I Introduction
Probability theory in the quantum realm is different in important ways from that of the classical world. Nevertheless, they both crucially rely on real numbers to represent probabilities of events. This makes sense as observations of quantum systems must still be interpreted trough classical means. However, in principle one can imagine a world governed by different physical laws where even the standard notion of a probability is different. In this paper we study a reasonable class of alternatives to the real unit interval as the set of allowed probabilities. We will establish that this quite general seeming class actually only contains (continuous products of) the real unit interval. This shows that any ‘reasonable’ enough physical theory must necessarily be based on probabilities represented by real numbers.
To find the right set of alternatives to the real unit interval we must first ask which structure it is we need to retain in order to still be able to speak of abstract probabilities. There are a variety of operations on the real unit interval that are used in their interpretation as probabilities. First of all, in order to be able to talk about coarsegraining the probabilities of mutually exclusive events, we must be able to take the sum of two probabilities as long as . Second, in order to represent the complement of an event we require the involution given by . The probability is the unique number such that . Axiomatizing this structure of a partially defined addition combined with an involution defines an effect algebra [foulis1994effect]. The unit interval of course also has a multiplication
. This operation is needed in order to talk about, for instance, joint distributions. An
effect monoid is an effect algebra with an associative biadditive (possibly noncommutative) multiplication, and hence axiomatizes these three interacting algebraic structures (addition, involution and multiplication) present in the unit interval.In order to define an analogue to Bayes’ theorem we would also need the division operation that is available in the unit interval: when
, then there is a probability such that (namely ). We actually will not require the existence of such a division operation, as it turns out to follow (nontrivially) from our final requirement.A property that sets the unit interval apart from, for instance, the rational numbers between and , is that is closed under taking limits. In particular, each ascending chain of probabilities has a supremum. In other words: the unit interval is complete. We have then arrived at our candidate for an abstract notion of the set of probabilities: an complete effect monoid.
Further motivation for the use of this structure as a natural candidate for the set of probabilities is its prevalence in effectus theory. This is a recent approach to categorical logic [cho2015introduction] and a general framework to deal with notions such as states, predicates, measurement and probability in deterministic, (classical) probabilistic and quantum settings [kentathesis, basthesis]. The set of probabilities in an effectus have the structure of an effect monoid. Examples of effectuses include any generalized probabilistic theory [barrett2007information], where the probabilities are the unit interval, but also any topos (and in fact any extensive category with final object), where the probabilities are the Boolean values [kentathesis].
An effectus defines the sum of some morphisms. In a effectus, this is strengthened to the existence of some countable sums (making it a partially additive category [arbib1980partially]). In such an effectus the probabilities form an complete effect monoid [kentathesis].
Effect monoids are of broader interest than only to study effectuses: examples of effect monoids include all Boolean algebras and unit intervals of partially ordered rings. Furthermore any effect monoid can be used to define a generalized notion of convex set and convex effect algebra (by replacing the usual unit interval by elements of the effect monoid, see [basthesis, 179 & 192] or [jacobs2011probabilities]).
This now raises the question of how close a probability theory based on an complete effect monoid is to regular probability theory.
Our main result is that complete effect monoids can always be embedded into a direct sum of an complete Boolean algebra and the unit interval of a boundedcomplete commutative unital Calgebra. The latter is isomorphic to for some basically disconnected compact Hausdorff space . If the effect monoid is directed complete, i.e. any directed set has a supremum, then it is even isomorphic to the direct sum of a complete Boolean algebra and the unit interval of a monotone complete commutative Calgebra.
This result basically states that any complete effect monoid can be split up into a sharp part (the Boolean algebra), and a convex probabilistic part (the commutative C*algebra). This then gives us from basic algebraic and ordertheoretic considerations a dichotomy between sharp and fuzzy logic.
As part of the proof of this embedding theorem we find an assortment of additional structure present in complete effect monoids: it has a partially defined division operation, it is a lattice, and multiplication must necessarily be normal (i.e. preserve suprema).
The classification also has some further nontrivial consequences. In particular, it shows that any complete effect monoid must necessarily be commutative.
We also use the classification to show that an complete effect monoid without zero divisors must either be trivial, , the twoelement Boolean algebra, , or the unit interval, . This gives a new characterization of the real unit interval as the unique complete effect monoid without zero divisors and more than two elements. We have hence retrieved the full structure of the unit interval from purely algebraic and ordertheoretic considerations.
In so far as the structure of an complete effect monoid is required for common actions involving probabilities, (coarsegraining, negations, joint distributions, limits) our results motivate the usage of real numbers in any hypothetical alternative physical theory.
Ii Preliminaries
Before we state the main results of this paper technically, we recall the definitions of the structures involved.
Definition 1.
An effect algebra (EA) is a set with distinguished element , partial binary operation (called sum) and (total) unary operation (called complement), satisfying the following axioms, writing whenever is defined and .

