory of Riesz spaces, we prove that the category of transition models (coalgebras) is dually equivalent with the category of Archimedean modal Riesz spaces (Theorem 5.1).
This result has a number of consequences. Firstly, Riesz modal logic characterizes bisimulation (Corollary 8.3). Secondly, we obtain a sound and complete axiomatizations of Riesz modal logic (Theorem 8.1). The axioms and inferences rules are depicted in Figure 4 in Section 8. While other simple probabilistic modal logics characterizing bisimulation have been completely axiomatic in the literature (see, e.g., the Markovian logic of [StonePrakash, KMP2013]), to the best of our knowledge, Riesz modal logic is the first probabilistic logic which, once extended with fixed–point operators, is sufficiently expressive to interpret other expressive logics such as probabilistic CTL.
Using duality theory, we can investigate properties of the final coalgebra (which, in the context of operational semantics, is understood as the collection of “behaviours” [Jacobs2016, KurzPHD]) by establishing results of the initial modal Riesz space, and vice versa. We prove some fundamental properties of the initial modal Riesz space in Section 6. These allow us, for instance, to prove that the final coalgebra is a compact Polish space (Theorem 7.3).
Riesz modal logic is, by design, a very simple formalism and lacks temporal operators needed to express many of the useful properties expressible in logics such as probabilistic CTL. As already mentioned, the required expressiveness can be achieved by extending Riesz modal logic with recursively defined operators, in the style of Kozen’s modal –calculus. This has been shown in, e.g., [MIO2012b, MioSimpsonFI2017, MIO2014a, Mio18]. Fixed–point definitions usually rely on the Knaster–Tarski theorem on complete lattices. In this context, by applying a theorem of Kantorovich in the theory of Riesz spaces, we prove a fundamental completion result (Theorem 4.9): every Archimedean modal Riesz space can be embedded in a Dedekind complete modal Riesz space. This, by duality, means that any topological Markov chain (coalgebra) can be embedded into a topological Markov chain having a state–space which is Stonean (i.e., the Stone–dual of a complete Boolean algebra).
1.5. Organization of this work
This article is organized as follows:
Section 2: Technical background. We provide the necessary background definitions and results. This section is quite lengthy but, hopefully, serves the purpose of keeping this article reasonably self–contained. Subsections 2.1 and 2.2 deal with basic notions from probability theory and coalgebra and can be safely skipped by readers familiar with these topics. Subsections 2.3, 2.4, 2.5 and 2.6 deal with the basic definitions and results of the theory of Riesz spaces. Once again, these can be safely ignored by readers familiar with this theory and consulted only when necessary.
Section 3: Riesz Modal Logic, Syntax and Transition Semantics. In this section we define the syntax and the transition semantics of Riesz modal logic. The latter is given in terms of topological Markov chains, which we refer to as Markov processes, defined in Section 2.2. We give several examples of formulas and, in Subsection 3.2, explain how Riesz modal logic can interpret other similar real–valued probabilistic modal logics that have appeared in the literature, including the Łukasiewicz modal logic of [MioThesis, MIO2014a, MioSimpsonFI2017].
Section 4: Modal Riesz spaces. In this section we introduce the notion of modal Riesz space, the algebraic counterpart of Riesz modal logic. In Subsection 4.1 we establish a completion theorem for modal Riesz spaces (Theorem 4.9). This result is likely of fundamental importance in the future development of fixed–point extensions of Riesz modal logic based on the Knaster–Tarski theorem. In Subsection 4.2 we comment on some similarities with the notion of state MV–algebra from [FM2009].
Section 5: Duality between Markov processes and modal Riesz spaces. In this section we establish our main technical result (Theorem 5.1): the categories of uniformly complete Archimedean modal Riesz spaces and that of Markov processes with coalgebra morphisms are dually equivalent.
Section 6: Initial algebra. In this section we give explicit constructions of the initial objects of several categories of modal Riesz spaces and establish some basic properties. We also leave open an important question which we could not answer so far (see Subsection 6.3).
Section 7: Final coalgebra. In this section we illustrate one application of the duality theory: it is possible to establish properties of the final coalgebra in the category of Markov processes by proving properties of the initial modal Riesz space. We prove that the state–space of the final coalgebra is a compact Polish space.
