c

Journal “Algebra and Discrete Mathematics”

**Quantum Boolean algebras**

**Rafael Díaz**

Communicated by V. A. Artamonov

A b s t r ac t . We introduce quantum Boolean algebras which are the analogue of the Weyl algebras for Boolean affine spaces.

We study quantum Boolean algebras from the logical and the set theoretical viewpoints.

**1.** **Introduction**

After Stone [20] and Zhegalkin [23], Boole’s main contribution to science [5] can be understood as the realization that the mathematics of logical phenomena is controlled — to a large extend — by the field Z2 ={0,1} with two elements; in contrast the mathematics of classical physical phenomena is controlled — to a large extend — by the field R of real numbers. The switch from Z2 to R corresponds with a deep ontological jump from logical to physical phenomena. The switch from R toCcorresponds to the jump from classical to quantum physics.

What makes the logic/physics jump possible is the fact that Z2 may
be regarded as an object of two different categories. On the one hand, it
is a field (Z2*,*+, .) with sum and product defined by making 0 the neutral
element and 1 the product unit. On the other hand, it is a set of truth
values with 0 and 1 representing falsity and truth, respectively. Indeed,
(Z2*,*∨,∧,(·)) is a Boolean algebra: a complemented distributive lattice
with minimum 0 and maximum 1. The operations∨,∧, and (·) correspond
with the logical connectives OR, AND, and NOT. The two viewpoints are

**2010 MSC:**06E75, 16S32, 81P10.

**Key words and phrases:**Boolean algebras, Weyl algebras, quantum logic.

related by the identities:*a*∨*b*=*a*+*b*+*ab, a*∧*b*=*ab, a*=*a*+ 1. These
identities, together with the inverse relation*a*+*b*= (a∧*b)*∨(a∧*b), allow*
us to switch back and forth from the algebraic to the logical viewpoint.

By and large, the logical and algebraic viewpoints have remained separated. In this work, in order to explore quantum-like phenomena in characteristic 2, we place ourselves at the jump. Our algebraic viewpoint is, in a sense, complementary to the quantum logic approach initiated by Birkhoff and von Neumann [3] based on the theory of lattices. For example, while the meet in quantum logic is a commutative connective, we propose in this work a quantum analogue for the meet which turns out to be non-commutative. The appearance of non-commutative operations is an essential feature of quantum mechanics [1, 7, 22].

We take as our guide the well-known fact that the quantization of
canonical phase space may be identified with the algebra of differential
operators on configuration space. In analogy with the real/complex case,
we introduce the algebra BDO* _{n}* of Boolean differential operators onZ

^{n}_{2}. We provide a couple of presentations by generators and relations of BDO

*n*, giving rise to the Boole-Weyl algebras BA

*n*and the shifted Boole-Weyl algebras SBA

*. We call these algebras the quantum Boolean algebras.*

_{n}We study the structural coefficients of BA*n* and SBA*n* in various bases.

Having introduced quantum Boolean algebras, we proceed to study them from the logical and set theoretical viewpoints. For us, the main difference between classical and quantum logic rest on the fact that classical observations, propositions, can be measured without, in principle, modifying the state of the system; quantum observations, in contrast, are quantum operators: the measuring process changes the state of the system.

Indeed, regardless of the actual state of the system, after measurement the system will be an eigenstate of the observable. Quantum observables are operators acting on the states of the system, and thus quite different to classical observables which are descriptions of the state of the system.

This work is organized as follows. In Section 2 we review some stan-
dard facts on regular functions on affine spaces overZ2. In Section 3
we introduce BDO*n*, the algebra of Boolean differential operators on
Z^{n}_{2}. In Section 4 we introduce the Boole-Weyl algebra BA*n* which is
a presentation by generators and relations of BDO*n*. We describe the
structural coefficients of BA*n* in several bases. In Section 5 we introduce
the shifted Boole-Weyl algebra SBA*n* which is another presentation by
generators and relations of BDO*n*, and describe the structural coefficients
of SBA*n*in several bases. In Section 6 we discuss the logical aspects of our
constructions: we introduce a quantum operational logic that generalizes

classical propositional logic, and for which Boolean differential operators
play a semantic role akin to that played by truth functions in classical
propositional logic. We use the theory of operads and props to describe
our results. In Section 7 we adopt a set theoretical viewpoint and show
that just as classical propositional logic is intimately related with PP(x),
the Boolean algebra of sets of subsets of *x, quantum operational logic is*
intimately related with PP(x⊔*x) the quantum Boolean algebra of sets*
of subsets of two disjoint copies of *x. In the final Section 8 we make some*
closing remarks and mention a few topics for future research.

**2.** **Regular functions on Boolean affine spaces**

Our main goal in this work is to study the Boolean analogue for the
Weyl algebras, and to describe those algebras from a logical and a set
theoretical viewpoints. Fixing a field*k, the Weyl algebra W**n* over *k*can
be identified with the *k-algebra of algebraic differential operators on the*
affine space A* ^{n}*(k) =

*k*

*. By definition [14, 19] the*

^{n}*k-algebra*

*k[*A

*] of regular functions on*

^{n}*k*

*is the*

^{n}*k-algebra of maps*

*f*:*k** ^{n}*→

*k*

such that there exists a polynomial *F* ∈*k[x*1*, . . . , x**n*] with*f*(a) =*F*(a)
for all*a*∈*k** ^{n}*. If

*k*is a field of characteristic zero, then the

*k-algebra of*regular functions on

*k*

*can be identified with*

^{n}*k[x*

_{1}

*, . . . , x*

*n*] the polynomial ring of over

*k. Let∂*1

*, . . . , ∂*

*n*be the derivations of

*k[x*1

*, . . . , x*

*n*] given by

*∂**i**x**j* =*δ**i,j* for *i, j*∈[n] ={1, . . . , n}. The *k-algebra DO**n* of differential
operators on*k** ^{n}* is the subalgebra of

End* _{k}*(k[x

_{1}

*, . . . , x*

*n*])

generated by*∂**i* and the operators of multiplication by*x**i* for*i*∈[n].