Commutativity: if , then and .

Zero: and .

Associativity: if and , then , , and .

For any , the complement is the unique element with .

If for some , then .
For we write whenever there is a with . This turns into a poset with minimum and maximum . The map is an order antiisomorphism. Furthermore if and only if . If , then the element with is unique and is denoted by [foulis1994effect].
A morphism between effect algebras is a map such that and whenever , and then . A morphism necessarily preserves the complement, , and the order: . A morphism is an embedding when it is also order reflecting: if then . Observe that an embedding is automatically injective. We say and are isomorphic and write when there exists an isomorphism (i.e. a bijective morphism whose inverse is a morphism too) from to . Note that an isomorphism is the same as a surjective embedding.
Example 2.
Let be an orthomodular lattice. Then is an effect algebra with the partial addition defined by and in that case . The complement, , is given by the orthocomplement, . The lattice order coincides with the effect algebra order (defined above). See e.g. [basmaster, Prop. 27].
Example 3.
Let be an ordered Abelian group (such as the selfadjoint part of a Calgebra). Then any interval where is a positive element of forms an effect algebra, with addition given by and in that case . The complement is defined by . The effect algebra order on coincides with the regular order on .
In particular, the set of effects of a unital Calgebra forms an effect algebra with , and .
Effect algebras have been studied extensively (to name a few: [chajda2009every, ravindran1997structure, gudder1996examples, gudder1996effect, jenvca2001blocks, dvurevcenskij2010every, foulis2010type]) and even found surprising applications such as in [staton2018effect, roumen2016cohomology, jacobs2015effect]. The following remark gives some categorical motivation to the definition of effect algebras.
Remark 4.
An effect algebra is a bounded poset: a partially ordered set with a minimal and maximal element. In [mayet1995classes] it is shown that any bounded poset can be embedded into a orthomodular poset . This is known as the Kalmbach extension [kalmbach1977orthomodular]. This extends to a functor from the category of bounded posets to the category of orthomodular posets, and this functor is in fact left adjoint to the forgetful functor going in the opposite direction [harding2004remarks]. This adjunction gives rise to the Kalmbach monad on the category of bounded posets. The Eilenberg–Moore category for the Kalmbach monad is isomorphic to the category of effect algebras, and hence effect algebras are in fact algebras over bounded posets [jenvca2015effect].
The category of effect algebras is both complete and cocomplete. There is also an algebraic tensor product of effect algebras that makes the category of effect algebras symmetric monoidal
[jacobs2012coreflections]. The monoids in the category of effect algebras resulting from this tensor product are called effect monoids, and they can be explicitly defined as follows:Definition 5.
An effect monoid (EM) is an effect algebra with an additional (total) binary operation , such that the following conditions hold for all .

Unit: .

Distributivity: if , then , ,
Or, in other words, the operation is biadditive.