Section 8: Applications of duality to Riesz modal logic. Another application of the duality theory is, of course, to establish properties of Riesz modal logic. In this section we prove that Riesz modal logic characterizes probabilistic bisimilarity and that the proof system of Figure 4, for proving semantic equality between pairs of Riesz modal logic formulas, is sound and complete.
Section 9: Other classes of models: In this section we show how our notion of Markov process (as given in Definition 2.7) is in fact very general in the sense that most other similar notions appeared in the literature can be embedded into Markov processes in our sense.
Section 10: Conclusions: In this section we present some final comments and direction for future research.
2. Technical background
2.1. Topology, measures and Riesz–Markov–Kakutani representation theorem
We denote by the category of compact Hausdorff spaces with continuous maps as morphisms. If is a compact Hausdorff space, we denote with the collection of Borel sets of , i.e., the smallest algebra of subsets of containing all open sets. A (Borel) subprobability measure on is a function such that , and for all countable sequences of pairwise disjoint Borel sets. The measure is a probability measure if .
A (sub–)probability measure on the compact Hausdorff space is Radon if for every Borel set , . In other words, a measure is Radon if the measure of every Borel set can be approximated to any degree of precision by compact subsets of . Most naturally occurring probability (sub–)measures are Radon. In particular, if is a Polish space, all (sub–)probability measures are Radon.
Given a set , we denote the collection of all functions by . If is a topological space, then denotes the subset of consisting of all continuous functions. We use and to denote the constant (continuous) functions defined as and , for all , respectively. Using the vector space operations of pointwise, both and can be given the structure of a vector space. Furthermore, the ordering () defined pointwise as is a lattice on both and .
Given a compact Hausdorff space and a (sub–)probability measure on , one can define the expectation functional as
(1) 
where the integral is well defined because any , being continuous and defined on a compact space, is measurable and bounded. One can then observe that:

[label=()]

is a linear map: , and , for all ,

is positive: if then , and

is decreasing: .
The latter inequality becomes an equality if is a probability measure.
The celebrated Riesz–Markov–Kakutani representation theorem statesthat, in fact, any such functional corresponds to a unique Radon subprobability (see [LAX]).
Theorem 2.1 ((Riesz–Markov–Kakutani)).
Let be a compact Hausdorff space. For every functional such that (i) is linear, (ii) is positive and (iii) , there exists one and only one Radon subprobability measure on such that .
Given a compact Hausdorff space we denote with the collection of all Radon subprobability measures on . Equivalently, by the Riesz–Markov–Kakutani theorem, we can identify with the collection of functionals
The set can be endowed with the weak* topology, the coarsest (i.e., having fewest open sets) topology such that, for all , the map , defined as , is continuous. The weak* topology on is compact and Hausdorff by the BanachAlaoglu theorem. Hence maps a compact Hausdorff space to the compact Hausdorff space . In fact becomes a functor on by defining, for any continuous map in , the continuous map as
(2) 
for all .
The functor is shown to be the underlying functor of a monad in [Keimel2009, §6], based Świrszcz’s proof of the probabilistic case [Swirszcz74, swirszcz75] (see also Giry’s work [giry1981]). However, we will not require the monad structure for the purposes of this article. Following [FurberJ14a], we call the Radon monad.
2.2. Markov Processes and Coalgebra
Informally, a (discretetime) Markov process consists of a set of states and a transition function that associates to each state
on the state space . This mathematical object is interpreted, given an initial state , as generating an infinite trajectory (or “computation”) in the state space , where is chosen randomly using the probability distribution . A slight variant of this model, allowing the generation of infinite as well as finite trajectories, uses transition functions associating to each state a subprobability distribution . The intended interpretation is that the computation will stop at state with probability , where is the total mass of , and will continue with probability following the (normalized) probability distribution .Example 2.2.
Consider the following Markov process having state space and transition function defined by: and depicted below:
From the state the computation progresses to itself with probability , to with probability and it halts with probability . From the state the computation progresses to with probability and it halts with probability .
