A characterization of ordered abstract probabilities

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 coarse-graining the probabilities of mutually exclusive events, we must be able to take the sum $x + y$ of two probabilities $x, y \in [0, 1]$ as long as $x + y \leq 1$ . Second, in order to represent the complement of an event we require the involution given by $x^{⊥} \equiv 1 - x$ . The probability $x^{⊥}$ is the unique number such that $x + x^{⊥} = 1$ . 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 $x \cdot y$

. 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 non-commutative) 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

x \leq y

, then there is a probability

z

such that

y \cdot z = x

(namely

z = x / y

). We actually will not require the existence of such a division operation, as it turns out to follow (non-trivially) from our final requirement.

A property that sets the unit interval $[0, 1]$ apart from, for instance, the rational numbers between $0$ and $1$ , is that $[0, 1]$ is closed under taking limits. In particular, each ascending chain of probabilities $x_{1} \leq x_{2} \leq \dots$ has a supremum. In other words: the unit interval is $ω$ -complete. We have then arrived at our candidate for an abstract notion of the set of probabilities: an $ω$ -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 ${0, 1}$ [kentathesis].

An effectus defines the sum of some morphisms. In a $ω$ -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 $ω$ -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 $ω$ -complete effect monoid is to regular probability theory.

Our main result is that $ω$ -complete effect monoids can always be embedded into a direct sum of an $ω$ -complete Boolean algebra and the unit interval of a bounded- $ω$ -complete commutative unital C $^{*}$ -algebra. The latter is isomorphic to $C (X)$ for some basically disconnected compact Hausdorff space $X$ . 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 C $^{*}$ -algebra.

This result basically states that any $ω$ -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 order-theoretic 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 $ω$ -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 non-trivial consequences. In particular, it shows that any $ω$ -complete effect monoid must necessarily be commutative.

We also use the classification to show that an $ω$ -complete effect monoid without zero divisors must either be trivial, ${0}$ , the two-element Boolean algebra, ${0, 1}$ , or the unit interval, $[0, 1]$ . This gives a new characterization of the real unit interval as the unique $ω$ -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 order-theoretic considerations.

In so far as the structure of an $ω$ -complete effect monoid is required for common actions involving probabilities, (coarse-graining, 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) $(E,, 0, ()^{⊥})$ is a set $E$ with distinguished element $0 \in E$ , partial binary operation (called sum) and (total) unary operation $x \mapsto x^{⊥}$ (called complement), satisfying the following axioms, writing $x ⊥ y$ whenever $x y$ is defined and $1 \equiv 0^{⊥}$ .

Commutativity: if $x ⊥ y$ , then $y ⊥ x$ and $x y = y x$ .
Zero: $x ⊥ 0$ and $x 0 = x$ .
Associativity: if $x ⊥ y$ and $(x y) ⊥ z$ , then $y ⊥ z$ , $x ⊥ (y z)$ , and $(x y) z = x (y z)$ .
For any $x \in E$ , the complement $x^{⊥}$ is the unique element with $x x^{⊥} = 1$ .
If $x ⊥ 1$ for some $x \in E$ , then $x = 0$ .

For $x, y \in E$ we write $x \leq y$ whenever there is a $z \in E$ with $x z = y$ . This turns $E$ into a poset with minimum $0$ and maximum $1$ . The map $x \mapsto x^{⊥}$ is an order anti-isomorphism. Furthermore $x ⊥ y$ if and only if $x \leq y^{⊥}$ . If $x \leq y$ , then the element $z$ with $x z = y$ is unique and is denoted by $y ⊖ x$ [foulis1994effect].

A morphism $f : E \to F$ between effect algebras is a map such that $f (1) = 1$ and $f (x) ⊥ f (y)$ whenever $x ⊥ y$ , and then $f (x y) = f (x) f (y)$ . A morphism necessarily preserves the complement, $f (x^{⊥}) = f (x)^{⊥}$ , and the order: $x \leq y ⟹ f (x) \leq f (y)$ . A morphism is an embedding when it is also order reflecting: if $f (x) \leq f (y)$ then $x \leq y$ . Observe that an embedding is automatically injective. We say $E$ and $F$ are isomorphic and write $E ≅ F$ when there exists an isomorphism (i.e. a bijective morphism whose inverse is a morphism too) from $E$ to $F$ . Note that an isomorphism is the same as a surjective embedding.

Example 2.

Let $(B, 0, 1, \land, \lor, ()^{⊥})$ be an orthomodular lattice. Then $B$ is an effect algebra with the partial addition defined by $x ⊥ y ⟺ x \land y = 0$ and in that case $x y = x \lor y$ . 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 $G$ be an ordered Abelian group (such as the self-adjoint part of a C $^{*}$ -algebra). Then any interval $[0, u]_{G} \equiv {a \in G; 0 \leq a \leq u}$ where $u$ is a positive element of $G$ forms an effect algebra, with addition given by $a ⊥ b ⟺ a + b \leq u$ and in that case $a b = a + b$ . The complement is defined by $a^{⊥} = u - a$ . The effect algebra order on $[0, u]_{G}$ coincides with the regular order on $G$ .