Associativity: .
We call an effect monoid commutative if for all ; an element of idempotent whenever ; elements , of orthogonal when ; and we denote the set of idempotents of by .
Example 6.
Any Boolean algebra , being an orthomodular lattice, is an effect algebra by Example 2, and, moreover, a commutative effect monoid with multiplication defined by . Conversely, any orthomodular lattice for which distributes over (and thus ) is a Boolean algebra.
Example 7.
The unit interval of any (partially) ordered unital ring (in which the sum and product of positive elements and are again positive) is an effect monoid.
Let, for example, be a compact Hausdorff space. We denote its space of continuous functions into the complex numbers by . This is a commutative unital Calgebra (and conversely by the Gel’fand theorem, any commutative Calgebra with unit is of this form) and hence its unit interval is a commutative effect monoid.
In [kentathesis, Ex. 4.3.9] and [basmaster, Cor. 51] two different noncommutative effect monoids are constructed.
Definition 8.
Let and be effect monoids. A morphism from to is a morphism of effect algebras with the added condition that for all . Similar to the case of effect algebras, an embedding is a morphism that is order reflecting. Also here an isomorphism of effect monoids is the same thing as a surjective embedding of effect monoids.
Example 9.
Any Boolean algebra is isomorphic to the set of clopens of its Stone space . This gives an effect monoid embedding from into .
Remark 10.
A physical or logical theory which has probabilities of the form can be seen as a theory with a natural notion of space, where probabilities are allowed to vary continuously over the space . Such a spatial theory is considered in for instance Ref. [moliner2017space].
Example 11.
Given two effect algebras/monoids and we define their direct sum as the Cartesian product with pointwise operations. This is again an effect algebra/monoid.
Example 12.
Let be an effect monoid and let be some idempotent. The subset is called the left corner by and is an effect monoid with and all other operations inherited from . Later we will see that is an isomorphism . Analogous facts hold for the right corner .
Definition 13.
Let be an effect algebra. A directed set is a nonempty set such that for all there exists a such that . is directed complete when for any directed set there is a supremum . It is complete if directed suprema of countable sets exist, or equivalently if any increasing sequence in has a supremum.
Remark 14.
A directed complete partially ordered set is often referred to by the shorthand dcpo. These structures lie at the basis of domain theory and are often encountered when studying denotational semantics of programming languages as they allow for a natural way to talk about fix points of recursion. Note that being complete is strictly weaker. For effect algebras we could have equivalently defined directed completeness with respect to downwards directed sets, as the complement is an order antiisomorphism.
Example 15.
Let be a complete Boolean algebra. Then is a complete effect monoid. If is complete as a Boolean algebra, then is directedcomplete as effect monoid.
Example 16.
Let be an extremally disconnected compact Hausdorff space, i.e. where the closure of every open set is open. Then is a directedcomplete effect monoid. If is a basically disconnected [gillman2013rings, 1H] compact Hausdorff space, i.e. where every cozero set has open closure, then is an complete effect monoid [gillman2013rings, 3N.5].
Iii Overview
The main results of the paper are the following theorems:
Theorem.
Let be an complete effect monoid. Then embeds into , where is an complete Boolean algebra, and , where is a basically disconnected compact Haussdorff space (see Theorem 75).
Theorem.
Let be a directedcomplete effect monoid. Then where is a complete Boolean algebra and for some extremallydisconnected compact Hausdorff space (see Theorem 76).
By Example 9, the Boolean algebra also embeds into a , and hence we could ‘coarsegrain’ the direct sums above and say that any complete effect monoid embeds into the unit interval of , where is the disjoint union of the topological spaces.
Other results that either follow directly from the above, or are proven along the way are the following: Let be an complete effect monoid. Then:

is a lattice.

is an effect divisoid [basthesis].

The multiplication in is normal: .

If is convex (as an effect algebra), then scalar multiplication is homogeneous: for any and .

is commutative.