This informal description readily translates to a formal definition for Markov processes having finite or countably infinite state space , also known as Markov chains. Sometimes, however, it is interesting to model Markov process having uncountable state spaces (e.g., ). When is uncountable, some technical assumptions must be considered. Typically, is assumed to be a topological or measurable space and is defined as a map from to the collection of (sub–)probability measures on satisfying certain convenient regularity assumption.
In this work we define Markov processes as follows.
Definition 2.3.
A Markov process is a pair such that is a compact Hausdorff topological space and is a continuous map.
Example 2.4 (Finite Markov chains).
Finite Markov chains, such as the one defined in the example 2.2 above, can be formalized as Markov processes in the sense of Definition 2.3. Indeed, the finite state space , endowed with the discrete topology, is a compact Hausdorff space. And the transition function is continuous (since is discrete). Observe that the space is isomorphic to the set of all subprobability distributions on .
Example 2.5 (Uncountable Markov process).
We define a Markov process having state space where, from the state , the computation progresses to with probability and it halts with probability . This is formalized by defining the transition function as follows:
where is the Dirac probability measure centred on . Note that , being the pointwise product of the continuous function ( and the continuous function (), is indeed continuous.
Remark 2.6.
While these examples are natural, Definition 2.3 might appear unnecessarily restrictive because several practically interesting classes of probabilistic systems do not have a state space endowed with a compact topology (e.g., and are not compact) and often the transition functions are not continuous (e.g., they are just Borel measurable). The choice of using the class of compact Hausdorff spaces and continuous transition functions in Definition 2.3 is mostly motivated by mathematical convenience since, as described later, this is the class of topological spaces appearing in the duality theory of Riesz spaces. We will explain in detail in Section 9 how this is not at all a restriction when it comes to Riesz modal logic.
The theory of coalgebra (for a comprehensive introduction see [Jacobs2016]) provides a convenient framework for formalizing the notion of morphism between Markov processes. The following is an equivalent reformulation of Definition 2.3 in coalgebraic terms and relies on the fact, discussed earlier, that is an endofunctor on the category .
Definition 2.7.
A Markov process is a coalgebra of the endofunctor in the category , i.e., it is a morphism in . A (coalgebra) morphism between the coalgebra and the coalgebra is a continuous function such that the following diagram commutes:
(3) 
Such a morphism will be denoted by .
Definition 2.8 (Category of Markov Processes).
We define the category of Markov processes to be where the objects are coalgebras in and morphisms are coalgebra morphisms.
It is a well known fact that is always a category, for any functor . In computer science, and in particular in the field of categorical semantics of programming languages, one specific coalgebra in plays an important role. This is (when it exists) the final object in , and is called the final coalgebra. The universal property that characterizes is that, for every other coalgebra , there exists one and only one coalgebra morphism in . This property allows to interpret the domain of as the space of all “behaviours” as follows: given any coalgebra , the behaviour of the state is the point . And two states are “behaviourally equivalent” if .
For this reason in Section 7 we study some properties the final Markov process, i.e., the final object in .
2.3. Riesz Spaces
This section contains the basic definitions and results related to Riesz spaces. We refer to [Luxemburg] for a comprehensive reference to the subject.
A Riesz space is an algebraic structure such that is a vector space over the reals, is a lattice and the induced order is compatible with addition in the sense that: (i) for all , if then , and (ii) if and is a non–negative real, then . Formally we have:
Definition 2.9 (Riesz Space).
The language of Riesz spaces is given by the (uncountable) signature where is a constant, , and are binary functions and is a unary function, for all . A Riesz space is a algebra satisfying the equations of Figure 1. We use the standard abbreviations of for and for .
Hence the family of Riesz spaces is a variety in the sense of universal algebra.
Example 2.10.
The most familiar example is the real line with its usual linear ordering, i.e., with and being the usual and operations. An important fact about this Riesz space is the following (see, e.g., [LvA2007]). Given two terms in the language of Riesz spaces, the equality holds in all Riesz spaces if and only if is true in . In the terminology of universal algebra one says that generates the variety of all Riesz spaces. In this sense plays in the theory of Riesz spaces a role similar to the twoelement Boolean algebra in the theory of Boolean algebras.
Example 2.11.