By definition, the Weyl algebra A*n* is the*k-algebra defined via gene-*
rators and relations as

*khx*_{1}*,. . ., x**n**, y*_{1}*, . . . , y**n*i/hx*i**x**j*−x*j**x**i**, y**i**y**j*−y*j**y**i**, y**i**x**j*−x*j**y**i**, y**i**x**i*−x*i**y**i*−1i.

where *khx*_{1}*, . . . , x**n**, y*_{1}*, . . . , y**n*iis the free associative *k-algebra generated*
by*x*_{1}*, . . . , x**n**, y*_{1}*, . . . , y**n*, andhx*i**x**j*−*x**j**x**i**, y**i**y**j* −*y**j**y**i**, y**i**x**j*−*x**j**y**i**, y**i**x**i*−
*x**i**y**i*−1i is the ideal generated by the relations *x**i**x**j* =*x**j**x**i* and*y**i**y**j* =
*y**j**y**i* for *i, j* ∈ [n], *y**i**x**j* = *x**j**y**i* for *i* 6= *j* ∈ [n], *y**i**x**i* = *x**i**y**i* + 1 for
*i* ∈ [n]. The Weyl algebra A*n* comes with a natural representation

A*n* → End*k*(k[x1*, . . . , x**n*]) sending *y**i* to *∂**i* and *x**i* to the operator of
multiplication by *x**i*. This representation induces an isomorphism of
algebras A*n*→DO*n*.

We proceed to study the analogue of the Weyl algebras for the Boolean
affine spacesA* ^{n}*(Z2) =Z

^{n}_{2}. First, we review some basic facts on regular functions onZ

^{n}_{2}. Let M(Z

^{n}_{2}

*,*Z2) be theZ2-algebra of all maps from Z

^{n}_{2}to Z2 with pointwise addition and multiplication. TheZ2-algebraZ2[A

*] of regular functions onZ*

^{n}

^{n}_{2}is the sub-algebra of M(Z

^{n}_{2}

*,*Z2) consisting of the maps

*f*:Z

^{n}_{2}→Z2 for which there exists a polynomial

*F*∈Z2[x1

*, . . . , x*

*n*] such that

*f*(a) =

*F*(a) for all

*a*∈ Z

^{n}_{2}. In this case Z2[A

*] is not a polynomial ring; instead we have the following result.*

^{n}**Lemma 1.** *There is an exact sequence of* Z2*-algebras*

0→ hx^{2}_{1}+*x*_{1}*, . . . ., x*^{2}* _{n}*+

*x*

*i →Z2[x*

_{n}_{1}

*, . . . , x*

*]→Z2[A*

_{n}*]→0*

^{n}*where*hx

^{2}

_{1}+

*x*

_{1}

*, . . . ., x*

^{2}

*+*

_{n}*x*

*i*

_{n}*is the ideal generated by the relationsx*

^{2}

*=*

_{i}*x*

_{i}*for* *i*∈[n].

Therefore the ringZ2[A* ^{n}*] of regular functions onZ

^{n}_{2}can be identified with the quotient ring

Z2[A* ^{n}*] =Z2[x

_{1}

*, . . . , x*

*n*]/hx

^{2}

_{1}+

*x*

_{1}

*, . . . ., x*

^{2}

*+*

_{n}*x*

*n*i.

Often we think ofZ^{n}_{2} as a ring, with coordinate-wise sum and product.

We identify Z* ^{n}*2 with P[n], the set of subsets of [n], via characteristic
functions:

*a*∈Z

^{n}_{2}is identified with the subset

*a*⊆[n] such that

*i*∈

*a*if and only if

*a*

*i*= 1. With this identification the product

*ab*of elements in Z

^{n}_{2}agrees with the intersection

*a*∩

*b*of the sets

*a*and

*b; the sum*

*a*+

*b*corresponds with the symmetric difference

*a*+

*b*= (a∪

*b)*\(a∩

*b);*

the element*a*+ (1, . . . ,1) is identified with the complement*a*of *a. Note*
that*a*∪*b*=*a*+*b*+*ab. We let PP[n] be the set of families of subsets of*
[n]. For *a*∈P[n], let *m** ^{a}*:Z

^{n}_{2}→Z2 be the characteristic function of the set {a} ⊆ P[n]. For

*a*∈ P[n] non-empty, let

*x*

*∈ Z2[x1*

^{a}*, . . . , x*

*n*] be the monomial

*x*

*=*

^{a}^{Q}

_{i∈a}*x*

*i*. Also set

*x*

^{∅}= 1. The monomial

*x*

*defines the characteristic function*

^{a}*x*

*: Z*

^{a}

^{n}_{2}→ Z2 of the set {b |

*a*⊆

*b} ⊆*P[n]. For

*a*∈P[n] non-empty, let

*w*

*∈Z2[x1*

^{a}*, . . . , x*

*n*] be given by

*w*

*=*

^{a}^{Q}

*(x*

_{i∈a}*i*+1).

Also set*w*^{∅} = 1. The monomial *w** ^{a}* defines the characteristic function

*w*

*:Z*

^{a}

^{n}_{2}→Z2 of the set {b|

*b*⊆

*a} ⊆*P[n].

Lemma 2 below follows from the definitions above and the M¨obius inversion formula [18], which can be stated as follows. Given maps

*f, g*: P[n]→*R, withR* a ring of characteristic 2, then
*f(b) =*^{X}

*a⊆b*

*g(a)* if and only if *g(b) =*^{X}

*a⊆b*

*f*(a).