If has no nontrivial zerodivisors (i.e. when , then or ), then is isomorphic to , or .
The paper is structured as follows. In Section IV we recover and prove some basic results regarding effect algebras/monoids. Then in Section V we will show that in any complete effect monoid , we can define a kind of partial division operation which turns it into a effect divisoid. Using this division we show that the multiplication must be normal. Then in Section VI we study idempotents that are either Boolean, meaning that all elements below must also be idempotents, or halvable, meaning that there is an such that . We establish that an complete effect monoid where is Boolean must be a Boolean algebra, while if is halvable then it must be convex. In Section VII we show that a maximal collection of orthogonal idempotents of can be found that consists of a mix of halvable and Boolean idempotents. The corner associated to such an idempotent will either be convex (if is halvable) or Boolean (if is Boolean). Using normality of multiplication we show that embeds into the direct sum of the corners associated to these idempotents. Letting be the direct sum of the Boolean corners, and be the direct sum of the convex corners, we see that embeds into , where is Boolean and is convex. In Section VIII, we recall some results regarding order unit spaces and use Yosida’s representation theorem to show that a convex complete effect monoid must be isomorphic to the unit interval of a . Then in Section IX we collect all the results and prove our main theorems. Finally in Section X we conclude and discuss some future work and open questions.
Iv Basic results
We do not assume any commutativity of the product in an effect monoid. Nevertheless, some commutativity comes for free.
Lemma 17.
For any in an effect monoid , we have .
Proof.
. Cancelling on both sides gives the desired equality. ∎
Lemma 18.
An element is an idempotent if and only if .
Proof.
. Hence if and only if . ∎
Lemma 19.
For with , we have
Proof.
Assume . Then , so that . Similarly . Hence . Similarly .
Now assume . Then immediately . The final implication (that ) is proven similarly. ∎
Lemma 20.
Let be an effect monoid with idempotent . Then for any .
Proof.
Clearly and so by Lemma 19 . Similarly and so . Thus , as desired. ∎
Corollary 21.
Let be an effect monoid with idempotent . The map is an isomorphism .
The following two lemmas are simple observations that will be used several times.
Lemma 22.
Let be elements of an effect algebra . If for some from , then (and ).
Proof.
Since and , we have , and so . Then , yielding and , so and . ∎
Lemma 23.
Let be an idempotent from an effect monoid , and let be elements below . If exists, then .
Proof.
Since , we have , and similarly, . But then , and hence . By Lemma 19 we then have . ∎
We defined directed set to mean upwards directed. Using the fact that is an order antiisomorphism, a directedcomplete effect algebra also has all infima of downwards directed (or ‘filtered’) sets (and similarly for countable infima in a complete effect algebra).
The partial sum and difference in an effect algebra have the following nontrivial interactions with suprema and infima.
Lemma 24.
Let be an element and a nonempty subset of an effect algebra . The following equations hold under the conditions printed next to them, by which we mean too that every sum, infimum, and supremum appearing on either side of such an equation exists (under those conditions):
if and exists  
if and exists  
if and exists  
if and exists  
if and exists  
if and exists 
Proof.
Note that gives an order isomorphism with inverse . Whence preserves and reflects all infima and suprema restricted to and . Surely, given elements from , and a subset of the interval , it is clear that any supremum (infimum) of in will be the supremum (infimum) of in too (using here that is nonempty). The converse does not always hold, but when has a supremum in , then this is the supremum in too (and when has an infimum in , then this is the infimum in too). These considerations yield the first four equations. For the latter two we just add the observation that gives an order reversing isomorphism . ∎
We can now prove a few basic yet useful facts of complete effect monoids. These lemmas deal with elements that are summable with themselves: elements such that which means that is defined. For we will use the notation for the fold sum of with itself (when it is defined). We study these selfsummable elements to be able to define a “” in some effect monoids later on.
Lemma 25.
For any in some effect monoid , the element is summable with itself.
Proof.
Since , and by Lemma 17, we see that indeed exists. ∎
Lemma 26.
The only element of an complete effect algebra for which the fold sum exists for all is zero.
Proof.
We have , and so . ∎
Lemma 27.
Let be a complete effect monoid with satisfying . Then .
Proof.
Since we have , and hence (because of Lemma 25) is summable with itself. But furthermore , and so .
Continuing in this fashion, we see that exists for every and . Hence, for any the sum exists so that by Lemma 26 . ∎
Lemma 28.
Let be a complete effect monoid. Suppose is summable with itself. Then .
Proof.
Write . As and we see that is defined. But this is true for all , and so by Lemma 26 . ∎
V Floors, ceilings and division
In this section we will see that any complete effect monoid has floors and ceilings. These are respectively the largest idempotent below an element and the smallest idempotent above an element. We will also construct a “division”: for we will find an element such that .
Then using ceilings and this division we will show that multiplication in a complete effect monoid is always normal, i.e. that for nonempty for which exists. This technical result will be frequently used in the remaining sections.
Definition 29.
Let be a (potentially infinite) family of elements from an effect algebra . We say that the sum exists if for every finite subset the sum exists and the supremum exists as well. In that case we write .
Lemma 30.
Given for an effect monoid , we have
for every natural number .
Proof.
From the computation
the result follows immediately. ∎
Corollary 31.
The sum exists for any element from an complete effect monoid .
Definition 32.
Given an element of an complete effect monoid
are called the ceiling of a and the floor of , respectively.
We list some basic properties of and in Proposition 40, after we have made the observations necessary to establish them.
Lemma 33.
Given an element of an effect monoid the fold sum exists for any .
Proof.
Corollary 34.
Given an element of an complete effect monoid , we have .
Proof.
Lemma 35.
We have for any element of an complete effect monoid .
Proof.
Lemma 36.
Let and be elements of an complete effect monoid with for all . Then .
Proof.
Since for all , we have
which implies that , and thus . ∎
Corollary 37.
Given elements of a complete effect monoid such that exists, and for all , we have .
Proof.
Writing , we have and for all , and so by Lemma 36. ∎
Proposition 38.
Given elements and of an complete effect monoid ,
Proof.
If , then also for all . Hence by Corollary 37 . ∎
Proposition 39.
The floor of an element of an complete effect monoid is an idempotent.
Proof.
Proposition 40.
Given an element of an complete effect monoid .