For an example of Riesz space whose order is not linear take the vector space with order defined pointwise: , for all . More generally, for every set , the set with operations defined pointwise is a Riesz space. Since Riesz spaces are algebras, other examples can be found by taking subalgebras. For instance, the collection of bounded functions is a Riesz subspace of . As another example, if is a topological space, then the set of continuous functions is another Riesz subspace of .
The following definitions are useful. Let be a Riesz space. An element is positive if . The set of all positive elements is called the positive cone and is denoted by . Given an element , we define , and . Note that , , and .
Definition 2.12 (Archimedean Riesz space).
An element of a Riesz space is infinitely small if there exists some such that , for all . Clearly, is infinitely small. The Riesz space is Archimedean if is the only infinitely small element in . Equivalently, is Archimedean if it satisfies the following (countably) infinitary rule:
All the Riesz spaces in Example 2.10 are Archimedean.
Example 2.13.
The vector space with the lexicographic order, defined as either or and , is not Archimedean. For instance, is infinitely small with respect to .
As usual in universal algebra, a homomorphism between Riesz spaces is a function preserving all operations. Therefore a Riesz homomorphism is a linear map preserving finite meets and joins.
Definition 2.14 (Ideals and Maximal Ideals).
A subset of a Riesz space is an ideal if it is the kernel of a homomorphism , i.e., , for some Riesz space . The sets and itself are trivially ideals. All other ideals are called proper. Ideals in can be partially ordered by inclusion. An ideal is called maximal if it is a proper ideal and there is no larger proper ideal .
The following alternative characterization of ideals (see, e.g., Section 3.9 of [vulikh]) is often much more simple to deal with.
Proposition 2.15.
Let be a Riesz space. A subset is an ideal if and only is a Riesz subspace of (i.e., closed under all operations) and furthermore, for all and , if and then .
Example 2.16.
Given any Riesz space , the collection of infinitely small elements (see Definition 2.12) is an ideal of .
We now introduce the important concept of a strong unit.
Definition 2.17 (Strong Unit).
An element is called a strong unit if it is positive (i.e., ) and for every there exists such that .
Example 2.18.
The real line has as strong unit. The space does not have a strong unit. Its subspace consisting of bounded functions has (the constant function) as strong unit. Similarly, let be a compact topological space and the Riesz space of continuous functions into . Since is compact, any function is bounded and therefore is a strong unit of .
We now introduce a notion of convergence in Riesz spaces which plays an important role in the duality theory of Riesz spaces.
Definition 2.19 (convergence and uniform Cauchy sequences).
Let be a Riesz space and be a positive element . We say that a sequence converges uniformly to , written , if for every positive real there exists a natural number such that , for all . We say that is a uniform Cauchy sequence if for every there exists a number such that , for all .
Clearly, if then is a uniform Cauchy sequence.
Definition 2.20 (uniform completeness).
A Riesz space is uniformly complete if for every uniform Cauchy sequence there exists such that . It is uniformly complete if it is uniformly complete, for all .
We now state important properties related to uniform completeness of Archimedean Riesz spaces with strong unit.
Theorem 2.21 ((45.5 in [Luxemburg])).
If is Archimedean and has strong unit , then is uniformly complete if and only if it is uniform complete.
Example 2.22.
The Riesz space is 1uniformly complete as the notion of uniform Cauchy sequence coincides with the usual notion of Cauchy sequence of reals. Since is a strong unit in it follows that is uniformly complete. Let be a compact Hausdorff space, the Riesz space of continuous functions and the constant function . Then is uniformly complete ([Luxemburg, Example 27.7, Theorem 43.1]). Once again, is uniformly complete because is a strong unit.
Theorem 2.23 ((Theorem 43.1 in [Luxemburg])).
Let be Archimedean with strong unit . Let be defined as:
(4) 
Then is a norm on , i.e., , and , for all and .
As a consequence, each Archimedean Riesz space with strong unit is a normed vector space and therefore can be endowed with the metric defined as . Accordingly, we say that a Riesz homomorphism between Archimedean spaces with strong units is continuous (resp. is an isometry) if it is continuous (resp. distance preserving) with respect to the metrics of and .
Importantly, on Archimedean spaces with strong unit, the notion of uniform convergence and convergence in the norm (i.e., in the metric ) coincide.