**Lemma 2.** *The following identities hold in* Z2[A* ^{n}*]:

1)*m** ^{a}*=

*x*

^{a}*w*

^{a}*.*2)

*x*

*=*

^{a}^{X}

*a⊆b*

*m*^{b}*.* 3) *m** ^{a}*=

^{X}

*a⊆b*

*x*^{b}*.* 4) *w** ^{b}*=

^{X}

*a⊆b*

*m*^{a}*.*
5)*m** ^{a}*=

^{X}

*a⊆b*

*w*^{b}*.* 6) *w** ^{b}* =

^{X}

*a⊆b*

*x*^{a}*.* 7) *x** ^{b}* =

^{X}

*a⊆b*

*w*^{a}*.* 8) *m*^{a}*m** ^{b}*=δ

*ab*

*m*

^{a}*.*9)

*x*

^{a}*x*

*=*

^{b}*x*

^{a∪b}*.*10)

*w*

^{a}*w*

*=*

^{b}*w*

^{a∪b}*.*

Note that Z2[A* ^{n}*] = M(Z

^{n}_{2}

*,*Z2), indeed a map

*f*:Z

^{n}_{2}→ Z2 can be written as

*f* = ^{X}

*f*(a)=1

*m** ^{a}*=

^{X}

*f*(a)=1

*x*^{a}*w** ^{a}*=

^{X}

*f*(a)=1

Y

*i∈a*

*x**i*

Y

*i∈a*

(x*i*+ 1)

= ^{X}

*f*(a)=1,b⊆a

*x** ^{a∪b}*=

^{X}

*f(a)=1,a⊆b*

*x*^{b}*.*

From Lemma 2 we see that there are several natural bases for the Z2-vector space

Z2[A* ^{n}*] =Z2[x

_{1}

*, . . . , x*

*n*]/hx

^{2}

_{1}+

*x*

_{1}

*, . . . ., x*

^{2}

*+*

_{n}*x*

*n*i,

namely we can pick{m* ^{a}*|

*a*∈P[n]},{x

*|*

^{a}*a*∈P[n]}, or{w

*|*

^{a}*a*∈P[n]}.

We use the following notation to write the coordinates of *f* ∈Z2[A* ^{n}*] in
each one of these bases

*f* = ^{X}

*a∈P[n]*

*f*(a)m* ^{a}*=

^{X}

*a∈P[n]*

*f** _{x}*(a)x

*=*

^{a}^{X}

*a∈P[n]*

*f** _{w}*(a)w

^{a}*.*

We obtain three linear maps *f* →*f*,*f* →*f**x* and *f* →*f**w* fromZ2[A* ^{n}*] to
M(Z

^{n}_{2}

*,*Z2). The coordinates

*f*,

*f*

*and*

_{x}*f*

*are connected, via the M¨obius inversion formula, by the relations:*

_{w}*f**x*(b) =^{X}

*a⊆b*

*f*(a), f(b) =^{X}

*a⊆b*

*f**x*(a), f*w*(b) =^{X}

*a⊆b*

*f*(a),

*f*(b) =^{X}

*a⊆b*

*f**w*(a), f*x*(a) =^{X}

*a⊆b*

*f**w*(b), f*w*(a) =^{X}

*a⊆b*

*f**x*(b).

The maps*f* →*f**x* and*f* →*f**w* fail to be ring morphisms. Instead we
have the identities:

(f g)*x*(c) = ^{X}

*a∪b=c*

*f**x*(a)g*x*(b) and (f g)*w*(c) = ^{X}

*a∪b=c*

*f**w*(a)g*w*(b).

We define a predicate*O* on finite sets as follows: given a finite set*a,*
then *Oa* holds if and only if the cardinality of *a*is an odd number. In
other words,*O* is the map from finite sets to Z2 such that*Oa*= 1 if and
only if the cardinality of*a*is odd.

**Example 3.** Let *C* ∈ PP[n]. An ordered *k-covering of* *a* ∈ P[n] by
elements of *C* is a tuple *c*_{1}*, . . . , c** _{k}* ∈

*C*such that

*c*

_{1}∪ · · · ∪

*c*

*=*

_{k}*a. Let*

*k-Cov*

*C*(a) be the set of

*k-coverings ofa*by elements of

*C. Then*

*a*∈P[n]

belongs to*C* if and only if|k-Cov*C*(a)|is odd for every*k*>1. Indeed, let
*f* ∈Z2[A* ^{n}*] be given by

*f* =^{X}

*c∈C*

*x** ^{c}* =

^{X}

*a∈P[n]*

1*C*(a)x^{a}*,*

where 1*C*: P[n]→Z2 is the characteristic function of*C. Sincef** ^{k}*=

*f*for every

*f*∈Z2[A

*], we have*

^{n}X

*a∈P[n]*

1*C*(a)x* ^{a}*=

*f*=

*f*

*=*

^{k}^{X}

*a∈P[n]*

X

*a*_{1}∪···∪a*k*=a

Y*k*

*i=1*

1*C*(a*i*)

*x*^{a}

= ^{X}

*a∈P[n]*

*O(k-Cov**C*(a))x^{a}*.*

We conclude that 1*C*(a) =*O(k-Cov**C*(a)), and thus *a*∈*C* if and only if

|k-Cov* _{C}*(a)|is odd.

**3.** **Differential operators on Boolean affine spaces**

Next we consider the algebra of differential operators on affine Boolean
spaces. Note that the partial derivatives*∂** _{i}*onZ2[x

_{1}

*, . . . , x*

*] do not descent to well-defined operators onZ2[A*

_{n}*]; indeed if we had such an operator, then 0 =*

^{n}*x*

*i*+

*x*

*i*=

*∂*

*i*

*x*

^{2}

*=*

_{i}*∂*

*i*

*x*

*i*= 1. The Boolean partial derivative

*∂**i**f*:Z^{n}_{2} →Z2 of a map *f*:Z^{n}_{2} →Z2 is given [6, 17] by

*∂**i**f(x) =f*(x+*e**i*) +*f*(x)

where *e**i* ∈Z^{n}_{2} is the vector with vanishing entries except at position *i.*

This definition yields well-defined operators *∂**i*: Z2[A* ^{n}*]→ Z2[A

*]. The*

^{n}operators*∂**i* are skew derivations; indeed they satisfy the twisted Leibnitz
identity

*∂** _{i}*(f g) = (∂

_{i}*f*)g+ (s

_{i}*f*)(∂

_{i}*g)*

where the shift operators *s**i*:Z2[A* ^{n}*] → Z2[A

*] are given by*

^{n}*s*

*i*

*f*(x) =

*f*(x+

*e*

*i*). Indeed:

*∂**i*(f g)(x) =*f*(x+*e**i*)g(x+*e**i*) +*f(x)g(x)*

= [f(x+*e**i*) +*f*(x)]g(x) +*f*(x+*e**i*)[g(x+*e**i*) +*g(x)]*

=*∂**i**f(x)g(x) +s**i**f(x)∂**i**g(x).*

The operators *∂**i* are nilpotent:

*∂*_{i}^{2}*f*(x) =*∂**i**f*(x+*e**i*) +*∂**i**f*(x) =*f(x) +f*(x+*e**i*) +*f*(x+*e**i*) +*f*(x) = 0.

**Definition 4.** TheZ2-algebra BDO*n* of Boolean differential operators
on Z^{n}_{2} is the Z2-subalgebra of End_{Z}_{2}(Z2[A* ^{n}*]) generated by

*∂*

*i*and the operators of multiplication by

*x*

*i*for

*i*∈[n].

**Theorem 5.** *The following identities hold for* *x**i**, ∂**i**, s**i* ∈ BDO*n* *and*
*i*∈[n]:

1. x^{2}* _{i}* =

*x*

*i*; 2. ∂

_{i}^{2}= 0; 3. s

^{2}

*= 1; 4. ∂*

_{i}*i*=

*s*

*i*+ 1;

5. ∂*i**s**i* =*s**i**∂**i*=*∂**i*; 6. s*i*=*∂**i*+ 1; 7. s*i**x**i* =*x**i**s**i*+*s**i*= (x*i*+ 1)s*i*;
8. ∂*i**x**i* =*x**i**∂**i*+*s**i*=*x**i**∂**i*+*∂**i*+ 1.

*Proof.* We have already shown that *x*^{2}* _{i}* =

*x*

*i*and

*∂*

_{i}^{2}= 0. For the other identities we have

• *s*^{2}_{i}*f*(x) =*s**i**f*(x+*e**i*) =*f*(x+*e**i*+*e**i*) =*f*(x);

• *∂**i**f*(x) =*f*(x+*e**i*) +*f(x) =s**i**f*(x) +*f*(x) = (s*i*+ 1)f(x);

•*s**i**∂**i**f*(x) =*∂**i**f(x*+*e**i*) =*f(x*+*e**i*+*e**i*) +*f(x*+*e**i*) =*f(x) +f*(x+*e**i*) =

*∂**i**f*(x);

• *∂*_{i}*s*_{i}*f*(x) =*s*_{i}*f*(x+*e** _{i}*) +

*s*

_{i}*f(x) =*

*f(x*+

*e*

*+*

_{i}*e*

*) +*

_{i}*f(x*+

*e*

*) =*

_{i}*f*(x) +

*f*(x+

*e*

*i*) =

*∂*

*i*

*f*(x);

•*s**i**x**i**f*(x) = (x*i*+1)f(x+e*i*) =*x**i**f*(x+e*i*)+f(x+e*i*) =*x**i**s**i**f*(x)+s*i**f*(x) =
(x_{i}*s** _{i}*+

*s*

*)f(x);*

_{i}• *s**i**f*(x) = *f*(x+*e**i*) = *f*(x +*e**i*) + *f(x) +* *f*(x) = *∂**i**f*(x) + *f(x) =*
(∂*i*+ 1)f(x);

• *∂** _{i}*(x

_{i}*f*)(x) =

*x*

_{i}*f*(x+

*e*

*) +*

_{i}*f(x*+

*e*

*) +*

_{i}*x*

_{i}*f*(x) =

*x*

*(f(x+*

_{i}*e*

*) +*

_{i}*f*(x)) +

*f*(x+

*e*

*i*), thus

• *∂**i*(x*i**f*) =*x**i**∂**i**f* +*f*(x+*e**i*) = (x*i**∂**i*+*s**i*)f = (x*i**∂**i*+*∂**i*+ 1)f.

The operator *∂**i* acts on the bases*m** ^{a}*,

*x*

*and*

^{a}*w*

*as follows:*

^{a}*∂**i**m** ^{a}*=

*m*

^{a+e}*+*

^{i}*m*

^{a}*,*

*∂**i**x** ^{a}*=

(*x** ^{a\i}* if

*i*∈

*a*

0 otherwise and *∂**i**w** ^{a}*=

(*w** ^{a\i}* if

*i*∈

*a*0 otherwise.

From these expressions we obtain that:

•*∂**i**f*(a) = 1 if and only if*f*(a)6=*f*(a+*e**i*), that is,

*∂**i**f* = ^{X}

*a∈P[n],f(a)6=f(a+e**i*)

*m*^{a}*.*

•(∂*i**f*)*x*(a) =*f**x*(a∪*i) if* *i /*∈*a, and (∂**i**f*)*x*(a) = 0 if *i*∈*a, that is,*

*∂**i**f* = ^{X}

*i∈a∈P[n]*

*f**x*(a)x^{a−i}*.*

•(∂*i**f*)*w*(a) =*f**w*(a∪*i) if* *i /*∈*a, and (∂**i**f*)*w*(a) = 0 if *i*∈*a, that is,*

*∂**i**f* = ^{X}

*i∈a∈P[n]*

*f**w*(a)w^{a−i}*.*

More generally one can show by induction, for*a, b*∈P[n], that:

*∂*^{b}*m** ^{a}*=

^{X}

*c⊆b*

*m*^{a+c}*,*

*∂*^{b}*x** ^{a}*=

(*x** ^{a\b}* if

*b*⊆

*a*

0 otherwise, and *∂*^{b}*w** ^{a}*=

(*w** ^{a\b}* if

*b*⊆

*a*0 otherwise.

By definition BDO*n*⊆End_{Z}_{2}(Z2[A* ^{n}*]) acts naturally onZ2[A

*], so we get a map*

^{n}BDO* _{n}*⊗

_{Z}

_{2}Z2[A

*]→Z2[A*

^{n}*].*

^{n}**Proposition 6.** *Consider maps* *D*: P[n]×P[n]→Z2 *and* *f*: P[n]→Z2*.*
1. *Let* *D*= ^{P}

*a,b∈P[n]*

*D(a, b)m*^{a}*∂** ^{b}* ∈BDO

*n*

*,*

*f*=

^{P}

*c∈P[n]*

*f*(c)m* ^{c}* ∈ Z2[A

*],*

^{n}*and*

*Df*=

^{P}

*a∈P[n]*

*Df*(a)m^{a}*. Then we have*
*Df*(a) =^{X}

*e⊆b*

*D(a, b)f(a*+*e).*

2. *Let* *D*= ^{P}

*a,b∈P[n]*

*D**x*(a, b)x^{a}*∂** ^{b}* ∈BDO

*n*

*,*

*f*=

^{P}

*c∈P[n]*

*f**x*(c)x* ^{c}* ∈Z2[A

*],*

^{n}*andDf*=

^{P}

*e∈P[n]*

*Df**x*(e)x^{e}*. Then*
*Df**x*(e) = ^{X}

*a,b⊆c*

*a∪(c\b)=e*

*D**x*(a, b)f*x*(c).

*Proof.*

1. Df = ^{X}

*a,b,c∈P[n]*

*D(a, b)f*(c)m^{a}*∂*^{b}*m** ^{c}* =

^{X}

*a,e⊆b,c*

*D(a, b)f(c)m*^{a}*m*^{c+e}

= ^{X}

*a,e⊆b*

*D(a, b)f(a*+*e)m** ^{a}*=

^{X}

*a∈P[n]*

X

*e⊆b*

*D(a, b)f(a*+*e)*

*m*^{a}*.*

2. Df = ^{X}

*a,b,c∈P[n]*

*D**x*(a, b)f*x*(c)x^{a}*∂*^{b}*x** ^{c}* =

^{X}

*a,b⊆c*

*D**x*(a, b)f*x*(c)x^{a∪c\b}

= ^{X}

*e∈P[n]*

X

*a,b⊆c*

*a∪(c\b)=e*

*D**x*(a, b)f*x*(c)

*x*^{e}*.*

**Theorem 7.** *For* *n*>1 *we have* BDO*n*= End_{Z}_{2}(Z2[A* ^{n}*]).

*Proof.* Note that
dim

End_{Z}_{2}(Z2[A* ^{n}*])

= dim(Z2[A* ^{n}*])dim(Z2[A

*]) = 2*

^{n}*2*

^{n}*= 2*

^{n}^{2n}

*.*The set {x

^{a}*∂*

*|*

^{b}*a, b*∈ P[n]} has 2

^{2n}elements and generates BDO

*n*as a vector space over Z2; thus it is enough to show that it is a linearly independent set. Suppose that

X

*a,b∈P[n]*

*f*(a, b)x^{a}*∂** ^{b}* =

^{X}

*b∈P[n]*

X

*a∈P[n]*

*f*(a, b)x^{a}

*∂** ^{b}* = 0.

Pick a minimal set*c*∈P[n] such that ^{P}

*a∈P[n]*

*f*(a, c)x* ^{a}*6= 0. We have
X

*a,b∈P[n]*

*f*(a, b)x^{a}*∂*^{b}

(x* ^{c}*) =

^{X}

*b∈P[n]*

X

*a∈P[n]*

*f*(a, b)x^{a}

*∂** ^{b}*(x

*)*

^{c}= ^{X}

*a∈P[n]*

*f*(a, c)x* ^{a}*= 0.

Therefore, since{x* ^{a}*|

*a*∈P[n]} is a basis forZ2[A

*], we have*

^{n}*f*(a, c) = 0 in contradiction with the fact

^{P}

_{a∈P[n]}*f*(a, c)x

*6= 0. We conclude that dim BDO*

^{a}

_{n}^{}= 2

^{2n}yielding the desired result.

Putting together Proposition 6 and Theorem 7 we get a couple of
explicit ways of identifying BDO* _{n}* with M

_{2}

*(Z2), the algebra of square matrices of size 2*

^{n}*with coefficients in Z2. Note that M*

^{n}_{2}

*(Z2) may be identified with M(P[n]×P[n],Z2). Moreover, we can identify M(P[n]× P[n],Z2) with the set of directed graphs with vertex set P[n] and without multiple edges as follows: given a matrix*

^{n}*M*∈M2

*(Z2) its associated graph has an edge from*

^{n}*b*to

*a*if and only if

*M*

*a,b*= 1. Let R : BDO

*n*→M

_{2}

*(Z2) be theZ2-linear map constructed as follows. Consider the bases*

^{n}{m^{a}*∂** ^{b}* |

*a, b*∈P[n]} for BDO

*n*and {m

*|*

^{a}*a*∈P[n]} forZ2[A

*].*

^{n}For *a, b* ∈ P[n], let R(m^{a}*∂** ^{b}*) be the matrix of

*m*

^{a}*∂*

*on the basis*

^{b}*m*

*. The action of*

^{a}*m*

^{a}*∂*

*on*

^{b}*m*

*is given by*

^{c}*m*

^{a}*∂*

^{b}*m*

*=*

^{c}*m*

^{a}^{P}

_{e⊆b}*m*

*= P*

^{c+e}*e⊆b**m*^{a}*m** ^{c+e}*=

*m*

*ifc+*

^{a}*a*⊆

*b*and zero otherwise. Therefore, the matrix R(m

^{a}*∂*

*) is given for*

^{b}*c, d*∈P[n] by the rule

R(m^{a}*∂** ^{b}*)

*=*

_{c,d}(1 if*c*=*a*and *d*+*a*⊆*b,*
0 otherwise.

**Example 8.** The graph of R(m^{{1,2}}*∂*^{{2,3}}) is show in Figure 1.

F i g u r e 1 . Graph of the matrix R(m^{{}^{1}^{,}^{2}^{}}*∂*^{{}^{2}^{,}^{3}^{}}).

For a second representation consider theZ2-linear map S : BDO*n*→
M_{2}* ^{n}*(Z2) constructed as follows. Consider the bases

{x^{a}*∂** ^{b}* |

*a, b*∈P[n]} for BDO

*n*and {x

*|*

^{a}*a*∈P[n]} forZ2[A

*].*

^{n}For *a, b*∈ P[n] let S(x^{a}*∂** ^{b}*) be the matrix of

*x*

^{a}*∂*

*on the basis*

^{b}*x*

*. The action of*

^{a}*x*

^{a}*∂*

*on*

^{b}*x*

*is given by*

^{c}*x*^{a}*∂*^{b}*x** ^{c}* =

(*x** ^{a∪c\b}* if

*b*⊆

*c,*0 otherwise.

Therefore, the matrix S(x^{a}*∂** ^{b}*) is given for

*c, d*∈P[n] by the rule S(x

^{a}*∂*

*)*

^{b}*=*

_{c,d}(1 if*c*=*a*∪*d*\*b* and *b*⊆*d,*
0 otherwise.

**Example 9.** The graph associated to the matrix S(m^{{1}}*∂*^{{3}}) is shown
in Figure 2.

F i g u r e 2 . Graph of the matrix S(m^{{}^{1}^{}}*∂*^{{}^{3}^{}}).

**4.** **Boole-Weyl Algebras**

First we motivate, from the viewpoint of canonical quantization, our
definition of Boole-Weyl algebras. Canonical phase space for a field *k*
of characteristic zero can be identified with the affine space *k** ^{n}*×

*k*

*. The Poisson bracket on*

^{n}*k[x*

_{1}

*, . . . , x*

*n*

*, y*

_{1}

*, . . . , y*

*n*] in canonical coordinates

*x*1

*, . . . , x*

*n*

*, y*1

*, . . . , y*

*n*on

*k*

*×*

^{n}*k*

*is given by*

^{n}{x*i**, x**j*}= 0, {y*i**, y**j*}= 0, {x*i**, y**j*}=*δ**i,j**.*

Equivalently, the Poisson bracket is given for*f, g*∈*k[x*1*, . . . , x**n**, y*1*, . . . , y**n*]
by

{f, g}=
X*n*

*i=1*

*∂f*

*∂x**i*

*∂g*

*∂y**i*

− *∂f*

*∂y**i*

*∂f*

*∂x**i*

*.*

Canonical quantization may be formulated as the problem of promoting
the commutative variables *x**i* and *y**j* into non-commutative operators *x*_{b}*i*

and *y*_{b}*j* satisfying the commutation relations:

[*x*_{b}*i**,x*_{b}*j*] = 0, [*y*_{b}*i**,y*_{b}*j*] = 0, [*y*_{b}*i**,x*_{b}*j*] =*δ**i,j**.*

Note that the free algebra generated by*x*_{b}*i* and*y*_{b}*j* subject to the above
relations is precisely what is called the Weyl algebra and its usually
denoted by *A**n*. Now let *k* = Z2 and consider the affine phase spaces

Z^{n}_{2} ×Z^{n}_{2}. Let *x*1*, . . . , x**n**, y*1*, . . . , y**n* be canonical coordinates on Z^{n}_{2} ×Z^{n}_{2}.
The analogue of the Poisson bracket

{*,*}:Z2[A^{2n}]⊗Z2[A^{2n}]→Z2[A^{2n}]
can be expressed for*f, g*∈Z2[A^{2n}] as

{f, g}=
X*n*

*i=1*

*∂f*

*∂x**i*

*∂g*

*∂y**i*

+ *∂f*

*∂y**i*

*∂f*

*∂x**i*

*,*
where _{∂x}^{∂}

*i* and_{∂y}^{∂}

*i* are the Boolean partial derivatives along the coordinates
*x**i* and *y**i*. Clearly, the full set of axioms for a Poisson bracket will not
longer hold, e.g. Boolean derivatives are skew derivations. Nevertheless,
the bracket is still determined by its values on the canonical coordinates:

{x*i**, x**j*}= 0,{y*i**, y**j*}= 0,{x*i**, y**j*}=*δ**i,j*. Canonical quantization consists
in promoting the commutative variables*x** _{i}* and

*y*

*to non-commutative operators*

_{j}*x*

_{b}

*i*and

*y*

_{b}

*j*satisfying the commutation relations:

[*x*_{b}*i**,x*_{b}*j*] = 0, [*y*_{b}*i**,y*_{b}*j*] = 0, [*y*_{b}*i**,x*_{b}*j*] = 0 for*i*6=*j,* and [*y*_{b}*i**,x*_{b}*i*]*s**i* = 1.

Note that in the last relation we use the twisted commutator
[f, g]*s**i* =*f g*+ (s*i**f*)g;

this choice is expected since the operators*y*_{b}*i* are skew derivations instead
of usual derivations. The relation [*y*_{b}*i**,x*_{b}*i*]*s**i* = 1 can be equivalently written
using commutators as [*y*_{b}*i**,x*_{b}*i*] =*y*_{b}*i*+ 1.

**Definition 10.** The Boole-Weyl algebra BA*n*is the quotient ofZ2hx_{1}*, . . . ,*
*x**n**, y*1*, . . . , y**n*i, the free associative Z2-algebra generated by *x*1*, . . . , x**n**,*
*y*_{1}*, . . . , y**n*, by the ideal

hx^{2}* _{i}* +

*x*

*i*

*, x*

*i*

*x*

*j*+

*x*

*j*

*x*

*i*

*, y*

*i*

*y*

*j*+

*y*

*j*

*y*

*i*

*, y*

^{2}

_{i}*, y*

*i*

*x*

*j*+

*x*

*j*

*y*

*i*

*, y*

*i*

*x*

*i*+

*x*

*i*

*y*

*i*+

*y*

*i*+ 1i.