In particular, the set of effects $[0, 1]_{C}$ of a unital C $^{*}$ -algebra $C$ forms an effect algebra with $a ⊥ b ⟺ a + b \leq 1$ , and $a^{⊥} = 1 - a$ .

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 $P$ can be embedded into a orthomodular poset $K (P)$ . 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 $(M,, 0,^{⊥}, \cdot)$ with an additional (total) binary operation $\cdot$ , such that the following conditions hold for all $a, b, c \in M$ .

Unit: $a \cdot 1 = a = 1 \cdot a$ .
Distributivity: if $b ⊥ c$ , then $a \cdot b ⊥ a \cdot c$ , $b \cdot a ⊥ c \cdot a$ ,

$a \cdot (b c) = (a \cdot b) (a \cdot c), and (b c) \cdot a = (b \cdot a) (c \cdot a) .$

Or, in other words, the operation $\cdot$ is bi-additive.
Associativity: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$ .

We call an effect monoid $M$ commutative if $a \cdot b = b \cdot a$ for all $a, b \in M$ ; an element $p$ of $M$ idempotent whenever $p^{2} \equiv p \cdot p = p$ ; elements $a$ , $b$ of $M$ orthogonal when $a \cdot b = b \cdot a = 0$ ; and we denote the set of idempotents of $M$ by $P (M)$ .

Example 6.

Any Boolean algebra $(B, 0, 1, \land, \lor, ()^{⊥})$ , being an orthomodular lattice, is an effect algebra by Example 2, and, moreover, a commutative effect monoid with multiplication defined by $x \cdot y = x \land y$ . Conversely, any orthomodular lattice for which $\land$ distributes over (and thus $\lor$ ) is a Boolean algebra.

Example 7.

The unit interval $[0, 1]_{R}$ of any (partially) ordered unital ring $R$ (in which the sum $a + b$ and product $a \cdot b$ of positive elements $a$ and $b$ are again positive) is an effect monoid.

Let, for example, $X$ be a compact Hausdorff space. We denote its space of continuous functions into the complex numbers by $C (X) \equiv {f : X \to C; f continuous}$ . This is a commutative unital C $^{*}$ -algebra (and conversely by the Gel’fand theorem, any commutative C $^{*}$ -algebra with unit is of this form) and hence its unit interval $[0, 1]_{C (X)} = {f : X \to [0, 1]}$ is a commutative effect monoid.

In [kentathesis, Ex. 4.3.9] and [basmaster, Cor. 51] two different non-commutative effect monoids are constructed.

Definition 8.

Let $M$ and $N$ be effect monoids. A morphism from $M$ to $N$ is a morphism of effect algebras with the added condition that $f (a \cdot b) = f (a) \cdot f (b)$ for all $a, b \in M$ . Similar to the case of effect algebras, an embedding $M \to N$ 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 $B$ is isomorphic to the set of clopens of its Stone space $X_{B}$ . This gives an effect monoid embedding from $B$ into $[0, 1]_{C (X_{B})}$ .

Remark 10.

A physical or logical theory which has probabilities of the form $[0, 1]_{C (X)}$ can be seen as a theory with a natural notion of space, where probabilities are allowed to vary continuously over the space $X$ . Such a spatial theory is considered in for instance Ref. [moliner2017space].

Example 11.

Given two effect algebras/monoids $E_{1}$ and $E_{2}$ we define their direct sum $E_{1} \oplus E_{2}$ as the Cartesian product with pointwise operations. This is again an effect algebra/monoid.

Example 12.

Let $M$ be an effect monoid and let $p \in M$ be some idempotent. The subset $p M \equiv {p \cdot a; a \in M}$ is called the left corner by $p$ and is an effect monoid with $(p \cdot a)^{⊥} \equiv p \cdot a^{⊥}$ and all other operations inherited from $M$ . Later we will see that $a \mapsto (p \cdot a, p^{⊥} \cdot a)$ is an isomorphism $M ≅ p M \oplus p^{⊥} M$ . Analogous facts hold for the right corner $M p \equiv {a \cdot p; a \in M}$ .

Definition 13.

Let $E$ be an effect algebra. A directed set $S \subseteq E$ is a non-empty set such that for all $a, b \in S$ there exists a $c \in S$ such that $a, b \leq c$ . $E$ is directed complete when for any directed set $S$ there is a supremum $⋁ S$ . It is $ω$ -complete if directed suprema of countable sets exist, or equivalently if any increasing sequence $a_{1} \leq a_{2} \leq \dots$ in $E$ 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 $ω$ -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 anti-isomorphism.

Example 15.

Let $B$ be a $ω$ -complete Boolean algebra. Then $B$ is a $ω$ -complete effect monoid. If $B$ is complete as a Boolean algebra, then $B$ is directed-complete as effect monoid.

Example 16.

Let $X$ be an extremally disconnected compact Hausdorff space, i.e. where the closure of every open set is open. Then $[0, 1]_{C (X)}$ is a directed-complete effect monoid. If $X$ is a basically disconnected [gillman2013rings, 1H] compact Hausdorff space, i.e. where every cozero set has open closure, then $[0, 1]_{C (X)}$ is an $ω$ -complete effect monoid [gillman2013rings, 3N.5].