Theorem 2.24 ((Theorem 43.1 in [Luxemburg])).
Let be an Archimedean Riesz space with strong unit . A sequence converges uniformly to if and only if converges in norm to . The space is uniformly complete if and only if it is complete as a metric space.
2.4. Riesz Spaces with a distinguished positive element
It is now convenient to extend the language of Riesz spaces with a new constant symbol for a positive element.
Definition 2.25.
A Riesz space with distinguished positive element is a pair where is a Riesz space and . A morphism between and is a Riesz homomorphism such that . If is a strong unit in we say that is unital.
When confusion might arise, we will stress the fact that a homomorphism preserves the distinguished positive elements (i.e., ) by saying that is a unital (Riesz) homomorphism. We write for the category having Riesz spaces with a distinguished positive element as objects and unital homomorphisms as morphisms. We write for the subcategory of whose objects are unital Riesz spaces.
Example 2.26.
The basic example is the real line . Since is a strong unit, this is in fact a unital Riesz space. Furthermore it follows easily from the result mentioned in Example 2.10 that generates the variety .
The following theorem (see, e.g., [Luxemburg, Thm 27.34]) expresses a key property of unital Riesz spaces.
Theorem 2.27.
Let be a unital Riesz space. Then, for every unital homomorphism , the ideal is maximal. Conversely, every maximal ideal in is of the form for a unique unital Riesz homomorphism .
Hence there is a onetoone correspondence between maximal ideals in unital Riesz spaces and homomorphisms into preserving the unit. Observe, once again (cf. Examples 2.10 and 2.26), how the Riesz space plays in the theory of unital Riesz spaces a role similar to two element Boolean algebra in the theory of Boolean algebras.
We say that a unital Riesz space is Archimedean if is Archimedean. We write for the category of Archimedean unital Riesz spaces with unital Riesz homomorphisms. We write for the category of Archimedean and uniformly complete unital Riesz spaces with unital Riesz homomorphisms.
Example 2.28.
Let be a compact Hausdorff space. Let be the constant function. Then is an Archimedean unital and uniformly complete Riesz space [Luxemburg, Example 27.7, Theorem 43.1].
The following results describe the property of being Archimedean for unital Riesz spaces.
Theorem 2.29.
Let be a unital Riesz space. An an element is infinitely small if and only if , for all . This means that the Archimedean rule (cf. Definition 2.12) can be equivalently reformulated as follows:
Furthermore, is Archimedean if and only if for every there exists a unital Riesz homomorphism such that .
Corollary 2.30.
Let and be unital Riesz spaces and a unital Riesz homomorphism. If is infinitely small then is also infinitely small.
Proof.
By assumption we have that for all the inequality holds. Since is a unital homomorphism we have that is monotone and . Hence we get
for all which means that is infinitely small. ∎
2.5. Yosida’s Theorem and Duality Theory of Riesz Spaces
In this section we assume familiarity with the basic notions from category theory regarding equivalences of categories and adjunctions. A standard reference is [maclane].
The celebrated Stone duality theorem states that any Boolean algebra is isomorphic to the Boolean algebra of clopen sets (or equivalently continuous functions where is given the discrete topology) of a unique (up to homeomorphism) Stone space, i.e., a compact Hausdorff and zero–dimensional topological space . Here is the collection of maximal (Boolean) ideals in endowed with the hull–kernel topology. In fact this correspondence can be made into a categorical equivalence between and .
A similar representation theorem, due to Yosida [yosida], states that every uniformly complete, unitary and Archimedean Riesz space is isomorphic to , the Riesz space of all continuous functions , of a unique (up to homeomorphism) compact Hausdorff space . This correspondence can be made into a categorical equivalence
(5) 
see, e.g., [westerbaan2016] for a detailed proof. In fact Yosida proved a more general result which can be conveniently formulated as an adjunction between and which restricts to the equivalence (5) on the subcategory . In the rest of this section we describe it as a unit–counit adjunction consisting of two functors:
and two natural transformations:
called unit and counit, respectively.
We first define the functor .