The floor of is an idempotent with . In fact, is the greatest idempotent below .

The ceiling of is the least idempotent above .

We have and .
Proof.
Point 3 follows from Lemma 30. Concerning point 1: by Proposition 39 we know that is an idempotent. Also, since , we clearly have . Now, if is an idempotent in with , then , and so . Whence is the greatest idempotent below . Point 2 now follows easily from 1, since is the dual of under the order antiisomorphism . ∎
Lemma 41.
for all summable elements and of an complete effect monoid (that is, is the supremum of and ).
Proof.
Since , we have , and similarly, . Let be an upper bound of and ; we claim that . Since and , we have and , and so by Lemma 19. Whence . ∎
Any complete effect monoid is a lattice effect algebra [rievcanova2000generalization]:
Theorem 42.
Any pair of elements and from an complete effect monoid has an infimum, , given by
Proof.
First order of business is showing that the sum exists for every . In fact, we’ll show that for all , by induction. Indeed, for , we have , and if for some , then . In particular, exists, and, moreover,
By a similar reasoning, we get
Already writing , we know at this point that and . It remains to be shown that defined above is the greatest lower bound of and . So let with and be given; we must show that .
As an intermezzo, we observe that . Indeed, we have , and , because exists. By Proposition 38 it follows that . Whence writing , we have using Lemma 20. Observing that and using Lemma 19 we also have . We then calculate and similarly .
Returning to the problem of whether , we have
Whence is the infimum of and . ∎
Corollary 43.
Any pair of elements and from an complete effect monoid has a supremum, , given by .∎
The presence of finite infima in complete effect monoids prevents certain subtleties around the existence of arbitrary suprema and infima.
Corollary 44.
Let be elements of an complete effect monoid , and let be a nonempty subset of .
Then has a supremum (infimum) in if and only if has a supremum (infimum) in , and these suprema (infima) coincide.
Proof.
It is clear that if has a supremum in , then this is also the supremum in . For the converse, suppose that has a supremum in , and let be an upper bound for in ; in order to show that is the supremum of in too, we must prove that . Note that is an upper bound for . Indeed, given we have , and , so . Moreover, one easily sees that using the fact that is nonempty. Whence is an upper bound for in , and so , making the supremum of in . Similar reasoning applies to infima of . ∎
Corollary 45.
Given an element and a nonempty subset of an complete effect monoid such that exists for all ,

the supremum exists iff exists, and in that case ;

the infimum exists iff exists, and in that case .
Proof.
The map , being an order isomorphism, preserves and reflects suprema and infima. Now apply Corollary 44. ∎
Definition 46.
Given elements of an complete effect monoid, set
Note that the sum exists, because for all .
Lemma 47.
Let be an element of an complete effect monoid .

.

for all summable with .

for all .

for all with .

.

The maps are order isomorphisms.