Iii Overview

The main results of the paper are the following theorems:

Theorem.

Let $M$ be an $ω$ -complete effect monoid. Then $M$ embeds into $M_{1} \oplus M_{2}$ , where $M_{1}$ is an $ω$ -complete Boolean algebra, and $M_{2} = [0, 1]_{C (X)}$ , where $X$ is a basically disconnected compact Haussdorff space (see Theorem 75).

Theorem.

Let $M$ be a directed-complete effect monoid. Then $M ≅ M_{1} \oplus M_{2}$ where $M_{1}$ is a complete Boolean algebra and $M_{2} = [0, 1]_{C (X)}$ for some extremally-disconnected compact Hausdorff space $X$ (see Theorem 76).

By Example 9, the Boolean algebra $M_{1}$ also embeds into a $[0, 1]_{C (X_{M_{1}})}$ , and hence we could ‘coarse-grain’ the direct sums above and say that any $ω$ -complete effect monoid embeds into the unit interval of $C (X_{M_{1}} + X)$ , 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 $M$ be an $ω$ -complete effect monoid. Then:

$M$ is a lattice.
$M$ is an effect divisoid [basthesis].
The multiplication in $M$ is normal: $a \cdot ⋁ S = ⋁ a \cdot S$ .
If $M$ is convex (as an effect algebra), then scalar multiplication is homogeneous: $λ (a \cdot b) = (λ a) \cdot b = a \cdot (λ b)$ for any $λ \in [0, 1]$ and $a, b \in M$ .
$M$ is commutative.
If $M$ has no non-trivial zero-divisors (i.e. when $a \cdot b = 0$ , then $a = 0$ or $b = 0$ ), then $M$ is isomorphic to $[0, 1]$ , ${0, 1}$ or ${0}$ .

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 $ω$ -complete effect monoid $M$ , 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 $p$ must also be idempotents, or halvable, meaning that there is an $a \in M$ such that $p = a a$ . We establish that an $ω$ -complete effect monoid where $1$ is Boolean must be a Boolean algebra, while if $1$ is halvable then it must be convex. In Section VII we show that a maximal collection of orthogonal idempotents of $M$ can be found that consists of a mix of halvable and Boolean idempotents. The corner $p M$ associated to such an idempotent will either be convex (if $p$ is halvable) or Boolean (if $p$ is Boolean). Using normality of multiplication we show that $M$ embeds into the direct sum of the corners associated to these idempotents. Letting $M_{1}$ be the direct sum of the Boolean corners, and $M_{2}$ be the direct sum of the convex corners, we see that $M$ embeds into $M_{1} \oplus M_{2}$ , where $M_{1}$ is Boolean and $M_{2}$ is convex. In Section VIII, we recall some results regarding order unit spaces and use Yosida’s representation theorem to show that a convex $ω$ -complete effect monoid must be isomorphic to the unit interval of a $C (X)$ . 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 $a \in M$ in an effect monoid $M$ , we have $a \cdot a^{⊥} = a^{⊥} \cdot a$ .

Proof.

$a^{2} (a^{⊥} \cdot a) = (a a^{⊥}) \cdot a = 1 \cdot a = a = a \cdot 1 = a \cdot (a a^{⊥}) = a^{2} (a \cdot a^{⊥})$ . Cancelling $a^{2}$ on both sides gives the desired equality. ∎

Lemma 18.

An element $p \in M$ is an idempotent if and only if $p \cdot p^{⊥} = 0$ .

Proof.

$p = p \cdot 1 = p \cdot (p p^{⊥}) = p^{2} p \cdot p^{⊥}$ . Hence $p = p^{2}$ if and only if $p \cdot p^{⊥} = 0$ . ∎

Lemma 19.

For $a, p \in M$ with $p^{2} = p$ , we have

p \cdot a = a ⟺ a \cdot p = a ⟺ a \leq p .

Proof.

Assume $a \leq p$ . Then $a \cdot p^{⊥} \leq p \cdot p^{⊥} = 0$ , so that $a \cdot p^{⊥} = 0$ . Similarly $p^{⊥} \cdot a = 0$ . Hence $a = a \cdot 1 = a \cdot (p p^{⊥}) = a \cdot p a \cdot p^{⊥} = a \cdot p$ . Similarly $p \cdot a = a$ .

Now assume $p \cdot a = a$ . Then immediately $a = p \cdot a \leq p \cdot 1 = p$ . The final implication (that $a \cdot p = a ⟹ a \leq p$ ) is proven similarly. ∎

Lemma 20.

Let $M$ be an effect monoid with idempotent $p \in M$ . Then $p \cdot a = a \cdot p$ for any $a \in M$ .

Proof.

Clearly $p \cdot a \leq p \cdot 1 = p$ and so by Lemma 19 $p \cdot a \cdot p = a \cdot p$ . Similarly $a \cdot p \leq p$ and so $p \cdot a \cdot p = p \cdot a$ . Thus $p \cdot a = p \cdot a \cdot p = a \cdot p$ , as desired. ∎

Corollary 21.

Let $M$ be an effect monoid with idempotent $p \in M$ . The map $e \mapsto (p \cdot e, p^{⊥} \cdot e)$ is an isomorphism $M ≅ p M \oplus p^{⊥} M$ .

The following two lemmas are simple observations that will be used several times.

Lemma 22.

Let $a \leq b$ be elements of an effect algebra $E$ . If $b b^{'} \leq a a^{'}$ for some $a^{'} \leq b^{'}$ from $E$ , then $a = b$ (and $a^{'} = b^{'}$ ).

Proof.

Since $a \leq a^{'}$ and $b \leq b^{'}$ , we have $a a^{'} \leq b b^{'}$ , and so $a a^{'} = b b^{'}$ . Then $0 = (b b^{'}) ⊖ (a a^{'}) = (b ⊖ a) (b^{'} ⊖ a^{'})$ , yielding $b ⊖ a = 0$ and $b^{'} ⊖ a^{'} = 0$ , so $b = a$ and $b^{'} = a^{'}$ . ∎

Lemma 23.

Let $p$ be an idempotent from an effect monoid $M$ , and let $a, b \leq p$ be elements below $p$ . If $a b$ exists, then $a b \leq p$ .

Proof.

Since $a \leq p$ , we have $a \cdot p^{⊥} = 0$ , and similarly, $b \cdot p^{⊥} = 0$ . But then $(a b) \cdot p^{⊥} = 0$ , and hence $(a b) \cdot p = a b$ . By Lemma 19 we then have $a b \leq p$ . ∎

We defined directed set to mean upwards directed. Using the fact that $a \mapsto a^{⊥}$ is an order anti-isomorphism, a directed-complete effect algebra also has all infima of downwards directed (or ‘filtered’) sets (and similarly for countable infima in a $ω$ -complete effect algebra).

The partial sum and difference $⊖$ in an effect algebra have the following non-trivial interactions with suprema and infima.

Lemma 24.

Let $x$ be an element and $S$ a non-empty subset of an effect algebra $E$ . 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):

$x ⋁ S$	$= ⋁ x S$	if $S \subseteq [0, x^{⊥}]_{E}$ and $⋁ S$ exists
$x ⋀ S$	$= ⋀ x S$	if $S \subseteq [0, x^{⊥}]_{E}$ and $⋀ x S$ exists
$(⋁ S) ⊖ x$	$= ⋁ s \in S s ⊖ x$	if $S \subseteq [x, 1]_{E}$ and $⋀ s \in S s ⊖ x$ exists
$(⋀ S) ⊖ x$	$= ⋀ s \in S s ⊖ x$	if $S \subseteq [x, 1]_{E}$ and $⋀ S$ exists
$x ⊖ ⋁ S$	$= ⋀ x ⊖ S$	if $S \subseteq [0, x]_{E}$ and $⋁ S$ exists
$x ⊖ ⋀ S$	$= ⋁ x ⊖ S$	if $S \subseteq [0, x]_{E}$ and $⋁ x ⊖ S$ exists

Proof.

Note that $a \mapsto x a$ gives an order isomorphism $[0, x^{⊥}]_{E} \to [x, 1]_{E}$ with inverse $a \mapsto a ⊖ x$ . Whence $x ()$ preserves and reflects all infima and suprema restricted to $[0, x^{⊥}]_{E}$ and $[x, 1]_{E}$ . Surely, given elements $a \leq b$ from $E$ , and a subset $S$ of the interval $[a, b]_{E}$ , it is clear that any supremum (infimum) of $S$ in $E$ will be the supremum (infimum) of $S$ in $[a, b]_{E}$ too (using here that $S$ is non-empty). The converse does not always hold, but when $S$ has a supremum in $[a, 1]_{E}$ , then this is the supremum in $E$ too (and when $S$ has an infimum in $[0, b]_{E}$ , then this is the infimum in $E$ too). These considerations yield the first four equations. For the latter two we just add the observation that $x ⊖ ()$ gives an order reversing isomorphism $[0, x]_{E} \to [0, x]_{E}$ . ∎

We can now prove a few basic yet useful facts of $ω$ -complete effect monoids. These lemmas deal with elements that are summable with themselves: elements $a$ such that $a ⊥ a$ which means that $a a$ is defined. For $n \in N$ we will use the notation $n a = a \dots a$ for the $n$ -fold sum of $a$ with itself (when it is defined). We study these self-summable elements to be able to define a “ $\frac{1}{2}$ ” in some effect monoids later on.

Lemma 25.

For any $a \in M$ in some effect monoid $M$ , the element $a \cdot a^{⊥}$ is summable with itself.

Proof.

Since $1 = 1 \cdot 1 = (a a^{⊥}) \cdot (a a^{⊥}) = a \cdot a a \cdot a^{⊥} a^{⊥} \cdot a a^{⊥} \cdot a^{⊥}$ , and $a \cdot a^{⊥} = a^{⊥} \cdot a$ by Lemma 17, we see that $a \cdot a^{⊥} a \cdot a^{⊥}$ indeed exists. ∎

Lemma 26.

The only element $a$ of an $ω$ -complete effect algebra $E$ for which the $n$ -fold sum $n a$ exists for all $n$ is zero.

Proof.

We have $a ⋁_{n} n a = ⋁_{n} a n a = ⋁_{n} (n + 1) a = ⋁_{n} n a$ , and so $a = 0$ . ∎

Lemma 27.

Let $M$ be a $ω$ -complete effect monoid with $a \in M$ satisfying $a^{2} = 0$ . Then $a = 0$ .

Proof.

Since $a^{2} = 0$ we have $a = a \cdot 1 = a \cdot (a a^{⊥}) = a \cdot a^{⊥}$ , and hence (because of Lemma 25) $a$ is summable with itself. But furthermore $(a a)^{2} = 4 a^{2} = 0$ , and so $(a a)^{2} = 0$ .

Continuing in this fashion, we see that $2^{n} a$ exists for every $n \in N$ and $(2^{n} a)^{2} = 0$ . Hence, for any $m \in N$ the sum $m a$ exists so that by Lemma 26 $a = 0$ . ∎

Lemma 28.

Let $M$ be a $ω$ -complete effect monoid. Suppose $a \in M$ is summable with itself. Then $⋀_{n} a^{n} = 0$ .

Proof.

Write $b \equiv ⋀_{n} a^{n}$ . As $(2 a)^{n} = 2^{n} a^{n}$ and $b \leq a^{n}$ we see that $2^{n} b$ is defined. But this is true for all $n$ , and so by Lemma 26 $b = 0$ . ∎

V Floors, ceilings and division

In this section we will see that any $ω$ -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 $a \leq b$ we will find an element $a / b$ such that $(a / b) \cdot b = a$ .

Then using ceilings and this division we will show that multiplication in a $ω$ -complete effect monoid is always normal, i.e. that $b \cdot ⋁ S = ⋁ b \cdot S$ for non-empty $S$ for which $⋁ S$ exists. This technical result will be frequently used in the remaining sections.

Definition 29.

Let $(x_{i})_{i \in I}$ be a (potentially infinite) family of elements from an effect algebra $E$ . We say that the sum $∑\vbox% \Large\displaylimitsi∈Ixi$ exists if for every finite subset $S \subseteq I$ the sum $∑\vbox% \Large\displaylimitsi∈Sxi$ exists and the supremum $⋁finite S⊆I∑\vbox\Large\displaylimitsi∈Sxi$ exists as well. In that case we write $∑\vbox% \Large\displaylimitsi∈Ixi≡⋁finite S⊆I∑\vbox\Large\displaylimitsi∈Sxi$ .

Lemma 30.

Given $a \in M$ for an effect monoid $M$ , we have

(a^{N})^{⊥} = a^{⊥} a^{⊥} \cdot a a^{⊥} \cdot a^{2} \dots a^{⊥} \cdot a^{N - 1}

for every natural number $N$ .

Proof.

From the computation

	$1$	$= a^{⊥} a$
		$= a^{⊥} (a^{⊥} a) \cdot a$
		$\equiv a^{⊥} a^{⊥} \cdot a a^{2}$
		$= a^{⊥} a^{⊥} \cdot a (a^{⊥} a) \cdot a^{2}$
		$\equiv a^{⊥} a^{⊥} \cdot a a^{⊥} \cdot a^{2} a^{3}$
		$⋮$
		$= (∑\vbox\huge% \displaylimitsN−1n=0a⊥⋅an)aN$

the result follows immediately. ∎

Corollary 31.

The sum $∑\vbox% \Large\displaylimits∞n=0a⊥⋅an$ exists for any element $a$ from an $ω$ -complete effect monoid $M$ .

Definition 32.

Given an element $a$ of an $ω$ -complete effect monoid $M$

⌈a⌉ ≡ ∑\vbox\huge\displaylimits∞n=0a⋅(a⊥)nand⌊a⌋ ≡ ∞⋀n=0an

are called the ceiling of a and the floor of $a$ , respectively.

We list some basic properties of $⌈ a ⌉$ and $⌊ a ⌋$ in Proposition 40, after we have made the observations necessary to establish them.

Lemma 33.

Given an element $a$ of an effect monoid $M$ the $n$ -fold sum $n (a^{⊥} \cdot a^{n - 1})$ exists for any $n \geq 1$ .

Proof.

Since $a$ and $a^{⊥}$ commute by Lemma 17, we compute

1 = 1n = (a⊥a)n = ∑\vbox\huge\displaylimitsnk=0(nk)((a⊥)k⋅an−k),

and in particular see that the sum $(\frac{n}{1}) (a^{⊥} \cdot a^{n - 1}) \equiv n (a^{⊥} \cdot a^{n - 1})$ exists. ∎

Corollary 34.

Given an element $a$ of an $ω$ -complete effect monoid $M$ , we have $⋀_{n} a^{⊥} \cdot a^{n} = 0$ .

Proof.

Write $b \equiv ⋀_{n} a^{⊥} \cdot a^{n}$ . Since the $n$ -fold sum $n (a^{⊥} \cdot a^{n - 1})$ exists by Lemma 33, and $b \leq a^{⊥} \cdot a^{n - 1}$ , the $n$ -fold sum $n b$ exists too. Whence $b = 0$ , by Lemma 26. ∎

Lemma 35.

We have $⌊ a ⌋ = ⌊ a ⌋ \cdot a = a \cdot ⌊ a ⌋$ for any element $a$ of an $ω$ -complete effect monoid $M$ .

Proof.

Since $⌊ a ⌋ \cdot a^{⊥} = (⋀_{n} a^{n}) \cdot a^{⊥} \leq ⋀_{n} a^{n} \cdot a^{⊥} = ⋀_{n} a^{⊥} \cdot a^{n} = 0$ by Lemma 17 and Corollary 34, we get $⌊ a ⌋ \cdot a^{⊥} = 0$ , and so $⌊ a ⌋ \cdot a = ⌊ a ⌋$ . The other identity has a similar proof. ∎

Lemma 36.

Let $a$ and $b_{1} \leq b_{2} \leq \dots$ be elements of an $ω$ -complete effect monoid $M$ with $a \cdot b_{n} = 0$ for all $n \in N$ . Then $a \cdot ⋁_{n} b_{n} = 0$ .

Proof.

Since $b_{n} = (a a^{⊥}) \cdot b_{n} = a \cdot b_{n} a^{⊥} \cdot b_{n} = a^{⊥} \cdot b_{n}$ for all $n \in N$ , we have

⋁_{n} b_{n} = ⋁_{n} a^{⊥} \cdot b_{n} \leq a^{⊥} \cdot ⋁_{n} b_{n} \leq ⋁_{n} b_{n},

which implies that $a^{⊥} \cdot ⋁_{n} b_{n} = ⋁_{n} b_{n}$ , and thus $a \cdot ⋁_{n} b_{n} = 0$ . ∎

Corollary 37.

Given elements $a, b_{1}, b_{2}, \dots$ of a $ω$ -complete effect monoid $M$ such that $∑\vbox% \Large\displaylimitsnbn$ exists, and $a \cdot b_{n} = 0$ for all $n \in N$ , we have $a⋅∑\vbox\Large\displaylimitsnbn=0$ .

Proof.

Writing $sN≡∑\vbox\Large\displaylimitsNn=1bn$ , we have $s_{1} \leq s_{2} \leq \dots$ and $a \cdot s_{n} = 0$ for all $n$ , and so $a⋅∑\vbox\Large\displaylimitsnbn≡a⋅⋁nsn=0$ by Lemma 36. ∎

Proposition 38.

Given elements $a$ and $b$ of an $ω$ -complete effect monoid $M$ ,

a \cdot b = 0 ⟹ a \cdot ⌈ b ⌉ = 0.

Proof.

If $a \cdot b = 0$ , then also $a \cdot b \cdot (b^{⊥})^{n} = 0$ for all $n$ . Hence by Corollary 37 $a⋅⌈b⌉≡a⋅∑\vbox\Large\displaylimits∞n=1b⋅(b⊥)n=0$ . ∎

Proposition 39.

The floor $⌊ a ⌋$ of an element $a$ of an $ω$ -complete effect monoid is an idempotent.

Proof.

By Lemma 18 it suffices to show that $⌊ a ⌋ \cdot ⌊ a ⌋^{⊥} = 0$ . Since $⌊ a ⌋ \cdot a^{⊥} = 0$ (by Lemma 35) we have $⌊ a ⌋ \cdot ⌈ a^{⊥} ⌉ = 0$ by Proposition 38, and so $⌊ a ⌋ \cdot ⌊ a ⌋^{⊥} = 0$ , because $⌊ a ⌋^{⊥} = ⌈ a^{⊥} ⌉$ (which follows from Lemma 30). ∎

Proposition 40.

Given an element $a$ of an $ω$ -complete effect monoid $M$ .

The floor $⌊ a ⌋$ of $a$ is an idempotent with $⌊ a ⌋ \leq a$ . In fact, $⌊ a ⌋$ is the greatest idempotent below $a$ .
The ceiling $⌈ a ⌉$ of $a$ is the least idempotent above $a$ .
We have $⌈ a ⌉^{⊥} = ⌊ a^{⊥} ⌋$ and $⌊ a ⌋^{⊥} = ⌈ a^{⊥} ⌉$ .

Proof.

Point 3 follows from Lemma 30. Concerning point 1: by Proposition 39 we know that $⌊ a ⌋$ is an idempotent. Also, since $⌊ a ⌋ = ⋀_{n} a^{n}$ , we clearly have $⌊ a ⌋ \leq a$ . Now, if $s$ is an idempotent in $M$ with $s \leq a$ , then $s = s^{n} \leq a^{n}$ , and so $s \leq ⋀_{n} a^{n} \equiv ⌊ a ⌋$ . Whence $⌊ a ⌋$ is the greatest idempotent below $a$ . Point 2 now follows easily from 1, since $⌈ \cdot ⌉$ is the dual of $⌊ \cdot ⌋$ under the order anti-isomorphism $(\cdot)^{⊥}$ . ∎

Lemma 41.

$⌈ a b ⌉ = ⌈ a ⌉ \lor ⌈ b ⌉$ for all summable elements $a$ and $b$ of an $ω$ -complete effect monoid $M$ (that is, $⌈ a b ⌉$ is the supremum of $⌈ a ⌉$ and $⌈ b ⌉$ ).

Proof.

Since $⌈ a b ⌉ \geq a b \geq a$ , we have $⌈ a b ⌉ \geq ⌈ a ⌉$ , and similarly, $⌈ a b ⌉ \geq ⌈ b ⌉$ . Let $u$ be an upper bound of $⌈ a ⌉$ and $⌈ b ⌉$ ; we claim that $⌈ a b ⌉ \leq u$ . Since $⌈ a ⌉ \leq u$ and $⌈ b ⌉ \leq u$ , we have $a \leq ⌈ a ⌉ \leq ⌊ u ⌋$ and $b \leq ⌈ b ⌉ \leq ⌊ u ⌋$ , and so $a b \leq ⌊ u ⌋$ by Lemma 19. Whence $⌈ a b ⌉ \leq ⌊ u ⌋ \leq u$ . ∎

Any $ω$ -complete effect monoid is a lattice effect algebra [rievcanova2000generalization]:

Theorem 42.

Any pair of elements $a$ and $b$ from an $ω$ -complete effect monoid $M$ has an infimum, $a \land b$ , given by

a∧b=∑\vbox\huge\displaylimits∞n=1an⋅bn where [ a1=aan+1=an⋅b⊥nb1=bbn+1=a⊥n⋅bn.

Proof.

First order of business is showing that the sum $∑\vbox% \Large\displaylimitsNn=1an⋅bn$ exists for every $N$ . In fact, we’ll show that $a⊖∑\vbox\Large\displaylimitsNn=1an⋅bn=aN+1$ for all $N$ , by induction. Indeed, for $N = 1$ , we have $a ⊖ a \cdot b = a \cdot b^{⊥} = a_{2}$ , and if $a⊖∑\vbox\Large\displaylimitsNn=1an⋅bn=aN+1$ for some $N$ , then $aN+2=aN+1⋅b⊥N+1=aN+1⊖aN+1⋅bN+1=(a⊖∑\vbox\Large\displaylimitsNn=1an⋅bn)⊖aN+1⋅bN+1=a⊖∑\vbox\Large\displaylimitsN+1n=1an⋅bn$ . In particular, $∑\vbox% \Large\displaylimits∞n=1an⋅bn$ exists, and, moreover,

a = ∞⋀m=1am  ∑\displaylimits∞n=1an⋅bn.

By a similar reasoning, we get

b = ∞⋀m=1bm  ∑\displaylimits∞n=1an⋅bn.

Already writing $a∧b≡∑\vbox\Large\displaylimits∞n=1an⋅bb$ , we know at this point that $a \land b \leq a$ and $a \land b \leq b$ . It remains to be shown that $a \land b$ defined above is the greatest lower bound of $a$ and $b$ . So let $ℓ \in M$ with $ℓ \leq a$ and $ℓ \leq b$ be given; we must show that $ℓ \leq a \land b$ .

As an intermezzo, we observe that $(⋀_{n} a_{n}) \cdot (⋀_{m} b_{m}) = 0$ . Indeed, we have $(⋀_{n} a_{n}) \cdot (⋀_{m} b_{m}) \leq ⋀_{n} a_{n} \cdot b_{n}$ , and $⋀_{n} a_{n} \cdot b_{n} = 0$ , because $∑\vbox% \Large\displaylimits∞n=1an⋅bn$ exists. By Proposition 38 it follows that $(⋀_{n} a_{n}) \cdot ⌈ ⋀_{m} b_{m} ⌉ = 0$ . Whence writing $p \equiv ⌈ ⋀_{m} b_{m} ⌉$ , we have $p \cdot ⋀_{n} a_{n} \equiv (⋀_{n} a_{n}) \cdot p = 0$ using Lemma 20. Observing that $⋀_{n} b_{n} \leq p$ and using Lemma 19 we also have $p^{⊥} \cdot ⋀_{n} b_{n} = 0$ . We then calculate $p \cdot a = p \cdot (⋀_{n} a_{n} a \land b) = p \cdot (a \land b)$ and similarly $p^{⊥} \cdot b = p^{⊥} \cdot (a \land b)$ .

Returning to the problem of whether $ℓ \leq a \land b$ , we have

ℓ = p \cdot ℓ p^{⊥} \cdot ℓ \leq p \cdot a p^{⊥} \cdot b = p \cdot (a \land b) p^{⊥} \cdot (a \land b) = a \land b .

Whence $a \land b$ is the infimum of $a$ and $b$ . ∎

Corollary 43.

Any pair of elements $a$ and $b$ from an $ω$ -complete effect monoid $M$ has a supremum, $a \lor b$ , given by $a \lor b = (a^{⊥} \land b^{⊥})^{⊥}$ .∎

The presence of finite infima in $ω$ -complete effect monoids prevents certain subtleties around the existence of arbitrary suprema and infima.

Corollary 44.

Let $a \leq b$ be elements of an $ω$ -complete effect monoid $M$ , and let $S$ be a non-empty subset of $[a, b]_{M}$ .

Then $S$ has a supremum (infimum) in $M$ if and only if $S$ has a supremum (infimum) in $[a, b]_{M}$ , and these suprema (infima) coincide.

Proof.

It is clear that if $S$ has a supremum in $M$ , then this is also the supremum in $[a, b]_{M}$ . For the converse, suppose that $S$ has a supremum $⋁ S$ in $[a, b]_{M}$ , and let $u$ be an upper bound for $S$ in $M$ ; in order to show that $⋁ S$ is the supremum of $S$ in $M$ too, we must prove that $⋁ S \leq u$ . Note that $b \land u$ is an upper bound for $S$ . Indeed, given $s \in S \subseteq [a, b]_{M}$ we have $s \leq b$ , and $s \leq u$ , so $s \leq b \land u$ . Moreover, one easily sees that $b \land u \in [a, b]_{M}$ using the fact that $S$ is non-empty. Whence $b \land u$ is an upper bound for $S$ in $[a, b]_{M}$ , and so $⋁ S \leq b \land u \leq u$ , making $⋁ S$ the supremum of $S$ in $M$ . Similar reasoning applies to infima of $S$ . ∎

Corollary 45.

Given an element $a$ and a non-empty subset $S$ of an $ω$ -complete effect monoid $M$ such that $a s$ exists for all $s \in S$ ,

the supremum $⋁ S$ exists iff $⋁ a S$ exists, and in that case $a ⋁ S = ⋁ a S$ ;
the infimum $⋀ S$ exists iff $⋀ a S$ exists, and in that case $a ⋀ S = ⋀ a S$ .

Proof.

The map $a () : [0, a^{⊥}]_{M} \to [a, 1]_{M}$ , being an order isomorphism, preserves and reflects suprema and infima. Now apply Corollary 44. ∎

Definition 46.

Given elements $a \leq b$ of an $ω$ -complete effect monoid, set

a/b ≡ ∑\vbox\huge\displaylimits∞n=0a⋅(b⊥)n.

Note that the sum exists, because $∑\vbox% \Large\displaylimitsNn=0a⋅(b⊥)n≤∑\vbox\Large\displaylimits∞n=0b⋅(b⊥)n≡⌈b⌉$ for all $N$ .

Lemma 47.

Let $b$ be an element of an $ω$ -complete effect monoid $M$ .

$b / b = ⌈ b ⌉$ .
$(a_{1} a_{2}) / b = a_{1} / b a_{2} / b$ for all summable $a_{1}, a_{2} \in M$ with $a_{1} a_{2} \leq b$ .
$(a \cdot b) / b = a \cdot ⌈ b ⌉$ for all $a \in M$ .
$(a / b) \cdot b = a$ for all $a \in M$ with $a \leq b$ .
${a \cdot b; a \in M} \equiv M b = [0, b]_{M} \equiv {a; a \in M; a \leq b}$ .
The maps $a \mapsto a \cdot b, b \cdot a : M ⌈ b ⌉ \to M b$ are order isomorphisms.

Proof.

Points 1 and 2 are easy, and left to the reader. Concerning 3, first note that

(a⋅b)/b ≡ ∑\vbox\huge\displaylimits∞n=0a⋅b⋅(b⊥)n ≤ a⋅∑\vbox\huge\displaylimits∞n=0b⋅(b⊥)n = a⋅⌈b⌉.

Thus $(a \cdot b) / b \leq a \cdot ⌈ b ⌉$ . Since similarly $(a^{⊥} \cdot b) / b \leq a^{⊥} \cdot ⌈ b ⌉$ , we get, using 2,

⌈ b ⌉ = b / b = (a \cdot b) / b (a^{⊥} \cdot b) / b \leq a \cdot ⌈ b ⌉ a^{⊥} \cdot ⌈ b ⌉ = ⌈

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Class Algebra for Ontology Reasoning

I Introduction

Ii Preliminaries

Definition 1.

Example 2.

Example 3.

Remark 4.

Definition 5.

Example 6.

Example 7.

Definition 8.

Example 9.

Remark 10.

Example 11.

Example 12.

Definition 13.

Remark 14.

Example 15.

Example 16.

Iii Overview

Theorem.

Theorem.

Iv Basic results

Lemma 17.

Proof.

Lemma 18.

Proof.

Lemma 19.

Proof.

Lemma 20.

Proof.

Corollary 21.

Lemma 22.

Proof.

Lemma 23.

Proof.

Lemma 24.

Proof.

Lemma 25.

Proof.

Lemma 26.

Proof.

Lemma 27.

Proof.

Lemma 28.

Proof.

V Floors, ceilings and division

Definition 29.

Lemma 30.

Proof.

Corollary 31.

Definition 32.

Lemma 33.

Proof.

Corollary 34.

Proof.

Lemma 35.

Proof.

Lemma 36.

Proof.

Corollary 37.

Proof.

Proposition 38.

Proof.

Proposition 39.

Proof.

Proposition 40.

Proof.

Lemma 41.

Proof.

Theorem 42.

Proof.