On objects, for a compact Hausdorff space , we define as the set of continuous realvalued functions on , equipped with the Riesz space operations defined pointwise from those on (see Example 2.11) and strong unit (). As discussed earlier (see Example 2.28) this is indeed a uniformly complete Archimedean and unital Riesz space. On continuous maps , we define , for all . This is easily proven to be a unital Riesz space morphism by the fact that the Riesz space operations are defined pointwise.
We now turn our attention to the description of the functor .
As in the Stone duality theorem, on objects in , the functor is defined as the spectrum of , i.e., the collection of all maximal ideals of (see Definition 2.14) equipped with the hull–kernel topology which can be defined as follows. A subset is closed in the hull–kernel topology if and only if there exists a (not necessarily maximal) ideal such that where . See, e.g., [Luxemburg, Theorem 36.4 (ii)] for a proof that is indeed a compact Hausdorff space. On maps, for a unital morphism we define, for every , .
We now turn our attention to the description of the unit map .
This is a collection of maps indexed by compact Hausdorff spaces. For a fixed compact Hausdorff space and we can define the map as which is easily seen to be a unital Riesz homomorphism. Therefore, by Theorem 2.27 the set is a maximal ideal in , i.e., . We then define as .
Lastly, we now proceed with the definition of the counit map .
This is a collection of morphisms in , or equivalently a collection of morphisms in , indexed by unital and Archimedean Riesz spaces . For a fixed such and we can define a function as , where is the homomorphism from Theorem 2.27. That is (see [Luxemburg, Thm 27.34]) the value is defined as the unique real number such that . The map is continuous, i.e., . We then define as .
The statement of Yosida’s theorem can then be formulated by the following two theorems (see [yosida, Theorems 1–3], also [Luxemburg, Theorems 45.3 and 45.4] and [westerbaan2016]).
Theorem 2.31.
Both and are functors. Both and are natural transformations. The quadruple is a unitcounit adjunction. The counit map is an isometric isomorphism between and its image in .
Proof.
The fact that is indeed a functor follows from elementary properties of composition of functions and identity maps. We now show is a functor. Let be a unital Riesz homomorphism and a maximal ideal. By Theorem 2.27 there is a unital Riesz morphism such that . The composite is a unital Riesz homomorphism , so
is a maximal ideal in . This shows is a function . We show it is continuous by showing that the preimage of a closed set is closed. Any closed set in is for some ideal . By [Luxemburg, Theorem 59.2 (iii)], is also an ideal. By elementary manipulations of the definitions, , which, as we started with an arbitrary closed set, proves the continuity of . By basic properties of the preimage mapping, preserves identity maps and reverses composition, and is therefore a contravariant functor, as required.
We now consider the unit. In [yosida, Theorem 4] Yosida shows that the mapping is a homeomorphism onto its image and has dense image. The compactness of then implies that the image of is closed, and therefore is all of , i.e., is a homeomorphism .
The proof of naturality, i.e., that for all continuous maps and for all , is done by expanding the definitions on each side, so is omitted. As a result, .
For the counit, Yosida shows that , and is a unital Riesz space homomorphism with normdense image [yosida, Theorems 1–2] (see also [Luxemburg, Theorem 45.3]).
The naturality of , i.e., that for all a unital Riesz homomorphism, , we have reduces to showing that , which is easily done using the linearity and unitality of and the definition of .
To show that is a right adjoint to , we only need to prove that the following diagrams commute:
where is a compact Hausdorff space and a unital Archimedean Riesz space.
For the first diagram, we want to show that if and , we have . Expanding the definitions shows us that this is equivalent to showing
(6) 
By the definition of , we have . Applying the definition of and elementary algebra then gives us the result.
For the second diagram, we want to show that for each and that . This can be proved simply by expanding the definitions.
We have therefore shown that , i.e., is a right adjoint to . ∎
Theorem 2.32.
When restricted to , the adjunction becomes an equivalence of categories. An object of is uniformly complete (i.e., it belongs to ) if and only if is a Riesz isomorphism.
Proof.
Yosida shows that Is a unital Riesz space isomorphism iff is uniformly complete in [yosida, Theorem 3] (see also [Luxemburg, Theorem 45.4]). As the (norm) unit ball of is exactly the inverse image of the unit ball of , we have that the embedding is also an isometry, and therefore is isomorphic to the Banach space completion of .
