nLab effect algebra



Higher algebra


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Caveat: There is an unrelated notion of “effect of a computation”; that is rather in proximity to the entry monad (in computer science).



In quantum mechanics a self-adjoint operator AA on the given Hilbert space such that its spectrum lies between 0 and 1 (hence a positive operator which is 1\leq 1) is sometimes called an effect or quantum effect (see e.g. (Ludwig, Kraus)). These operators generalize projection operators and may be thought of as quantum observables with “unsharp” or “fuzzy” value.

The notion of effect algebra (due to (Foulis-Bennet 94)) is an abstraction of the structure exhibited by the collection of such effect operators.



A partial commutative monoid? (PCM) consists of a set MM with a zero element 0M0 \in M and a partial binary operation :M×MM\vee : M \times M \to M satisfying the three requirements below. They involve the notation xyx \perp y for: xyx \vee y is defined; in that case x,yx, y are called orthogonal.

  1. Commutativity: xyx\perp y implies yxy\perp x and xy=yxx\vee y=y\vee x.

  2. Associativity: yzy\perp z and x(yz)x\perp(y\vee z) implies xyx\perp y and (xy)z(x\vee y)\perp z and x(yz)=(xy)zx\vee (y\vee z)=(x\vee y)\vee z.

  3. Zero: 0x0\perp x and 0x=x0\vee x=x

(Foulis-Bennet 94 p.22)

In a PCM, we define: xy: z.xz=yx \le y:\Leftrightarrow \exists_z. x \vee z = y. This is a preorder on any PCM.


A PCM is preordered by \le.


Reflexivity is immediate from the Zero axiom, and transitivity follows easily from Associativity.


A generalized effect algebra is a PCM (E,0,)(E, 0, \vee) such that:

  1. Cancellation Law: If aba \perp b, aca \perp c and ab=aca \vee b = a \vee c then b=cb = c.

  2. Positivity Law: If aba \perp b and ab=0a \vee b = 0 then a=b=0a = b = 0.

In a generalized effect algebra, we define: yx=z:y=xzy\ominus x=z:\Leftrightarrow y=x\vee z (which exists iff xyx \le y, and is unique by the Cancellation Law).


A generalized effect algebra is partially ordered by \le.


Suppose xyx \le y and yxy \le x. Let xa=yx \vee a = y and yb=xy \vee b = x. Then x(ab)=x=x0x \vee (a \vee b) = x = x \vee 0, and so ab=0a \vee b = 0 by the Cancellation Law. Therefore, a=b=0a = b = 0 and so x=yx = y.


An effect algebra is a PCM (E,0,)(E,0,\vee) with an orthocomplement. The latter is a unary operation () :EE(-)^\perp :E\to E satisfying:

  1. Orthocomplement Law. x Ex^\perp\in E is the unique element in EE with xx =1x\vee x^\perp=1, where 1=0 1=0^\perp.

  2. Zero-One Law. x1x=0x\perp 1\Rightarrow x=0.

For such an effect algebra one defines: xy:=(x y ) x\wedge y:=(x^\perp\vee y^\perp)^\perp (Foulis-Bennet 94 p. 23)


A structure (E,0,)(E, 0, \vee) is an effect algebra iff it is a generalized effect algebra with a greatest element, in which case that greatest element is 1=0 1 = 0^\perp.


Let (E,0,)(E, 0, \vee) be an effect algebra. Then EE is a generalized effect algebra since:

  1. Cancellation Law. If ab=aca \vee b = a \vee c then ab(ab) =ac(ab) =1a \vee b \vee (a \vee b)^\perp = a \vee c \vee (a \vee b)^\perp = 1, and so b=c=(a(ab) ) b = c = (a \vee (a \vee b)^\perp)^\perp.

  2. Positivity Law. If ab=0a \vee b = 0 then (ab)1(a \vee b) \perp 1, hence a1a \perp 1 and b1b \perp 1 by Associativity. Thus, a=b=0a = b = 0 by the Zero-One Law.

1 is the greatest elements since, for any xx, we have xx =1x \vee x^\perp = 1 and so x1x \leq 1.

Conversely, let (E,0,)(E, 0, \vee) be a generalized effect algebra with greatest element 1. Define x =1xx^\perp = 1 \ominus x for all xx. Then:

  1. Orthocomplement Law. x x^\perp is the unique element such that xx =1x \vee x^\perp = 1 by definition.

  2. Zero-One Law. If x1x \perp 1, then 1x11 \leq x \vee 1, so x1=1x \vee 1 = 1. Thus, x1=01x \vee 1 = 0 \vee 1, and so x=0x = 0 by the Cancellation Law.


If we consider ()y:up(y)down(y )(-)\ominus y:up(y)\to down(y^\perp) and ()y:down(y )up(-)\vee y:down(y^\perp)\to up as functors between posets we have adjunctions

((y)(y)(y)((-\wedge y)\dashv (-\ominus y)\dashv (-\vee y)

Hence these functors are a frobenius pair.

(Foulis-Bennet 94 p.25)



Let EE and FF be effect algebras. A morphism of effect algebras f:EFf : E \rightarrow F is a function such that:

  1. f(1) = 1

  2. If xyx \perp y then f(x)f(y)f(x) \perp f(y) and f(xy)=f(x)f(y)f(x \vee y) = f(x) \vee f(y).

We write EA\mathbf{EA} for the category of effect algebras and morphisms of effect algebras.


(1) effect algebra of predicates

(2) The real unit inteval [0,1][0,1] with \vee being addition of real numbers is an effect algebra since [0,1][0,1] is a pcm with zero object 00 and commutative, associative addition of real numbers and xyx\perp y iff x+y1x+y\le 1. The orthocomplement of x[0,1]x\in [0,1] is given by x =1x\x^\perp=1-x.

(3) Let DD denote the discrete-probability-distribution monad on SetSet which sends a set XX to the collection

D(X):={r 1x 1++r nx n|x iX,r i[0,1],Σ ir i=1}D(X):=\{r_1 x_1+\dots +r_n x_n|x_i\in X, r_i\in [0,1], \Sigma_i r_i=1\}

of formal convex combinations of elements of XX and let Kl(D)Kl(D) denote the Kleisli category of DD which has as objects (just) sets and a morphism f:XYf:X\to Y in Kl(D)Kl(D) is a function f:XD(Y)f:X\to D(Y) which can be interpreted as a Markov chain where the probability of the transition xx ix\to x_i is the coefficient r i[0,1]r_i\in [0,1] in the the convex sum f(x)=r 1x 1++r nx nf(x)=r_1 x_1+\dots+r_n x_n. Kl(D)Kl(D) has as coproducts coproducts of SetSet. A predicate on XKl(D)X\in Kl(D) is hence a function p:XD(X+X)p:X\to D(X+X) and [id X,id X]p=id X[id_X,id_X]\circ p =id_X means that p(x)D(X+X)p(x)\in D(X+X) is a convex combination of elements of the form k 1x,k 2xX+Xk_1 x, k_2 x\in X+X such that we have p(x)=φ(x)k 1x+ψ(x)k 2xp(x)=\varphi(x)k_1 x +\psi(x)k_2 x with φ(x),ψ(x)[0,1]\varphi(x),\psi(x)\in [0,1] such that φx+ψ(x)=1\varphi{x}+\psi(x)=1. Hence p(x)p(x) can be written as p(x)=φ(x)k 1x+(1φ(x))k 2xp(x)=\varphi(x)k_1 x + (1-\varphi(x))k_2 x. In particular a predicate is (uniquely determined by) a function φ:X[0,1]\varphi:X\to [0,1] to the unit interval. In this view the orthocomplemet of φ(x)\varphi(x) is the function x1φ(x)x\mapsto 1-\varphi(x) which is point-wise the orthocomplement of the unit interval in the second example.

(4) In the category HilbHilb of Hilbert spaces the coproduct coincides with the product and hence is a biproduct. In this case a predicate p:XXXp:X\to X\otimes X on a Hilbert space XX has the form p=<p 1,p 2>p=\lt p_1,p_2\gt of a pair of maps and [id X,id X]p[id_X,id_X]\circ p is equivalent to p 1+p 2=id Xp_1 +p_2=id_X where ++ is point-wise addition. In particular p 1p_1 and p 2p_2 determine each other uniquely.

And now comes the eponymous feature: The category HilbHilb is a dagger category and the dagger morphism () :Hilb opHilb(-)^\dagger:Hilb^{op}\to Hilb is the identity on objects and complex conjugation on morphisms. An endomorphism f:XXf:X\to X is called to be a positive endomorphism if there is a gg such that f=g gf=g^\dagger\circ g and a predicate on XX is called to be an effect (on XX) if p 1p_1 and p 2p_2 are positive. Another name for effect is “unsharp predicate”; in this terminology a “sharp predicate” is a subset of the set of projections onto XX.

(5) In a C*^\ast-algebra the elements between 0 and 1 form an effect algebra with (1a)(1-a) as the complement of aa.

(6) As a special case, we obtain the effect algebra of a von Neumann algebra. In general, this is not a lattice. De Groote defines a spectral order on self-adjoint operators which makes the collection of effects a boundedly complete lattice?. However, this is not the canonical order on an effect algebra, as defined above.

Relation to D-Posets

A parallel concept in the literature is that of D-poset (sometimes called D-lattice), originally introduced for the same purpose of studying fuzzy or quantum logics. These first appeared in Kôpka 92 Chovanec-Kôpka 95.


A partial binary operation \ominus on a poset (P,)(P, \leq) is called a difference operation (or simply difference) on PP iff:

(1) abbaa \leq b \leftrightarrow b \ominus a is defined,

(2) babb \ominus a \leq b,

(3) b(ba)=ab \ominus (b \ominus a) = a,

(4) if abca \leq b \leq c implies that cbcac \ominus b \leq c \ominus a and (ca)(cb)=ba(c \ominus a) \ominus (c \ominus b) = b \ominus a.

A D-poset is a poset (P,,,1)(P, \leq, \ominus, 1) with a difference operation and greatest element 1P1 \in P.

Any effect algebra is automatically a D-poset under the difference c:=baac=bc := b \ominus a \iff a \oplus c = b, well-defined by the cancellation property of generalized effect algebras. Ultimately this determines an isomorphism of categories between D-posets and effect algebras.

Relation to Boolean algebras

Every Boolean algebra is an effect algebra, with aba\bot b whenever ab=0a \wedge b=0. Then the partial monoid of the effect algebra is a restriction of the join of the Boolean algebra. The orthocomplement is a =¬aa^\bot=\neg a.

Starting from a powerset Boolean algebra, for example, we have aba\bot b when aa and bb are disjoint, and aba\vee b is the disjoint union.

A function is a Boolean algebra homomorphism if and only if it is a morphism of effect algebras.

In fact the finite Boolean algebras form a dense subcategory of the category EA\mathbf{EA} of effect algebras. This means that every effect algebra is a canonical colimit of finite Boolean algebras.

Since the powerset functor FinSetFinBool op\mathbf{FinSet}\to \mathbf{FinBool}^{\mathrm{op}} is an equivalence of categories (Stone duality), this density property also means that we have a full and faithful functor T:EA[FinSet,Set]T:\mathbf{EA}\to [\mathbf{FinSet},\mathbf{Set}]. This functor can be given explicitly as finding the tests of an effect algebra. A test is a sequence of orthogonal elements that sum to 11. Hence

T(E)(n)={(a 1,,a n)|a 1a n=1} T(E) (n) = \{ (a_1,\dots,a_n)\ |\ a_1\vee\dots \vee a_n=1\}

For example, starting from the unit interval, T([0,1])(n)T([0,1])(n) gives the (n1)(n-1)-simplex.

quantum probability theoryobservables and states


On quantum effect operators:

The notion of effect algebra is due to

  • D. J. Foulis, M. K. Bennet, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1 331–1 352.

Discussion of effect algebras in the context of categorical logic is in

Discussion in the context of quantum logic is in section 6 of

  • Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli, Quantum Logic and Nonclassical Logics, p. 127 in Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) Handbook of Quantum Logic and Quantum Structures: Quantum Logic, 2009 North Holland

A survey of the use of effect algebras in quantum mechanics is in

  • Teiko Heinosaari, Mario Ziman, Guide to Mathematical Concepts of Quantum Theory (arXiv:0810.3536)

    also appeared as:

    The Mathematical Language of Quantum Theory, Cambridge University Press (2011)

  • Hans de Groote, On a canonical lattice structure on the effect algebra of a von Neumann algebra arXiv:0410018

Discussion in relation to presheaves on FinSet op{}^{op} (hence in the classifying topos for objects) and density of Boolean algebras is in

  • Sam Staton, Sander Uijlen, Effect algebras, presheaves, non-locality and contextuality (pdf)

D-posets were first introduced in:

  • F. Kôpka,_D-posets of fuzzy sets_, Tatra Mountains Mathematical Publications, vol. 1, no. 1, pp. 83-87, 1992.

  • F. Chovanec and F. Kôpka, D-lattices, International Journal of Theoretical Physics, vol. 34, no. 8, pp. 1297–1302, 1995.

Last revised on December 1, 2023 at 08:27:10. See the history of this page for a list of all contributions to it.