**Theorem 11.** *The map*Z2hx1*, . . . , x**n**, y*1*, . . . , y**n*i →End_{Z}_{2}(Z2[A* ^{n}*])

*sen-*

*ding*

*x*

*i*

*to the operator of multiplication by*

*x*

*i*

*, and*

*y*

*i*

*to*

*∂*

*i*

*, descends to*

*an isomorphism*BA

*n*→End

_{Z}

_{2}(Z2[A

*])*

^{n}*of*Z2

*-algebras.*

*Proof.* By Theorem 5 the given map descends. By definition it is a
surjective map BA*n*→ BDO*n*= End_{Z}_{2}(Z2[A* ^{n}*]). Moreover, this map is
an isomorphisms since dim BA

*n*

= dim End_{Z}_{2}(Z2[A* ^{n}*])

^{}. Indeed using the commutation relations it is easy to check that the natural map

Z2[x_{1}*, . . . , x**n*]/hx^{2}* _{i}* +

*x*

*i*i ⊗Z2[y

_{1}

*, . . . , y*

*n*]/hy

^{2}

*i →BA*

_{i}*n*

is surjective. If ^{P}

*a,b∈P[n]*

*f*(a, b)x* ^{a}*⊗

*y*

*is in the kernel of the latter map, then the Boolean differential operator*

^{b}^{P}

*a,b∈P[n]*

*f*(a, b)x^{a}*∂** ^{b}* would vanish,
and therefore the coefficients

*f(a, b) must vanish as well. Thus*

dim(BA*n*) = dim(Z2[x_{1}*, . . . , x**n*]/hx^{2}* _{i}* +

*x*

*i*i)dim(Z2[y

_{1}

*, . . . , y*

*n*]/hy

_{i}^{2}i)

= 2* ^{n}*2

*= dim(Z2[A*

^{n}*])dim(Z2[A*

^{n}*]) = dim(End*

^{n}_{Z}

_{2}(Z2[A

*])).*

^{n}**Theorem 12.** The mapZ2hx_{1}*, . . . , x**n**, y*_{1}*, . . . , y**n*i →End_{Z}_{2}(Z2[A* ^{n}*]) sen-
ding

*x*

*i*to the operator of multiplication by

*w*

*i*=

*x*

*i*+ 1, and

*y*

*i*to the operator

*∂*

*, descends to an isomorphism BA*

_{i}*→End*

_{n}_{Z}

_{2}(Z2[A

*]) of Z2-algebras.*

^{n}*Proof.* Follows from the fact that *w**i* and *∂**j* satisfy exactly the same
relation as*x**i* and *∂**j*.

**Corollary 13.** Any identity in BA*n*involving*x**i*and*∂**j* has an associated
identity involving *w**i* and *∂**j* obtained by replacing*x**i* by *w**i*.

**Lemma 14.** *Fora, b, c, d*∈P[n]*the following identities hold in* BW_{n}*:*
*1.* *y*^{b}*m** ^{c}* =

^{P}

*b*1⊆b2⊆b

*m*^{c+b}^{2}*y*^{b}^{1}*.*
*2.* *m*^{a}*y*^{b}*m*^{c}*y** ^{d}*=

^{P}

*d⊆e*

*e\d⊆a+c⊆b*

*m*^{a}*y*^{e}*.*
*3.* *y*^{b}*x** ^{c}* =

^{P}

*k*_{1}⊆k2⊆b∩c

*x*^{c\k}^{2}*y*^{b\k}^{1}*.*
*4.* *x*^{a}*y*^{b}*x*^{c}*y** ^{d}* =

^{P}

*a⊆e,d⊆f*

*c(a, b, c, d, e, f*)x^{e}*y*^{f}*, where* *c(a, b, c, d, e, f*) =
*O{k*1⊆*k*2 ⊆*b*∩*c*|*a*∪(c\*k*2) =*e, b*\*k*1 =*f* \*d}.*

*Proof.* 1. By Theorem 11 it is enough to show that the differential opera-
tors associated with both sides of the equation are equal. Consider the
operator of multiplication by *f*: Z^{n}_{2} → Z2 and let *g*:Z^{n}_{2} → Z2 be any
other map. The twisted Leibnitz rule *∂**i*(f g) = (∂*i**f*)g+ (s*i**f)∂**i**g*can be
extended, since*s**i* and *∂**i* commute, to the identity:

*∂** ^{b}*(f g) =

^{X}

*b*1⊔b2=b

(s^{b}^{2}*∂*^{b}^{1}*f*)∂^{b}^{2}*g,*
thus the following identity holds in BA*n*:

*y*^{b}*f* = ^{X}

*b*1⊔b2=b

*s*^{b}^{2}(∂^{b}^{1}*f*)y^{b}^{2} for*f* ∈Z2[x1*, . . . , x**n*].

In particular we obtain that
*y*^{b}*m** ^{c}*=

^{X}

*b*_{1}⊔b2⊆b

*s*^{b}^{2}*s*^{b}^{1}(m* ^{c}*)y

^{b}^{2}=

^{X}

*b*_{1}⊔b2⊆b

*m*^{c+b}^{1}^{+b}^{2}*y*^{b}^{2} = ^{X}

*b*_{1}⊆b2⊆b

*m*^{c+b}^{2}*y*^{b}^{1}*.*

2. We have

*m*^{a}*y*^{b}*m*^{c}*y** ^{d}*=

^{X}

*b*_{1}⊆b2⊆b

*m*^{a}*m*^{c+b}^{2}*y*^{b}^{1}*y** ^{d}*=

^{X}

*b*_{1}⊆b2⊆b

*δ*_{a,c+b}_{2}*m*^{a}*y*^{b}^{1}^{⊔d}

= ^{X}

*b*_{1}⊆a+c⊆b

*m*^{a}*y*^{b}^{1}^{⊔d}= ^{X}

*d⊆e*

*e\d⊆a+c⊆b*

*m*^{a}*y*^{e}*.*

where the last identity follows from the fact that*b*_{2}=*a*+*c*and*e*=*b*_{1}⊔*d.*

3. From the relations*y**i**x**j* =*x**j**y**i* for *i*6= *j* and *y**i**x**i* =*x**i**y**i*+*y**i*+ 1
we can argue as follows. If a letter*y**i* is placed just to the left of a*x**j* we
can move it to the right, since these letters commute. If instead we have
a product*y**i**x**i*, then three options arises:

a) *y**i* moves to the right of*x**i*;
b) *y**i* absorbs*x**i*;

c) *x**i* and *y**i* annihilate each other leaving an 1.

Call*k*_{1} the set of indices for which c) occurs, and *k*_{2} the set of indices
for which either b) or c) occur. Then*k*_{1} ⊆*k*_{2}⊆*b*∩*c*and the set for which
option a) occurs is*b*∩*c*\*k*2. Thus the desired identity is obtained. 4. We
have

*x*^{a}*y*^{b}*x*^{c}*y** ^{d}*=

^{X}

*k*_{1}⊆k2⊆b∩c

*x*^{a∪c\k}^{2}*y*^{(b\k}^{1}^{)⊔d}= ^{X}

*a⊆e,d⊆f*

*c(a, b, c, d, e, f*)x^{e}*y*^{f}*,*
where*c(a, b, c, d, e, f*) =O{k_{1}⊆*k*_{2}⊆*b∩c*|*a∪(c\k*_{2}) =*e, b\k*_{1}=*f*\*d}.*

**Example 15.** *y*^{{1}}*m*^{{1}} =*m*^{{1}}+*m*^{∅}+*m*^{{∅}}*y*^{{1}}; *m*^{{1}}*y*^{{1}}*m*^{{1}}*y*^{{1}} =
*m*^{{1}}*y*^{{1}}; *y*^{{1}}*m*^{{1,2}} =*m*^{{1,2}}+*m*^{{2}}+*m*^{{2}}*y*^{{1}}; *m*^{{2}}*y*^{{1}}*m*^{{1,2}}*y*^{{1}} =
*m*^{{2}}*y*^{{1}};*y*^{{1,2}}*m*^{{1,2,3}}=*m*^{{1,2,3}}+*m*^{{2,3}}+*m*^{{1,3}}+*m*^{{1}}+*m*^{{2,3}}*y*^{{1}}+
*m*^{{3}}*y*^{{1}}+*m*^{{1,3}}*y*^{{2}}+*m*^{{3}}*y*^{{2}}+*m*^{{3}}*y*^{{1,2}};*m*^{{3}}*y*^{{1,2}}*m*^{{1,2,3}}*y*^{{1}} =
*m*^{{3}}*y*^{{1,2}}.

**Example 16.** For *i*∈[k] assume given *A**i* ∈PP[n] and *f**i* =^{P}_{a∈A}_{i}*y** ^{a}*.
Then

*f*1· · ·*f**k*= ^{X}

*b∈P[n]*

*O{(a*1*, . . . , a**k*)∈*A*1×*. . .*×*A**k* |*a*1⊔ · · · ⊔*a**k* =*b}y*^{b}*.*

In particular, for *A*∈PP[n] and*f* =^{P}_{a∈A}*y** ^{a}*, we get that

*f*

*=*

^{k}^{X}

*b∈P[n]*

*O{a*_{1}*, . . . , a** _{k}*∈

*A*|

*a*

_{1}⊔ · · · ⊔

*a*

*=*

_{k}*b}y*

^{b}*.*For example, if

*A*= P[n] then for

*k*>2 we have

*f** ^{k}*=

^{X}

*b∈P[n]*

*O{a*_{1}*, . . . , a** _{k}*∈P[n]|

*a*

_{1}⊔· · ·⊔a

*=*

_{k}*b}y*

*=*

^{b}^{X}

*b∈P[n]*

(k^{|b|} mod 2)y^{b}*,*
thus*f** ^{k}*=

*f*if

*k*is odd and

*f*

*= 1 if*

^{k}*k*is even.

From Lemma 2 we see that there are several natural basis for BA*n*,
namely:

{m^{a}*y** ^{b}*|a, b∈P[n]}, {x

^{a}*y*

*|*

^{b}*a, b*∈P[n]}, {w

^{a}*y*

*|*

^{b}*a, b*∈P[n]}.

We write the coordinates of *f* ∈BA*n* in these bases as:

*f* = ^{X}

*a,b∈P[n]*

*f**m*(a, b)m^{a}*y** ^{b}* =

^{X}

*a,b∈P[n]*

*f**x*(a, b)x^{a}*y** ^{b}* =

^{X}

*a,b∈P[n]*

*f**w*(a, b)w^{a}*y*^{b}*.*
These coordinates systems are connected by the relations:

*f**x*(b, c) =^{X}

*a⊆b*

*f**m*(a, c), f*m*(b, c) =^{X}

*a⊆b*

*f**x*(a, c), f*w*(b, c) =^{X}

*a⊆b*

*f**m*(a, c),
*f**m*(b, c) =^{X}

*a⊆b*

*f**w*(a, c), f*x*(a, c) =^{X}

*a⊆b*

*f**w*(b, c), f*w*(a) =^{X}

*a⊆b*

*f**x*(b, c).

**Theorem 17.***Forf, g*∈BA*n**the following identities hold fora, e, h*∈P[n]:

1) (f g)*m*(a, e) = ^{P}

*b,c,d⊆e*

*e\d⊆a+c⊆b*

*f**m*(a, b)g*m*(c, d).

2) (f g)*x*(e, h) = ^{P}

*a⊆e,b,c,d⊆h*

*c(a, b, c, d, e, h)f**x*(a, b)g*x*(c, d),*where*

*c(a, b, c, d, e, h) =O*^{n}*k*_{1}⊆*k*_{2} ⊆*b*∩*c*|*a*∪(c\k_{2}) =*e, b*\*k*_{1} =*h*\*d*^{o}*.*
*Proof.* 1. Let *f* = ^{P}_{a,b∈P[n]}*f**m*(a, b)m^{a}*y** ^{b}*,

*g*=

^{P}

_{c,d∈P[n]}*g*

*m*(c, d)m

^{c}*y*

*, then*

^{d}*f g*= ^{X}

*a,b,c,d∈P[n]*

*f** _{m}*(a, b)g

*(c, d)m*

_{m}

^{a}*y*

^{b}*m*

^{c}*y*

^{d}= ^{X}

*a,b,c,d,e∈P[n]*

*f**m*(a, b)g*m*(c, d) ^{X}

*d⊆e*

*e\d⊆a+c⊆b*

*m*^{a}*y*^{e}

= ^{X}

*d⊆e,e\d⊆a+c⊆b*

*f** _{m}*(a, b)g

*(c, d)m*

_{m}

^{a}*y*

^{e}*.*