We already saw in Theorem 2.31 that Yosida proved that is always an isomorphism for a compact Hausdorff space. Therefore is an adjoint equivalence when restricted to [maclane, §IV.4]. ∎
The functor maps (not necessarily uniformly complete) Archimedean unital Riesz spaces to uniformly complete ones. In fact, Yosida showed that embeds densely in . Therefore is isomorphic to the completion of in its norm (from Theorem 2.23).
Definition 2.33.
The uniform Archimedean and unital Riesz space is called the uniform completion of and is simply denoted by . We always identify with the (isomorphic) dense subRiesz space of .
Proposition 2.34.
For every , the two spaces and are homeomorphic. Furthermore, for every and unital homomorphism there exists a unique unital Riesz homomorphism extending .
2.6. Dedekind Complete Riesz Spaces
We conclude this section by discussing Dedekind–complete Riesz spaces. We refer to [vulikh, §4] for a detailed introduction. Dedekind–complete Riesz spaces, due to their order–completeness properties, play a role when fixed–point extensions of Riesz modal logic, based on the Knaster–Tarski theorem, are considered (see, e.g., [MIO2012b, MioSimpsonFI2017, MIO2014a, Mio18]).
Definition 2.35.
Let be a lattice. The lattice is complete if for every subset there exist in both a least upper bound () and a greatest lower bound (). The lattice is Dedekind–complete if, for every bounded subset , there exist in both and .
It follows from this definition that every complete lattice is Dedekind–complete, but the converse is not true. The real numbers is an example of Dedekind–complete lattice which is not complete since suprema and infima of unbounded sets do not exist in .
Definition 2.36 (Dedekind–complete Riesz space).
The Riesz space is called Dedekind–complete if its underlying lattice is Dedekind–complete.
The following is an important property of Dedekind–complete Riesz spaces (see, e.g., Theorem 25.1 of [Luxemburg]).
Theorem 2.37.
Every Dedekind–complete Riesz space is Archimedean and uniformly complete.
In particular, if the Dedekind–complete Riesz space has a strong unit , the Yosida duality described in Section 2.5, can be applied. Therefore every Dedekind–complete Riesz space with strong unit is isomorphic to the space of real–valued functions of a unique (up to homeomorphism) compact Hausdorff space . To the order–theoretic property of Dedekind–completeness corresponds, via Yosida duality, a topological property of the dual space: is extremally disconnected.
Definition 2.38.
A topological space is extremally disconnected if the closure of every open set is clopen. An extremally disconnected space that is also compact and Hausdorff is called Stonean.
It is well–known that a Stonean space is (up to homeomorphism) the Stone dual of a unique (up to isomorphism) complete Boolean algebra. Hence we get (see, e.g., Chapter IV of [vulikh]) the following:
Proposition 2.39.
Let be a Dedekind–complete Riesz space with strong unit . Then is isomorphic to for a unique (up to homeomorphism) Stonean space .
The following theorem, due to Yudin (see [vulikh, Thm IV.11.1]), is the Riesz space equivalent of the DedekindMacNeille completion theorem in the theory of lattices. It states that it is possible to embed an arbitrary unital Archimedean Riesz space into a essentially minimal Dedekind–complete unital Riesz space, called the Dedekind–completion of , preserving all suprema and infima existing in .
Theorem 2.40 ((Dedekind completion)).
For every Archimedean unital Riesz space there exists a Dedekind complete Archimedean and unital space , called the Dedekind completion of , such that:

embeds in , so we can just write ,

is order–dense in , i.e., for every there exists such that ,

existing suprema and infima in are preserved in . This means that for every and (sup existing and taken in ) then in too.

is the smallest Dedekind complete space satisfying the two properties above.
3. Riesz Modal Logic, Syntax and Transition Semantics
In this section we formally introduce Riesz modal logic for Markov processes.
Definition 3.1 (Syntax).
The set of formulas Form is generated by the following grammar:
The semantics of a formula , interpreted over a Markov process (see Definition 2.7), is a continuous function defined as follows.
Definition 3.2 (Semantics).
Let be a Markov process. The semantics (or interpretation) of a formula relative to the Markov process is the continuous function defined by induction on as follows: