Quantum Boolean algebras

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.

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Дата:	2017
Автор:	Diaz, R.
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Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2017
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Quantum Boolean algebras / R. Diaz // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 106-143. — Бібліогр.: 23 назв. — англ.

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spelling	irk-123456789-1562582019-06-19T01:27:06Z Quantum Boolean algebras Diaz, R. 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. 2017 Article Quantum Boolean algebras / R. Diaz // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 106-143. — Бібліогр.: 23 назв. — англ. 1726-3255 2010 MSC:06E75, 16S32, 81P10. http://dspace.nbuv.gov.ua/handle/123456789/156258 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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.
format	Article
author	Diaz, R.
spellingShingle	Diaz, R. Quantum Boolean algebras Algebra and Discrete Mathematics
author_facet	Diaz, R.
author_sort	Diaz, R.
title	Quantum Boolean algebras
title_short	Quantum Boolean algebras
title_full	Quantum Boolean algebras
title_fullStr	Quantum Boolean algebras
title_full_unstemmed	Quantum Boolean algebras
title_sort	quantum boolean algebras
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/156258
citation_txt	Quantum Boolean algebras / R. Diaz // Algebra and Discrete Mathematics. — 2017. — Vol. 24, № 1. — С. 106-143. — Бібліогр.: 23 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT diazr quantumbooleanalgebras
first_indexed	2025-07-14T08:42:57Z
last_indexed	2025-07-14T08:42:57Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 24 (2017). Number 1, pp. 106–143 c© Journal “Algebra and Discrete Mathematics” Quantum Boolean algebras Rafael Díaz Communicated by V. A. Artamonov Abstract. 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 to C corresponds 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. R. Díaz 107 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 BDOn of Boolean differential operators on Zn 2 . We provide a couple of presentations by generators and relations of BDOn, giving rise to the Boole-Weyl algebras BAn and the shifted Boole-Weyl algebras SBAn. We call these algebras the quantum Boolean algebras. We study the structural coefficients of BAn and SBAn 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 over Z2. In Section 3 we introduce BDOn, the algebra of Boolean differential operators on Zn 2 . In Section 4 we introduce the Boole-Weyl algebra BAn which is a presentation by generators and relations of BDOn. We describe the structural coefficients of BAn in several bases. In Section 5 we introduce the shifted Boole-Weyl algebra SBAn which is another presentation by generators and relations of BDOn, and describe the structural coefficients of SBAn in several bases. In Section 6 we discuss the logical aspects of our constructions: we introduce a quantum operational logic that generalizes 108 Quantum Boolean algebras 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 Wn over k can be identified with the k-algebra of algebraic differential operators on the affine space An(k) = kn. By definition [14, 19] the k-algebra k[An] of regular functions on kn is the k-algebra of maps f : kn → k such that there exists a polynomial F ∈ k[x1, . . . , xn] with f(a) = F (a) for all a ∈ kn. If k is a field of characteristic zero, then the k-algebra of regular functions on kn can be identified with k[x1, . . . , xn] the polynomial ring of over k. Let ∂1, . . . , ∂n be the derivations of k[x1, . . . , xn] given by ∂ixj = δi,j for i, j ∈ [n] = {1, . . . , n}. The k-algebra DOn of differential operators on kn is the subalgebra of Endk(k[x1, . . . , xn]) generated by ∂i and the operators of multiplication by xi for i ∈ [n]. By definition, the Weyl algebra An is the k-algebra defined via gene- rators and relations as k〈x1,. . ., xn, y1, . . . , yn〉/〈xixj−xjxi, yiyj−yjyi, yixj−xjyi, yixi−xiyi−1〉. where k〈x1, . . . , xn, y1, . . . , yn〉 is the free associative k-algebra generated by x1, . . . , xn, y1, . . . , yn, and 〈xixj − xjxi, yiyj − yjyi, yixj − xjyi, yixi − xiyi − 1〉 is the ideal generated by the relations xixj = xjxi and yiyj = yjyi for i, j ∈ [n], yixj = xjyi for i 6= j ∈ [n], yixi = xiyi + 1 for i ∈ [n]. The Weyl algebra An comes with a natural representation R. Díaz 109 An → Endk(k[x1, . . . , xn]) sending yi to ∂i and xi to the operator of multiplication by xi. This representation induces an isomorphism of algebras An → DOn. We proceed to study the analogue of the Weyl algebras for the Boolean affine spaces An(Z2) = Zn 2 . First, we review some basic facts on regular functions on Zn 2 . Let M(Zn 2 ,Z2) be the Z2-algebra of all maps from Zn 2 to Z2 with pointwise addition and multiplication. The Z2-algebra Z2[An] of regular functions on Zn 2 is the sub-algebra of M(Zn 2 ,Z2) consisting of the maps f : Zn 2 → Z2 for which there exists a polynomial F ∈ Z2[x1, . . . , xn] such that f(a) = F (a) for all a ∈ Zn 2 . In this case Z2[An] is not a polynomial ring; instead we have the following result. Lemma 1. There is an exact sequence of Z2-algebras 0 → 〈x2 1 + x1, . . . ., x2 n + xn〉 → Z2[x1, . . . , xn] → Z2[An] → 0 where 〈x2 1 +x1, . . . ., x2 n +xn〉 is the ideal generated by the relations x2 i = xi for i ∈ [n]. Therefore the ring Z2[An] of regular functions on Zn 2 can be identified with the quotient ring Z2[An] = Z2[x1, . . . , xn]/〈x2 1 + x1, . . . ., x2 n + xn〉. Often we think of Zn 2 as a ring, with coordinate-wise sum and product. We identify Zn 2 with P[n], the set of subsets of [n], via characteristic functions: a ∈ Zn 2 is identified with the subset a ⊆ [n] such that i ∈ a if and only if ai = 1. With this identification the product ab of elements in Zn 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 ma : Zn 2 → Z2 be the characteristic function of the set {a} ⊆ P[n]. For a ∈ P[n] non-empty, let xa ∈ Z2[x1, . . . , xn] be the monomial xa = ∏ i∈a xi. Also set x∅ = 1. The monomial xa defines the characteristic function xa : Zn 2 → Z2 of the set {b \| a ⊆ b} ⊆ P[n]. For a ∈ P[n] non-empty, let wa ∈ Z2[x1, . . . , xn] be given by wa = ∏ i∈a(xi+1). Also set w∅ = 1. The monomial wa defines the characteristic function wa : Zn 2 → Z2 of the set {b \| b ⊆ a} ⊆ P[n]. Lemma 2 below follows from the definitions above and the Möbius inversion formula [18], which can be stated as follows. Given maps 110 Quantum Boolean algebras f, g : P[n] → R, with R a ring of characteristic 2, then f(b) = ∑ a⊆b g(a) if and only if g(b) = ∑ a⊆b f(a). Lemma 2. The following identities hold in Z2[An]: 1) ma = xawa. 2) xa = ∑ a⊆b mb. 3) ma = ∑ a⊆b xb. 4) wb = ∑ a⊆b ma. 5) ma = ∑ a⊆b wb. 6) wb = ∑ a⊆b xa. 7) xb = ∑ a⊆b wa. 8) mamb =δabm a. 9) xaxb = xa∪b. 10) wawb = wa∪b. Note that Z2[An] = M(Zn 2 ,Z2), indeed a map f : Zn 2 → Z2 can be written as f = ∑ f(a)=1 ma = ∑ f(a)=1 xawa = ∑ f(a)=1 ∏ i∈a xi ∏ i∈a (xi + 1) = ∑ f(a)=1,b⊆a xa∪b = ∑ f(a)=1,a⊆b xb. From Lemma 2 we see that there are several natural bases for the Z2-vector space Z2[An] = Z2[x1, . . . , xn]/〈x2 1 + x1, . . . ., x2 n + xn〉, namely we can pick {ma \| a ∈ P[n]}, {xa \| a ∈ P[n]}, or {wa \| a ∈ P[n]}. We use the following notation to write the coordinates of f ∈ Z2[An] in each one of these bases f = ∑ a∈P[n] f(a)ma = ∑ a∈P[n] fx(a)xa = ∑ a∈P[n] fw(a)wa. We obtain three linear maps f → f , f → fx and f → fw from Z2[An] to M(Zn 2 ,Z2). The coordinates f , fx and fw are connected, via the Möbius inversion formula, by the relations: fx(b) = ∑ a⊆b f(a), f(b) = ∑ a⊆b fx(a), fw(b) = ∑ a⊆b f(a), f(b) = ∑ a⊆b fw(a), fx(a) = ∑ a⊆b fw(b), fw(a) = ∑ a⊆b fx(b). R. Díaz 111 The maps f → fx and f → fw fail to be ring morphisms. Instead we have the identities: (fg)x(c) = ∑ a∪b=c fx(a)gx(b) and (fg)w(c) = ∑ a∪b=c fw(a)gw(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 c1, . . . , ck ∈ C such that c1 ∪ · · · ∪ ck = a. Let k-CovC(a) be the set of k-coverings of a by elements of C. Then a ∈ P[n] belongs to C if and only if \|k-CovC(a)\| is odd for every k > 1. Indeed, let f ∈ Z2[An] be given by f = ∑ c∈C xc = ∑ a∈P[n] 1C(a)xa, where 1C : P[n] → Z2 is the characteristic function of C. Since fk = f for every f ∈ Z2[An], we have ∑ a∈P[n] 1C(a)xa = f = fk = ∑ a∈P[n] ( ∑ a1∪···∪ak=a k∏ i=1 1C(ai) ) xa = ∑ a∈P[n] O(k-CovC(a))xa. We conclude that 1C(a) = O(k-CovC(a)), and thus a ∈ C if and only if \|k-CovC(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 on Z2[x1, . . . , xn] do not descent to well-defined operators on Z2[An]; indeed if we had such an operator, then 0 = xi + xi = ∂ix 2 i = ∂ixi = 1. The Boolean partial derivative ∂if : Zn 2 → Z2 of a map f : Zn 2 → Z2 is given [6, 17] by ∂if(x) = f(x + ei) + f(x) where ei ∈ Zn 2 is the vector with vanishing entries except at position i. This definition yields well-defined operators ∂i : Z2[An] → Z2[An]. The 112 Quantum Boolean algebras operators ∂i are skew derivations; indeed they satisfy the twisted Leibnitz identity ∂i(fg) = (∂if)g + (sif)(∂ig) where the shift operators si : Z2[An] → Z2[An] are given by sif(x) = f(x + ei). Indeed: ∂i(fg)(x) = f(x + ei)g(x + ei) + f(x)g(x) = [f(x + ei) + f(x)]g(x) + f(x + ei)[g(x + ei) + g(x)] = ∂if(x)g(x) + sif(x)∂ig(x). The operators ∂i are nilpotent: ∂2 i f(x) = ∂if(x + ei) + ∂if(x) = f(x) + f(x + ei) + f(x + ei) + f(x) = 0. Definition 4. The Z2-algebra BDOn of Boolean differential operators on Zn 2 is the Z2-subalgebra of EndZ2(Z2[An]) generated by ∂i and the operators of multiplication by xi for i ∈ [n]. Theorem 5. The following identities hold for xi, ∂i, si ∈ BDOn and i ∈ [n]: 1. x2 i = xi; 2. ∂2 i = 0; 3. s2 i = 1; 4. ∂i = si + 1; 5. ∂isi = si∂i = ∂i; 6. si = ∂i + 1; 7. sixi = xisi + si = (xi + 1)si; 8. ∂ixi = xi∂i + si = xi∂i + ∂i + 1. Proof. We have already shown that x2 i = xi and ∂2 i = 0. For the other identities we have • s2 i f(x) = sif(x + ei) = f(x + ei + ei) = f(x); • ∂if(x) = f(x + ei) + f(x) = sif(x) + f(x) = (si + 1)f(x); • si∂if(x) = ∂if(x + ei) = f(x + ei + ei) + f(x + ei) = f(x) + f(x + ei) = ∂if(x); • ∂isif(x) = sif(x + ei) + sif(x) = f(x + ei + ei) + f(x + ei) = f(x) + f(x + ei) = ∂if(x); • sixif(x) = (xi+1)f(x+ei) = xif(x+ei)+f(x+ei) = xisif(x)+sif(x) = (xisi + si)f(x); • sif(x) = f(x + ei) = f(x + ei) + f(x) + f(x) = ∂if(x) + f(x) = (∂i + 1)f(x); • ∂i(xif)(x) = xif(x + ei) + f(x + ei) + xif(x) = xi(f(x + ei) + f(x)) + f(x + ei), thus • ∂i(xif) = xi∂if + f(x + ei) = (xi∂i + si)f = (xi∂i + ∂i + 1)f . R. Díaz 113 The operator ∂i acts on the bases ma, xa and wa as follows: ∂im a = ma+ei + ma, ∂ix a = { xa\i if i ∈ a 0 otherwise and ∂iw a = { wa\i if i ∈ a 0 otherwise. From these expressions we obtain that: • ∂if(a) = 1 if and only if f(a) 6= f(a + ei), that is, ∂if = ∑ a∈P[n],f(a) 6=f(a+ei) ma. • (∂if)x(a) = fx(a ∪ i) if i /∈ a, and (∂if)x(a) = 0 if i ∈ a, that is, ∂if = ∑ i∈a∈P[n] fx(a)xa−i. • (∂if)w(a) = fw(a ∪ i) if i /∈ a, and (∂if)w(a) = 0 if i ∈ a, that is, ∂if = ∑ i∈a∈P[n] fw(a)wa−i. More generally one can show by induction, for a, b ∈ P[n], that: ∂bma = ∑ c⊆b ma+c, ∂bxa = { xa\b if b ⊆ a 0 otherwise, and ∂bwa = { wa\b if b ⊆ a 0 otherwise. By definition BDOn ⊆ EndZ2(Z2[An]) acts naturally on Z2[An], so we get a map BDOn ⊗Z2 Z2[An] → Z2[An]. Proposition 6. Consider maps D : P[n] × P[n] → Z2 and f : P[n]→Z2. 1. Let D = ∑ a,b∈P[n] D(a, b)ma∂b ∈ BDOn, f = ∑ c∈P[n] f(c)mc ∈ Z2[An], and Df = ∑ a∈P[n] Df(a)ma. Then we have Df(a) = ∑ e⊆b D(a, b)f(a + e). 114 Quantum Boolean algebras 2. Let D = ∑ a,b∈P[n] Dx(a, b)xa∂b ∈ BDOn, f = ∑ c∈P[n] fx(c)xc ∈ Z2[An], and Df = ∑ e∈P[n] Dfx(e)xe. Then Dfx(e) = ∑ a,b⊆c a∪(c\b)=e Dx(a, b)fx(c). Proof. 1. Df = ∑ a,b,c∈P[n] D(a, b)f(c)ma∂bmc = ∑ a,e⊆b,c D(a, b)f(c)mamc+e = ∑ a,e⊆b D(a, b)f(a + e)ma = ∑ a∈P[n] (∑ e⊆b D(a, b)f(a + e) ) ma. 2. Df = ∑ a,b,c∈P[n] Dx(a, b)fx(c)xa∂bxc = ∑ a,b⊆c Dx(a, b)fx(c)xa∪c\b = ∑ e∈P[n] ( ∑ a,b⊆c a∪(c\b)=e Dx(a, b)fx(c) ) xe. Theorem 7. For n > 1 we have BDOn = EndZ2(Z2[An]). Proof. Note that dim ( EndZ2(Z2[An]) ) = dim(Z2[An])dim(Z2[An]) = 2n2n = 22n. The set {xa∂b \| a, b ∈ P[n]} has 22n elements and generates BDOn as a vector space over Z2; thus it is enough to show that it is a linearly independent set. Suppose that ∑ a,b∈P[n] f(a, b)xa∂b = ∑ b∈P[n] ( ∑ a∈P[n] f(a, b)xa ) ∂b = 0. Pick a minimal set c ∈ P[n] such that ∑ a∈P[n] f(a, c)xa 6= 0. We have ( ∑ a,b∈P[n] f(a, b)xa∂b ) (xc) = ∑ b∈P[n] ( ∑ a∈P[n] f(a, b)xa ) ∂b(xc) = ∑ a∈P[n] f(a, c)xa = 0. R. Díaz 115 Therefore, since {xa \| a ∈ P[n]} is a basis for Z2[An], we have f(a, c) = 0 in contradiction with the fact ∑ a∈P[n] f(a, c)xa 6= 0. We conclude that dim ( BDOn ) = 22n yielding the desired result. Putting together Proposition 6 and Theorem 7 we get a couple of explicit ways of identifying BDOn with M2n(Z2), the algebra of square matrices of size 2n with coefficients in Z2. Note that M2n(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 M ∈ M2n(Z2) its associated graph has an edge from b to a if and only if Ma,b = 1. Let R : BDOn → M2n(Z2) be the Z2-linear map constructed as follows. Consider the bases {ma∂b \| a, b ∈ P[n]} for BDOn and {ma \| a ∈ P[n]} for Z2[An]. For a, b ∈ P[n], let R(ma∂b) be the matrix of ma∂b on the basis ma. The action of ma∂b on mc is given by ma∂bmc = ma ∑ e⊆b mc+e =∑ e⊆b mamc+e = maifc + a ⊆ b and zero otherwise. Therefore, the matrix R(ma∂b) is given for c, d ∈ P[n] by the rule R(ma∂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. Figure 1. Graph of the matrix R(m{1,2}∂{2,3}). For a second representation consider the Z2-linear map S: BDOn → M2n(Z2) constructed as follows. Consider the bases {xa∂b \| a, b ∈ P[n]} for BDOn and {xa \| a ∈ P[n]} for Z2[An]. For a, b ∈ P[n] let S(xa∂b) be the matrix of xa∂b on the basis xa. The action of xa∂b on xc is given by xa∂bxc = { xa∪c\b if b ⊆ c, 0 otherwise. 116 Quantum Boolean algebras Therefore, the matrix S(xa∂b) is given for c, d ∈ P[n] by the rule S(xa∂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. Figure 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 kn × kn. The Poisson bracket on k[x1, . . . , xn, y1, . . . , yn] in canonical coordinates x1, . . . , xn, y1, . . . , yn on kn × kn is given by {xi, xj} = 0, {yi, yj} = 0, {xi, yj} = δi,j . Equivalently, the Poisson bracket is given for f, g ∈k[x1, . . . , xn, y1, . . . , yn] by {f, g} = n∑ i=1 ∂f ∂xi ∂g ∂yi − ∂f ∂yi ∂f ∂xi . Canonical quantization may be formulated as the problem of promoting the commutative variables xi and yj into non-commutative operators x̂i and ŷj satisfying the commutation relations: [x̂i, x̂j ] = 0, [ŷi, ŷj ] = 0, [ŷi, x̂j ] = δi,j . Note that the free algebra generated by x̂i and ŷj subject to the above relations is precisely what is called the Weyl algebra and its usually denoted by An. Now let k = Z2 and consider the affine phase spaces R. Díaz 117 Zn 2 × Zn 2 . Let x1, . . . , xn, y1, . . . , yn be canonical coordinates on Zn 2 × Zn 2 . The analogue of the Poisson bracket { , } : Z2[A2n] ⊗ Z2[A2n] → Z2[A2n] can be expressed for f, g ∈ Z2[A2n] as {f, g} = n∑ i=1 ∂f ∂xi ∂g ∂yi + ∂f ∂yi ∂f ∂xi , where ∂ ∂xi and ∂ ∂yi are the Boolean partial derivatives along the coordinates xi and yi. 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: {xi, xj} = 0, {yi, yj} = 0, {xi, yj} = δi,j . Canonical quantization consists in promoting the commutative variables xi and yj to non-commutative operators x̂i and ŷj satisfying the commutation relations: [x̂i, x̂j ] = 0, [ŷi, ŷj ] = 0, [ŷi, x̂j ] = 0 for i 6= j, and [ŷi, x̂i]si = 1. Note that in the last relation we use the twisted commutator [f, g]si = fg + (sif)g; this choice is expected since the operators ŷi are skew derivations instead of usual derivations. The relation [ŷi, x̂i]si = 1 can be equivalently written using commutators as [ŷi, x̂i] = ŷi + 1. Definition 10. The Boole-Weyl algebra BAn is the quotient of Z2〈x1, . . . , xn, y1, . . . , yn〉, the free associative Z2-algebra generated by x1, . . . , xn, y1, . . . , yn, by the ideal 〈x2 i + xi, xixj + xjxi, yiyj + yjyi, y2 i , yixj + xjyi, yixi + xiyi + yi + 1〉. Theorem 11. The map Z2〈x1, . . . , xn, y1, . . . , yn〉 → EndZ2(Z2[An]) sen- ding xi to the operator of multiplication by xi, and yi to ∂i, descends to an isomorphism BAn → EndZ2(Z2[An]) of Z2-algebras. Proof. By Theorem 5 the given map descends. By definition it is a surjective map BAn → BDOn = EndZ2(Z2[An]). Moreover, this map is an isomorphisms since dim ( BAn ) = dim ( EndZ2(Z2[An]) ) . Indeed using the commutation relations it is easy to check that the natural map Z2[x1, . . . , xn]/〈x2 i + xi〉 ⊗ Z2[y1, . . . , yn]/〈y2 i 〉 → BAn 118 Quantum Boolean algebras is surjective. If ∑ a,b∈P[n] f(a, b)xa ⊗ yb is in the kernel of the latter map, then the Boolean differential operator ∑ a,b∈P[n] f(a, b)xa∂b would vanish, and therefore the coefficients f(a, b) must vanish as well. Thus dim(BAn) = dim(Z2[x1, . . . , xn]/〈x2 i + xi〉)dim(Z2[y1, . . . , yn]/〈y2 i 〉) = 2n2n = dim(Z2[An])dim(Z2[An]) = dim(EndZ2(Z2[An])). Theorem 12. The map Z2〈x1, . . . , xn, y1, . . . , yn〉 → EndZ2(Z2[An]) sen- ding xi to the operator of multiplication by wi = xi + 1, and yi to the operator ∂i, descends to an isomorphism BAn → EndZ2(Z2[An]) of Z2-algebras. Proof. Follows from the fact that wi and ∂j satisfy exactly the same relation as xi and ∂j . Corollary 13. Any identity in BAn involving xi and ∂j has an associated identity involving wi and ∂j obtained by replacing xi by wi. Lemma 14. For a, b, c, d ∈ P[n] the following identities hold in BWn: 1. ybmc = ∑ b1⊆b2⊆b mc+b2yb1. 2. maybmcyd = ∑ d⊆e e\d⊆a+c⊆b maye. 3. ybxc = ∑ k1⊆k2⊆b∩c xc\k2yb\k1. 4. xaybxcyd = ∑ a⊆e,d⊆f c(a, b, c, d, e, f)xeyf , where c(a, b, c, d, e, f) = O{k1 ⊆ k2 ⊆ b ∩ c \| a ∪ (c \ k2) = e, b \ k1 = 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 : Zn 2 → Z2 and let g : Zn 2 → Z2 be any other map. The twisted Leibnitz rule ∂i(fg) = (∂if)g + (sif)∂ig can be extended, since si and ∂i commute, to the identity: ∂b(fg) = ∑ b1⊔b2=b (sb2∂b1f)∂b2g, thus the following identity holds in BAn: ybf = ∑ b1⊔b2=b sb2(∂b1f)yb2 for f ∈ Z2[x1, . . . , xn]. R. Díaz 119 In particular we obtain that ybmc = ∑ b1⊔b2⊆b sb2sb1(mc)yb2 = ∑ b1⊔b2⊆b mc+b1+b2yb2 = ∑ b1⊆b2⊆b mc+b2yb1 . 2. We have maybmcyd = ∑ b1⊆b2⊆b mamc+b2yb1yd = ∑ b1⊆b2⊆b δa,c+b2mayb1⊔d = ∑ b1⊆a+c⊆b mayb1⊔d = ∑ d⊆e e\d⊆a+c⊆b maye. where the last identity follows from the fact that b2 = a + c and e = b1 ⊔ d. 3. From the relations yixj = xjyi for i 6= j and yixi = xiyi + yi + 1 we can argue as follows. If a letter yi is placed just to the left of a xj we can move it to the right, since these letters commute. If instead we have a product yixi, then three options arises: a) yi moves to the right of xi; b) yi absorbs xi; c) xi and yi annihilate each other leaving an 1. Call k1 the set of indices for which c) occurs, and k2 the set of indices for which either b) or c) occur. Then k1 ⊆ k2 ⊆ b∩ c and the set for which option a) occurs is b ∩ c \ k2. Thus the desired identity is obtained. 4. We have xaybxcyd = ∑ k1⊆k2⊆b∩c xa∪c\k2y(b\k1)⊔d = ∑ a⊆e,d⊆f c(a, b, c, d, e, f)xeyf , where c(a, b, c, d, e, f)=O{k1 ⊆k2 ⊆b∩c \| a∪(c\k2)=e, b\k1 =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 Ai ∈ PP[n] and fi = ∑ a∈Ai ya. Then f1 · · · fk = ∑ b∈P[n] O{(a1, . . . , ak) ∈ A1 × . . . × Ak \| a1 ⊔ · · · ⊔ ak = b}yb. 120 Quantum Boolean algebras In particular, for A ∈ PP[n] and f = ∑ a∈A ya, we get that fk = ∑ b∈P[n] O{a1, . . . , ak ∈ A \| a1 ⊔ · · · ⊔ ak = b}yb. For example, if A = P[n] then for k > 2 we have fk = ∑ b∈P[n] O{a1, . . . , ak ∈ P[n] \| a1⊔· · ·⊔ak = b}yb = ∑ b∈P[n] (k\|b\| mod 2)yb, thus fk = f if k is odd and fk = 1 if k is even. From Lemma 2 we see that there are several natural basis for BAn, namely: {mayb\|a, b ∈ P[n]}, {xayb \| a, b ∈ P[n]}, {wayb \| a, b ∈ P[n]}. We write the coordinates of f ∈ BAn in these bases as: f = ∑ a,b∈P[n] fm(a, b)mayb = ∑ a,b∈P[n] fx(a, b)xayb = ∑ a,b∈P[n] fw(a, b)wayb. These coordinates systems are connected by the relations: fx(b, c) = ∑ a⊆b fm(a, c), fm(b, c) = ∑ a⊆b fx(a, c), fw(b, c) = ∑ a⊆b fm(a, c), fm(b, c) = ∑ a⊆b fw(a, c), fx(a, c) = ∑ a⊆b fw(b, c), fw(a) = ∑ a⊆b fx(b, c). Theorem 17. For f, g ∈BAn the following identities hold for a, e, h∈P[n]: 1) (fg)m(a, e) = ∑ b,c,d⊆e e\d⊆a+c⊆b fm(a, b)gm(c, d). 2) (fg)x(e, h) = ∑ a⊆e,b,c,d⊆h c(a, b, c, d, e, h)fx(a, b)gx(c, d), where c(a, b, c, d, e, h) = O { k1 ⊆ k2 ⊆ b∩c \| a∪(c\k2) = e, b\k1 = h\d } . Proof. 1. Let f = ∑ a,b∈P[n] fm(a, b)mayb, g = ∑ c,d∈P[n] gm(c, d)mcyd, then fg = ∑ a,b,c,d∈P[n] fm(a, b)gm(c, d)maybmcyd = ∑ a,b,c,d,e∈P[n] fm(a, b)gm(c, d) ∑ d⊆e e\d⊆a+c⊆b maye = ∑ d⊆e,e\d⊆a+c⊆b fm(a, b)gm(c, d)maye. R. Díaz 121 2. Let f = ∑ a,b∈P[n] fx(a, b)xayb and g = ∑ c,d∈P[n] gx(c, d)xcyd, then fg = ∑ a,b,c,d∈P[n] fx(a, b)gx(c, d)xaybxcyd = ∑ a,b,c,d∈P[n] ∑ a⊆e,d⊆h fx(a, b)gx(c, d)c(a, b, c, d, e, h)xeyh, where c(a, b, c, d, e, h)=O{k1 ⊆k2 ⊆b∩c \|a∪(c\k2) = e, b\k1 =h\d}. Example 18. Let xryr = ∑ a,b∈P[n] fm(a, b)mayb and xsys = ∑ a,b∈P[n] gm(a, b)mayb. Then (fg)2 m(a, e) = ∑ b,c,d⊆e e\d⊆a+c⊆b fm(a, b)gm(c, d). For a non-vanishing summand we must have that a = b = r, c = d = s, and s ⊆ e. The conditions e \ s ⊆ r + s ⊆ r implies that s ⊆ r and e \ s ⊆ r \ s, thus e ⊆ r. We conclude that (fg)2 m(a, e) = 1 if and only if s ⊆ r, a = r and s ⊆ e ⊆ r. Thus xryrxsys = 0 if s * r. For s ⊆ r we get xryrxsys = ∑ s⊆e⊆r xrye. In particular we get that (xryr)n = xryr. Example 19. Let f = ∑ a,b∈P[n] mayb = ∑ a,b∈P[n] fm(a, b)mayb. We have f2 m(a, e) = ∑ b,c,d⊆e e\d⊆a+c⊆b 1 = O { b, c, d \| d ⊆ e, e \ d ⊆ a + c ⊆ b } . Note that if a + c is not equal to [n], then there are an even number of choices for b, thus we can assume that c = a and b = [n]. The condition e \ d ⊆ a + c = [n] becomes trivial, and therefore f2(a, e) = OP[\|e\|] = 0 if e 6= ∅ and f2(a, e) = 1 if e = ∅. Therefore we have f2 = ∑ a∈P[n] ma. Example 20. Let r = ∑ i∈[n] x{i}y{i} = ∑ a,b∈P[n] rx(a, b)xayb ∈BAn, then r2 x(e, f) = ∑ a⊆e,b,c,d⊆f c(a, b, c, d, e, f)rx(a, b)rx(c, d), where c(a, b, c, d, e, f) = O{k1 ⊆ k2 ⊆ b ∩ c \| a ∪ (c \ k2) = e, b \ k1 = f \ d}. Clearly \|a\| = \|b\| = \|c\| = \|d\| = 1, a = b, and c = d. Moreover, we have \|b ∩ c\| 6 1. If \|b ∩ c\| = 1, then a = b = c = d = e = {i} for some i ∈ [n]. If k1 = ∅, then there are 122 Quantum Boolean algebras two options for k2 leading to a vanishing coefficient. Thus we may assume that k1 = k2 = {i} and then necessarily f = {i}. Thus we conclude that r2 x({i}, {i}) = 1. If instead \|b ∩ c\| = 0, then k1 = k2 = ∅, a ∪ c = e and b = f \ d. Let i 6= j and suppose that a = b = {i} and c = d = {j}. Then e = f = {i, j} and r2 x({i, j}, {i, j}) = 1. All together we conclude that r2 = ∑ i∈[n] x{i}y{i} + ∑ i6=j∈[n] x{i,j}y{i,j}. Example 21. From Corollary 13 we see that if s = ∑ i∈[n] w{i}y{i} then s2 = ∑ i∈[n] w{i}y{i} + ∑ i6=j∈[n] w{i,j}y{i,j}. Equivalently, if s = ∑ i∈[n] y{i} + ∑ i∈[n] x{i}y{i} then s2 = ∑ i∈[n] y{i} + ∑ i∈[n] x{i}y{i} + ∑ i6=j∈[n] y{i,j} + ∑ i6=j∈[n] x{i,j}y{i,j} + ∑ i6=j∈[n] ( x{i} + x{j} ) y{i,j}. 5. A shifted presentation So far, the operators ∂i have played the main role. In this section we take an alternative viewpoint and let the operators si be the main characters. Recall that the Boolean partial derivatives and the Boolean shift operators are related by the identities yi = si + 1 and si = yi + 1. For a, b ∈ P[n], set ya = ∏ i∈a yi and sa = ∏ i∈a si. We have yb = ∏ i∈b yi = ∏ i∈b (si + 1) = ∑ a⊆b sa, and by the Möbius inversion formula sb = ∑ a⊆b ya. Proposition 22. Consider maps D : P[n]×P[n] → Z2 and f : P[n] → Z2. 1. Let D = ∑ a,b∈P[n] D(a, b)masb ∈ BDOn, f = ∑ c∈P[n] f(c)mc ∈ Z2[An], and Df = ∑ a∈P[n] Df(a)ma. Then we have Df(a) = ∑ b∈P[n] D(a, b)f(a + b). R. Díaz 123 2. Let D = ∑ a,b∈P[n] Dx(a, b)xasb ∈ BDOn, f = ∑ c∈P[n] fx(c)xc ∈ Z2[An], and Df = ∑ d∈P[n] Dfx(d)xd. Then Dfx(d) = ∑ a,e⊆b∩c a∪(c\e)=d Dx(a, b)fx(c). Proof. 1. Df = ∑ a,b,c∈P[n] D(a, b)f(c)masbmc = ∑ a,b,c D(a, b)f(c)mamb+c = ∑ a,b D(a, b)f(a + b)ma = ∑ a∈P[n] ( ∑ b∈P[n] D(a, b)f(a + b) ) ma. 2. Df = ∑ a,b,c∈P[n] Dx(a, b)fx(c)xasbxc = ∑ a,e⊆b∩c Dx(a, b)fx(c)xa∪c\e = ∑ d∈P[n] ( ∑ a,e⊆b∩c a∪(c\e)=d Dx(a, b)fx(c) ) xd. Proposition 22 and Theorem 7 provide a couple of explicit ways of identifying BDOn with M2n(Z2) the algebra of square matrices of size 2n with coefficients in Z2. Consider the Z2-linear map R: BDOn → M2n(Z2) sending masb to R(masb) the matrix of masb on the basis ma. The action of masb on mc is given by masbmc = ma if c = a + b and 0 otherwise. Therefore, the matrix R(masb) is given for c, d ∈ P[n] by the rule R(masb)c,d = { 1 if c = a and d = a + b 0 otherwise. Example 23. The graph of the matrix R(masb)c,d is shown in Figure 3. Figure 3. Graph of the matrix R(m{1,2}∂{2,3}). For a second representation consider Z2-linear the map S: BDOn → M2n(Z2) sending xasb to S(xasb), the matrix of xasb on the basis xa. The action of xasb on xc is given by xasbxc = ∑ e⊆b∩c xa∪c\e. Therefore, the matrix S(xasb) is given for c, d ∈ P[n] by the rule S(xasb)c,d = { 1 if O{e ⊆ b ∩ d \| c = a ∪ d \ e} 0 otherwise 124 Quantum Boolean algebras Example 24. The graph of the matrix S(x{1,2}s{1,3}) is shown in Figure 4. Figure 4. Graph of the matrix R(m{1,2}∂{2,3}). Definition 25. The shifted Boole-Weyl algebra SBAn is the quotient of Z2〈x1, . . . , xn, s1, . . . , sn〉, the free associative Z2-algebra generated by x1, . . . , xn, s1, . . . , sn, by the ideal 〈x2 i + xi, xixj + xjxi, sisj + sjsi, s2 i + 1, sixj + xjsi, sixi + xisi + si〉. Theorem 26. The map Z2〈x1, . . . , xn, s1, . . . , sn〉 → EndZ2(Z2[An]) sen- ding xi to the operator of multiplication by xi, and si to the shift operator in the i-direction, descends to an isomorphism SBAn → EndZ2(Z2[An]) of Z2-algebras. Proof. One can check that the map Z2〈x1, . . . , xn, s1, . . . , sn〉 → Z2〈x1, . . . , xn, y1, . . . , yn〉 sending xi to xi and si to yi + 1 descends to an algebra isomorphisms SBAn → BAn. The result then follows from Theorem 11. Theorem 27. The map Z2〈x1, . . . , xn, s1, . . . , sn〉 → EndZ2(Z2[An]) sen- ding xi to the operator of multiplication by wi = xi + 1, and si to the shift operator in the i-direction, descends to an isomorphism SBAn → EndZ2(Z2[An]) of Z2-algebras. Proof. Follows from the fact that wi and sj satisfy exactly the same relation as xi and sj . Corollary 28. Any identity in SBAn involving xi and sj has an associ- ated identity involving wi and sj obtained by replacing xi by wi. Lemma 29. For a, b, c, d ∈ P[n] the following identities hold in SBAn: 1. sbmc = mb+csb. 2. masbmcsd = δa,b+cm asb+d. 3. sbxc = ∑ k⊆b∩c xc\ksb. 4. xasbxcsd = ∑ e⊆b∩c xa∪c\esb+d. R. Díaz 125 Proof. 1. For any f ∈ Z2[An] we have (sbmcf)(x) = mc(x + b)f(x + b) = mb+c(x)f(x + b) = mb+csbf(x), thus sbmc = mb+csb. 2. masbmcsd = mamb+csbsd = δa,b+cm b+csb+d. 3. From the identity sixi = xisi + si we see that as si pass to the right of xi, it may or may not absorb xi. The set k ⊆ b ∩ c is the set of indices for which xi is absorbed by si. 4. xasbxcsd = ∑ e⊆b∩c xaxc\esbsd = ∑ e⊆b∩c xa∪c\esb+d. Example 30. 1. s[n]mc = mcs[n], c = [n] \ c. 2. mcs[n]mcsd = mcsd. 3. s[n]xc = ∑ k⊆c xks[n]. 4. xas[n]xcsd = ∑ k⊆c xa∪ksd. From Lemma 2 we see that there are several natural basis for BAn, namely: {masb\|a, b ∈ P[n]}, {xasb \| a, b ∈ P[n]}, {wasb \| a, b ∈ P[n]}. We write the coordinates of f ∈ SBAn in these bases as: f = ∑ a,b∈P[n] fm,s(a, b)masb = ∑ a,b∈P[n] fx,s(a, b)xasb = ∑ a,b∈P[n] fw,s(a, b)wasb. These coordinates systems are connected by the relations: fx,s(b, c)= ∑ a⊆b fm,s(a, c), fm,s(b, c)= ∑ a⊆b fx,s(a, c), fw,s(b, c)= ∑ a⊆b fm,s(a, c), fm,s(b, c) = ∑ a⊆b fw,s(a, c), fx,s(a, c)= ∑ a⊆b fw,s(b, c), fw,s(a)= ∑ a⊆b fx,s(b, c). Theorem 31. For f, g ∈ SBAn the following identities hold for a, b, e, h ∈ P[n]: 1. (fg)m,s(a, b) = ∑ c∈P[n] fm,s(a, c)gm,s(a + c, b + c). 2. (fg)x,s(e, h) = ∑ a⊆e,b,c∈P[n] O{k ⊆b∩c \| a∪c\k =e}fx,s(a, b)gx,s(c, b+h). 126 Quantum Boolean algebras Proof. 1. Let f = ∑ a,b∈P[n] fm,s(a, b)masb, g = ∑ c,d∈P[n] gm,s(c, d)mcsd, then fg = ∑ a,b,c,d∈P[n] fm,s(a, b)gm,s(c, d)masbmcsd = ∑ b,c,d∈P[n] fm,s(b + c, b)gm,s(c, d)mb+csb+d = ∑ e,f∈P[n] ( ∑ b+c=e b+d=f fm,s(b + c, b)gm,s(c, d) ) mesf = ∑ e,f∈P[n] ( ∑ b∈P[n] fm,s(e, b)gm,s(e + b, f + b) ) mesf . 2. Let f = ∑ a,b∈P[n] fx,s(a, b)xasb, g = ∑ c,d∈P[n] gx,s(c, d)xcsd, then fg = ∑ a,b,c,d∈P[n] fx,s(a, b)gx,s(c, d)xasbxcsd = ∑ a,b,c,d∈P[n],k⊆b∩c fx,s(a, b)gx,s(c, d)xa∪c\ksb+d = ∑ e,h∈P[n] ( ∑ a,b,c,d∈P[n] k⊆b∩c a∪c\k=e,b+d=h fx,s(a, b)gx,s(c, d) ) xesh = ∑ e,h∈P[n] ( ∑ a⊆e,b,c∈P[n] k⊆b∩c a∪c\k=e fx,s(a, b)gx,s(c, b + h) ) xesh = ∑ e,h∈P[n] ( ∑ a⊆e,b,c∈P[n] O{k ⊆b∩c \|a∪c\k =e}fx,s(a, b)gx,s(c, b + h) ) xesh. Example 32. Suppose that f = ∑ a,b∈P[n] fm,s(a, b)masb and g =∑ c,d∈P[n] gm,s(c, d)mcsd are actually regular functions on Zn 2 , i.e. fm,s(a, b) = 0 if b 6= ∅, and gm,s(c, d) = 0 if d 6= ∅. A non-vanishing term in the expression (fg)m,s(a, b) = ∑ c∈P[n] fm,s(a, c)gm,s(a + c, b + c) must have c = ∅, and then we must also have that c = ∅ + c = ∅, and a + c = a + ∅ = a. Thus in this case the product fg is, as expected, just the pointwise product of functions on Zn 2 . R. Díaz 127 Example 33. Let f = ∑ a,b∈P[n] fm,s(a, b)masb, and suppose that g =∑ c,d∈P[n] gm,s(c, d)mcsd is such that gm,s(c, d) = 0 if c 6= [n]. Then a non-vanishing summand in the formula (fg)m,s(a, b) = ∑ c∈P[n] fm,s(a, c)gm,s(a + c, b + c) can only arise for c = a. Therefore (fg)m,s(a, b) = fm,s(a, a)gm,s([n], b+a). For example, we have ( ∑ a∈P[n] masa )( ∑ d∈P[n] m[n]sd ) = ∑ a,b∈P[n] masb. As another example consider f = ∑ a,b∈P[n] masb and g = m[n]s[n]. In this case we get that: ( ∑ a,b∈P[n] masb )( m[n]s[n] ) = ∑ a∈P[n] masa. 6. Quantum operational logic In this section we study quantum Boolean algebras from a logical viewpoint. Propositional logic may be approached from a myriad of viewpoints. Here we take a revisionist approach bias towards the theory of operads and props. We believe this approach may be of interest in itself, and is certainly pretty convenient for our current purposes as it would readily generalize to cover quantum operational logic. We assume the reader to be familiar with the language of operads and props [4,12, 13,15,16]. First we review the basic principles of classical propositional logic [6] which may be summarized as: • On the syntactic side, propositions are words in a certain language. Propositions are either simple or composite. Let x be the finite set of simple propositions, and P(x) be the set of all propositions. Composite propositions are obtained from the simple propositions using the logical connectives. There are several options for the choice of connectives, the most common ones being ∨, ∧, →, ¬. • On the semantics side, a truth function p̂ : Zx 2 → Z2 is associated to each proposition p ∈ P(x), where Zx 2 is the set of maps from x to Z2. The map P(x) → Z2[Ax] sending a proposition p to its truth function p̂ ∈ Z2[Ax] = M(Zx 2 ,Z2) is such that: 128 Quantum Boolean algebras – â is evaluation at a, i.e. âf = f(a) for a ∈ x, and f ∈ Zx 2 . – p̂ ∨ q = p̂ ∨ p̂, p̂ ∧ q = p̂ ∧ p̂, p̂ → q = p̂ → p̂, ¬̂p = ¬p̂, where the action of the connectives on truth functions comes from the corresponding operations on Z2. The map P(x) → Z2[Ax] is surjective, and there is a systematic procedure to tell when two propositions have the same associated truth function. For our purposes, it is convenient to describe P(x) using the binary connectives product . and sum +, and the constants 0, 1. In logical terms the product . is the logical conjunction, + is the exclusive or, and 0 and 1 represent falsity and truth, respectively. P(x) is defined recursively as the set of words in the symbols a ∈ x, ., +, 0, 1, (, ) such that: • x ⊆ P(x), 0 ∈ P(x), and 1 ∈ P(x). • If p, q ∈ P(x), then (pq) and (p + q) are also in P(x). We defined recursively the notion of sub-words in P(x). For all p, q, r ∈ P(x) set: • p is a sub-word of p; • p is a sub-word of (pq) and (p + q); • if p is a sub-word of q and q is a sub-word r, then p is a sub-word of r. Next we define an equivalence relation R(x), also denoted by ∼, on P(x). Given p, q ∈ P(x) we set: pR(x)q if and only if p̂ = q̂. The relation R(x) can be defined in syntactic terms as follows. Propositions p and q are related if and only if either p = q or there exists a sequence p1, . . . , pk, for some k > 1, such that p1 = p, and pk = q, and pi+1 is obtained from pi by replacing a sub-word of pi by an equivalent word according to the following relations valid for all propositions p, q, r ∈ P(x) : • Associativity and commutativity for . and +: p(qr) ∼ (pq)r, pq ∼ qp, (p + q) + r ∼ p + (q + r), p + q ∼ q + p. • Distributivity: p(q + r) ∼ pq + pr. • Additive and multiplicative units: 0 + p ∼ p and 1p ∼ p. • Additive nilpotency: p + p ∼ 0. • Multiplicative idempotency: pp ∼ p. Let Set be the category of sets, and set be the full subcategory of finite sets. Let Z( · ) 2 : set◦ → set R. Díaz 129 be the functor sending a finite set x to the free Boolean algebra generated by x, i.e. Zx 2 ≃ P(x); and sending a map f : x → y to the map Zy 2 → Zx 2 sending g ∈ Zy 2 to g ◦ f . Let Z2[A( · )] : set → set be the functor given by Z2[A( · )] = Z( · ) 2 ◦ Z( · ) 2 = ZZ( · ) 2 2 , i.e. Z2[Ax] is the algebra of regular Boolean functions on the affine space Zx 2 . Recall that an operad O in Set is given by a sequence of sets {O(n)}n∈N, together with right actions of the permutations groups O(n) × Sn → O(n) and composition maps ck : O(k) × O(n1) × · · · × O(nk) → O(n1 + · · · nk) which satisfy the equivariance, associativity, and unity axioms [16]. Any set X determines the endomorphism operad EndX defined by the sequence {EndX(n)}n∈N = {M(Xn, X)}n∈N. A permutation α ∈ Sn acts on a map f : Xn → X by fα(x1, . . . , xn) = f(xα−11, . . . , xα−1n). The composition maps arise as follows: M(Xk, X) × M(Xn1 , X) × · · · × M(Xn1 , X) ≃ M(Xk, X) × M(Xn1+···+nk , Xk) → M(Xn1+···+nk , Xk), where ≃ stands for the natural isomorphism, and the last arrow is com- position of maps. Theorem 34. The sequence {Z2[An]}n∈N defines an operad equivalent to the endomorphism operad of Z2 in set. Proof. The result follows from the identifications Z2[An] = ZZn 2 2 = M(Zn 2 ,Z2) = EndZ2(n). A S-collection is a sequence of sets X = {Xn}n∈N such that Xn comes with a right Sn-action. A sequence of sets A = {An}n∈N generates a S-collection with free Sn actions, namely the sequence A × S = {An × Sn}n∈N. 130 Quantum Boolean algebras A S-collection X generates the free operad FX = {FXn}n∈N described in [16]. For an S-collection of the form A × S, with A any sequence of sets, the free operad FA := F (A × S) generated by A × S admits the following description. FAn is the set of all pairs (t, α) such that: • t is a A-decorated planar rooted tree with n marked incoming leaves and one outgoing vertex, namely the root. A-decorated means that a choice of an element in Ak is made for each vertex in the tree with incoming valence k (other than the marked incoming leaves.) • A numbering of the marked incoming leaves, i.e. a bijection from the marked incoming leaves to [n]. • The action of Sn on FAn permutes the numberings of the leaves. • The operadic compositions is given by the grafting of trees. Let P be the free operad in Set generated by the following sequence: A0 = {0, 1}, A2 = {+, . } and Ak = ∅ for k 6= 0, 2. Proposition 35. For x ∈ set, the set of all propositions P(x) is equal to the free P-algebra generated by x. Proof. By definition the free P-algebra generated by x is given by P(x) = ∞∐ n=0 P(n) ×Sn xn, where P(n) ×Sn xn is the quotient of P(n) × xn by the relations: (fα, a1, . . . , an) ∼ (f, aα−11, . . . , aα−1n) for f ∈ P(n), α ∈ Sn, and(a1, . . . , an) ∈ xn. Since P is the free operad generated by the sequences of sets A, the free algebra can equivalently be described as the set of pairs (t, f) where: • t is a A-decorated planar rooted tree with n marked incoming leaves. Thus actually t is a binary tree with branches corresponding to + and . , the sum a product symbols. • f is a map from the n marked incoming leaves to x. It is clear that such pairs (t, f) are in bijective correspondence with proposition in P(x). R. Díaz 131 We also denote by P the functor P : set → Set sending x to the set of propositions P(x), and f : x → y to its unique extension P(f) : P(x) → P(y) sending x to y via f , and respecting the logical connectives. Let Req be the category of equivalence relations. Objects in Req are pairs (X, R) where X is a set and R is an equivalence relation on X. A morphism f : (X, R) → (Y, S) in Req is a map f : X → Y such that fR ⊆ S. We have a functor Req → Set sending (X, R) to the quotient set X/R. Proposition 36. The pair (P,R) constructed above defines a functor (P,R) : set → Req sending x to (P(x),R(x)). Thus we obtain the quotient functor P/R : set → set. Proposition 37. The functors P/R and Z2[A( · )] are naturally isomor- phic, both as set valued functors and as Boolean algebras valued functors. Proof. It follows from Lemma 1 that P/R(x) and Z2[Ax] are naturally isomorphic Boolean algebras. Proposition 38. The operads {P/R[n]}n∈N and {Z2[An]}n∈N are iso- morphic. Proof. We know from Proposition 37 that P/R[n] and Z2[An] are isomor- phic Boolean algebras. The operadic operations on P/R[n] arise from the operations of substitution and renaming of variables on propositional for- mulae. It is well-known that the formation of truth functions behaves well with respect to the operation of substitution and renaming of variables on propositional formulae. Thus {P/R[n]}n∈N and {Z2[An]}n∈N agree also as operads. Recall that Z2[Ax] is a poset with f 6 g if and only if f(a) 6 g(a) for all a ∈ Zx 2 . Classical propositional logic main concern is the pre-order ⊢ of entailment on P(x). The entailment relation ⊢ can be defined semantically as follows: for p, q ∈ P(x) set p ⊢ q if and only if p̂ 6 q̂, or equivalently, p ⊢ q if and only if there is r ∈ P(x) such that p̂ = q̂r̂. The entailment relation ⊢ can be defined syntactically as follows: p ⊢ q if and only if there exists r ∈ P(x) such that p ∼ qr. Next we move from the propositional settings to the operational settings, always within a Boolean context. Recall from the introduction 132 Quantum Boolean algebras that quantum observables are operators instead of propositions. In analogy with the classical case we identify operators with words in a certain language. On the semantic side truth functions are replaced by Boolean differential operators. We think of quantum operational logic as arising from the following principles: 1) Simple-composite operators. On the syntactic side operators are words in a certain language. For a set x we let x̃ = {ã \| a ∈ x} be a set disjoint from x whose elements are of the form ã for a ∈ x. Given x, the set O(x) of operators is obtained from the set of simple operators x ⊔ x̃ ⊆ O(x) using the binary connectives product . and sum +, and the constants 0, 1. Explicitly, O(x) is defined recursively as the set of words in the symbols a ∈ x ⊔ x̃, ., +, 0, 1, (, ) such that: • x ⊔ x̃ ⊆ O(x), 0 ∈ O(x), and 1 ∈ O(x). • If p, q ∈ O(x), then (pq) and (p + q) are also in O(x). 2) The logical interpretation of the connectives . and +. The product pq generalizes the classical connective AND, but there is also an ordering behind it: the operator pq may be interpreted as “act with the operator q, ANDTHEN act with the operator p”. The connective + corresponds to the exclusive or XOR. The constants 0 and 1 stand for the null operator and the identity operator, respectively. The logical interpretation is RESETTO0 and LEAVEASIS, respectively. 3) The algebra BDOx = EndZ2(Z2[Ax]) of Boolean differential ope- rators on Zx 2 . On the semantic side BDOx is thought as the quantum analogue for the Boolean algebra Z2[Ax] of truth functions. Just as we have a map from propositions to truth functions, we have a map (̂ · ) : O(x) → EndZ2(Z2[Ax]) from operators to Boolean differential operators given by: • â is the operator of multiplication by a, for a ∈ x. • ̂̃a is the Boolean partial derivative ∂a along a, for a ∈ x. • p̂ + q = p̂ + q̂ and p̂q = p̂q̂ for p, q ∈ O(x). • 0̂ is the operator identically equal to 0, and 1̂ is the identity operator. We think of the composition ◦ of operators in EndZ2(Z2[Ax]) as the quantum analogue of the meet ∧, or equivalently the product, on Z2[Ax] = M(Zx 2 ,Z2). Indeed ◦ is an extension of the classical meet. Consi- der the inclusion map Z2[Ax] → EndZ2(Z2[Ax]) sending f ∈ Z2[Ax] to the operator of multiplication by f . This map is additive and multiplicative, thus showing that the quantum structures are, as they should, an exten- sion of the classical ones. The map O(x) → EndZ2(Z2[Ax]) turns out to be R. Díaz 133 surjective, and there is a well-defined procedure to tell when two operators are assigned the same Boolean differential operator, which we proceed to introduce. Sub-words are defined in O(x) just as in propositional logic. We define an equivalence relation R(x), also denoted by ∼, on O(x). For p, q ∈ O(x) we set: pR(x)q if and only if p̂ = q̂. R(x) is defined syntactically as follows: p and q are related if and only if either p = q or there exists a sequence p1, . . . , pk, for some k > 1, such that p1 = p, and pk = q, and pi+1 is obtained from pi by replacing a sub-word of pi by an equivalent word according to the following relations: • Associativity for the product: p(qr) ∼ (pq)r. • Associativity and commutativity for +: (p + q) + r ∼ p + (q + r), p + q ∼ q + p. • Distributivity: p(q + r) ∼ pq + pr. • Additive and multiplicative units: 0 + p ∼ p and 1p ∼ p • Additive nilpotency: p + p ∼ 0. • Multiplicative idempotency and nilpotency: aa ∼ a and ãã ∼ 0, for a ∈ x. • Commutation relations: ba ∼ ab and ãb̃ ∼ b̃ã for a, b ∈ x, b̃a ∼ ab̃ for a 6= b ∈ x, and ãa ∼ aã + ã + 1 for a ∈ x. Let B be the category of finite sets and bijections. We recall that an S-collection as defined above is the same as a functor from B to Set, also known as a combinatorial species following the Montreal school initiated by Joyal [2,11]. Note that we may indeed regard O,R, and EndZ2(Z2[A( · )]) as functors B → Set. The pair (O,R) defines a functor (O,R) : B → Req. Proposition 39. The functors O/R and EndZ2(Z2[A( · )]) are naturally isomorphic as Boole-Weyl algebra valued functors. Proof. It follows from Theorem 11 that for any finite set x we have natural isomorphisms O/R(x) ≃ EndZ2(Z2[Ax]) respecting the structures of Boolean algebras on both sides. 134 Quantum Boolean algebras Next we define the entailment pre-order ⊢ on O(x). First we define a pre-order on EndZ2(Z2[Ax]). Given S, T ∈ EndZ2(Z2[Ax]) we set S 6 T if and only if there exists R such that S = TR. For example, if S = πA and T = πB are projections onto the subspaces A and B of Z2[Ax], respectively, then πA 6 πB if and only if A ⊆ B, since πA = πBR implies that πBπA = πBπBR = πBR = πA, and thus A ⊆ B. Entailment is defined semantically as follows, for p, q ∈ O(x) we set p ⊢ q if and only if there is r ∈ O(x) such that p̂ = q̂r̂. Equivalently, the entailment relation ⊢ can be defined syntactically as: p ⊢ q if and only if there is r ∈ O(x) such that p ∼ qr. As expected, entailment on operators is an extension of entailment on propositions. Proposition 40. The (P( · ), ⊢ ), (O( · ), ⊢ ), (Z2[A( · )], 6 ) and (EndZ2(Z2[A( · )]),6) may be regarded as functors from B to the cate- gory of pre-ordered sets. Moreover, these functors fit into the following commutative diagram of natural transformations: (P( · ), ⊢) // �� (O( · ), ⊢) �� (Z2[A( · )],6) // (EndZ2(Z2[A( · )]),6) where the top horizontal arrow is the natural inclusion map, the bottom horizontal arrow sends f to the operator of multiplication by f , and the vertical arrows are the valuation maps from propositions and operators to truth functions and Boolean differential operators, respectively. Let Z2-Vect be the category of vector spaces and linear transformation over Z2. We recall from [15] that a prop P in Z2-Vect is a strict symmetric monoidal category such that: • The objects are the natural numbers N. • The monoidal structure is given on objects by n ⊗ m = m + n. • The symmetric monoidal structure is enriched over Z2-Vect. In particular, the set of morphisms P (n, m) are Z2-vector spaces, and the product f ⊗ g on morphisms is bilinear. R. Díaz 135 Each vector space V over Z2 determines the prop EndV of endomor- phisms, given by EndV (n, m) = HomZ2(V ⊗n, V ⊗m). As we saw in Proposition 39, propositional logic give us a syntactical presentation of the endomorphism operad of Z2 in the category of sets. Next we extend this result to quantum operational logic which give us a syntactical presentation of the diagonal part of the endomorphism prop of Z2[A1] in the category Z2-vect. It is easy to check that if P is a prop, then its diagonal part P (n, n) is naturally an S-collection, with the right action P (n, n) × Sn → P (n, n) given by fα = α ◦ f ◦ α−1. Theorem 41. The functor O/R is isomorphic to the diagonal part of the prop EndZ2[A1] as Boole-Weyl algebra valued functors. Proof. We have the following chain of natural isomorphism O/R[n] ≃ EndZ2(Z2[An]) ≃ EndZ2(Z2[A1]⊗n) = HomZ2(Z2[A1]⊗n,Z2[A1]⊗n) = EndZ2[A1](n, n) respecting the structures of Boole-Weyl algebras, where the first iso- morphism comes from Theorem 11, and the second isomorphism comes from the fact that fact that Z2[An] ≃ Z2[A1]⊗n. These isomorphisms are Sn-equivariant since one can check for α ∈ Sn that α ◦ xi ◦ α−1 = xα−1i and α ◦ yi ◦ α−1 = yα−1i. 7. Set theoretical viewpoint The link between classical propositional logic and the algebra of sets arises as follows. Recall that there is a map P(x) → M(Zx 2 ,Z2) sending each proposition to its truth function. Since M(Zx 2 ,Z2) can be identified with PP(x) we obtain a map P(x) → PP(x) assigning to each proposition p a set p̂ of subsets of x. Moreover, the logical connectives intertwine nicely with the set theoretical operations on subsets, namely: p̂ + q = (p̂ ∪ q̂) \ (p̂ ∩ q̂), p̂q = p̂ ∧ q = p̂ ∩ q̂, ¬̂p = p̂, p̂ ∨ q = p̂ ∪ q̂p̂ → q = p̂ ∪ q̂. 136 Quantum Boolean algebras We stress the, often overlooked, fact that classical propositional logic des- cribes the set theoretical operations present in PP(x) that are common to all sets of the form P(y), i.e. the extra algebraic structures present in P(y) when y = P(x) play no significative role in the logic/set theory relation outlined above. Thus whereas the axioms characterizing the algebras P(x) have been massively studied, the algebraic structures characterizing Pn(x), for n > 2, have seldom attracted any attention. We proceed to consider the analogue statements in the quantum operational scenario for sets of the form [n]. It should be clear, however, that similar constructions apply for arbitrary finite sets. As in the classical case we have a map On → EndZ2(Z2[An]) sending operators (words in a certain language) to Boolean differential operators. As shown in Section 4 it is possible to identify EndZ2(Z2[An]) with the Boolean-Weyl algebra BAn, and with the shifted Boolean-Weyl algebra SBAn. Moreover, we described several explicit bases for these algebras. For example, each f ∈ BAn can be written in an unique way as: f = ∑ a,b∈P[n] f(a, b)xayb. Thus Boolean differential operators can be identified with maps from P[n] × P[n] to Z2, and we get the identifications: EndZ2(Z2[An]) ≃ BDOn ≃ BAn ≃ M(P[n] × P[n],Z2) ≃ P(P[n] × P[n]) ≃ PP([n] ⊔ [n]). We adopt the following conventions. We identify [n] ⊔ [n] with the set [n, ñ] = {1, 2, . . . , n, 1̃, 2̃, . . . , ñ}. Given a ⊆ [n] we let ã = {̃i \| i ∈ a} be the corresponding subset of [ñ] = {1̃, 2̃, . . . , ñ}. An element a ∈ P[n, ñ] will be written as a = a1 ⊔ ã2 with a1, a2 ∈ P[n]. Note that we have a natural map π : P[n, ñ] → P[n] × P[n] given by π(a) = (π1(a), π2(a)) = (a1, a2). We use indices without tilde to denote monomials of regular functions, and indices with tilde to denote Boolean derivatives or shift operators. The identification EndZ2(Z2[An]) = PP[n, ñ] allows us to give set theoretical interpretations to the algebraic structures on Boolean differential operators. Unlike the classical set theoretical structures, the quantum operational structures are not defined for arbitrary sets of the form P(y), quite to the contrary, they very much depend on the fact that y = P[n, ñ]. R. Díaz 137 Below we consider pairs (A, M) where A is a Z2-algebra and M is an A-module. A morphism (f, g) : (A1, M1) → (A2, M2) between such pairs, is given by Z2-linear maps f : A1 → A2 and g : M1 → M2 such that f is an algebra morphism, and g(am) = f(a)g(m) for all a ∈ A, m ∈ M . The addi- tive structure +: PP[n, ñ] × PP[n, ñ] → PP[n, ñ] on PP[n, ñ] is given by A + B = A ∪ B \ (A ∩ B). We consider several isomorphic products ◦, •, ⋆, and ∗ on PP[n, ñ] displaying different combinatorial properties. The products correspond with the various bases for BAn and SBAn. We first introduce the product ◦ on PP[n, ñ]. Theorem 42. There are maps ◦ : PP[n, ñ] × PP[n, ñ] → PP[n, ñ] and ◦ : PP[n, ñ] × PP[n] → PP[n], turning PP[n, ñ] into a Z2-algebra and PP[n] into a module over PP[n, ñ], such that the pair (PP[n, ñ], PP[n]) is isomorphic to ( EndZ2(M(Zn 2 ,Z2)), M(Zn 2 ,Z2) ) via the maps A → ∑ a∈A ma1∂a2 and F → ∑ a∈F ma. Proof. From Theorem 17 and Proposition 6 we see that the desired products ◦ are constructed as follows. For A, B ∈ PP[n, ñ], the product AB ∈ PP[n, ñ] is given by A ◦ B = { a ∈ P[n, ñ] ∣∣∣O{b ∈ P[n], c ∈ B \|c2 ⊆ a2, a1 ⊔ b̃ ∈ A, a2 \ c2 ⊆ a1 + c1 ⊆ b} } . Let A ∈ PP[n, ñ] and F ∈ PP[n], then AF ∈ PP[n] is given by A ◦ F = { a ∈ P[n] \| O{b ⊆ c ∈ P[n] \| a ⊔ c̃ ∈ A, a + b ∈ F} } . We provide a few applications of Theorem 42. Example 43. In PP[3, 3̃] we have {{1, 2, 2, 3}} ◦ {{1, 3, 1, 2}} = {{1, 2, 1, 2}, {1, 2, 1, 2, 3}}. Indeed a ∈ {{1, 2, 2, 3}}◦{{1, 3, 1, 2}} if there is a odd number of suitable pairs b, c. Note that c = {1, 3, 1, 2}, {1, 2} ⊆ a2, a1 ⊔ b̃ = {1, 2, 2, 3}, and thus necessarily a1 = {1, 2} and b = {2, 3}. Moreover, we must have a2 \ {1, 2} ⊆ {1, 2} + {1, 3} ⊆ {2, 3}, that is, a2 \ {1, 2} ⊆ {2, 3}. Thus either a2 = {1, 2} or a2 = {1, 2, 3} yielding the desired result. 138 Quantum Boolean algebras Example 44. For A ∈ PP[n] set A′ = π−1 2 (A). Let F ∈ PP[n], then A′ ◦ F = {a ∈ P[n] \| O{b ⊆ c ∈ P[n] \| c̃ ∈ A, a + b ∈ F}}. Note that∑ a∈P[n] ma = 1 and thus: (∑ a∈A ∂a ) ◦ (∑ b∈F mb ) = ∑ a∈A′◦F ma. Example 45. For A ∈ PP[n] let Â = {a ∈ P[n, ñ] \| a1 = a2 ∈ A}. Let F ∈ PP[n], then Â ◦ F = {a ∈ P[n] \| O{e ⊆ a \| a + e ∈ F}} and therefore (∑ a∈A ma∂a ) ◦ (∑ b∈F mb ) = ∑ a∈Â◦F ma. Next we introduce the product • on PP[n, ñ]. Theorem 46. There are maps • : PP[n, ñ] × PP[n, ñ] → PP[n, ñ] and • : PP[n, ñ] × PP[n] → PP[n] such that the pair (PP[n, ñ], PP[n]) is isomorphic to ( EndZ2(M(Zn 2 ,Z2)), M(Zn 2 ,Z2) ) via the maps A → ∑ a∈A xa1∂a2 and F → ∑ a∈F xa. Proof. From Theorem 17 and Proposition 6 the desired products • are constructed as follows. For A, B ∈ PP[n, ñ], the product A • B ∈ PP[n, ñ] is such that a ∈ A • B if and only if O{b ∈ A, c ∈ B, k1 ⊆ k2 ∣∣∣b1 ⊆ a1, c1 ⊆ a2, k2 ⊆ b2 ∩ c1, b1 ∪ (c1 \ k2) = a1, b2 \ k1 = a2 \ c2}. Let A ∈ PP[n, ñ] and F ∈ PP[n], then A • F ∈ PP[n] is given by A • F = { a ∈ P[n] \| O{b ∈ A, c ∈ F ∣∣∣b2 ⊆ c, b1 ∪ (c \ b2) = a} } . We provide a few applications of Theorem 46. Example 47. In PP[3, 3̃] we have {{1, 3, 2}} • {{2, 1}} = {{1, 2, 3, 1, 2}, {1, 3, 1, 2}, {1, 3, 1}}. Indeed, we must have b = {1, 3, 2} and c = {2, 1}, and thus there are three options for k1 ⊆ k2 ⊆ [2], namely ∅ ⊆ ∅, ∅ ⊆ {2} and {2} ⊆ {2}, giving rise to the sets {1, 2, 3, 1, 2}, {1, 3, 1, 2}, {1, 3, 1}, respectively. R. Díaz 139 Example 48. For A ∈ PP[n] set A′ = π−1({∅} × A). Let F ∈ PP[n], then A′ • F = {a ∈ P[n] ∣∣∣O{b ∈ A, c ∈ F \| b ⊆ c, c \ b = a}}. Therefore we get that (∑ a∈A ∂a ) ◦ (∑ b∈F xb ) = ∑ a∈A′•F xa. Example 49. For A ∈ PP[n] let Â = {a ∈ P[n, ñ] \| a1 = a2 ∈ A}. Let F ∈ PP[n], then Â • F = {a ∈ F \| O{b ∈ A \| b ⊆ a}} and therefore (∑ a∈A xa∂a ) ◦ (∑ b∈F xb ) = ∑ a∈Â•F xa = ∑ a∈F O{b ∈ A \| b ⊆ a}xa. In particular we have P̂[n] • P[n] = {∅}, and thus ( ∑ a∈P[n] xa∂a ) ◦ ( ∑ b∈P[n] xb ) = 1. Next we introduce the product ⋆ on PP[n, ñ]. Theorem 50. There are maps ⋆ : PP[n, ñ] × PP[n, ñ] → PP[n, ñ] and ⋆ : PP[n, ñ] × PP[n] → PP[n], turning PP[n, ñ] into a Z2-algebra and PP[n] into a module over PP[n, ñ], such that the pair (PP[n, ñ], PP[n]) is isomorphic to ( EndZ2(M(Zn 2 ,Z2)), M(Zn 2 ,Z2) ) via the maps A → ∑ a∈A ma1sa2 and F → ∑ a∈F ma. Proof. From Theorem 31 and Proposition 22 we see that the desired products ⋆ are constructed as follows. For A, B ∈ PP[n, ñ], the product A ⋆ B ∈ PP[n, ñ] is given by A ⋆ B = {a ∈ P[n, ñ] \| O{b ∈ P[n] \| a1 ⊔ b̃ ∈ A, (a1 + b) ⊔ ˜(a2 + b) ∈ B}}. Let A ∈ PP[n, ñ] and F ∈ PP[n], then A ⋆ F ∈ PP[n] is given by A ⋆ F = { a ∈ P[n] \| O{b ∈ P[n] \| a ⊔ b̃ ∈ A, a + b ∈ F} } . We provide a few applications of Theorem 50. 140 Quantum Boolean algebras Example 51. In PP[3, 3̃] we have {{1,2,3,3}}⋆{{1,2,2,3}}={{1,2,3,2}}. From the equation a1 ⊔ b̃ ∈ A we see that a1 = {1, 2, 3} and b = {3}. Also we must have a1 + {3} = {1, 2}, which holds, and a2 + {3} = {2, 3} which implies that a2 = {2}. Example 52. For A ∈ PP[n] set A′ = π−1 2 (A). Let F ∈ PP[n], then A′ ⋆ F = {a ∈ P[n] ∣∣∣O{b ∈ A \| a + b ∈ F}}. Therefore (∑ a∈A sa ) ◦ (∑ b∈F mb ) = ∑ a∈A′⋆F ma, in particular, (∑ a∈A sa ) ◦ (∑ b∈P[n] mb ) = OA ∑ a∈P[n] ma. Example 53. For A ∈ PP[n] set Â = {a ∈ P[n, ñ] \| a1 = a2 ∈ A}. Let F ∈ PP[n], then Â ⋆ F = ∅ if ∅ 6∈ F and Â ⋆ F = A if ∅ ∈ F . Therefore (∑ a∈A masa ) ◦ (∑ b∈F mb ) = c ∑ a∈A ma, where c = 1 if ∅ ∈ F , and c = 0 if ∅ 6∈ F . Example 54. Let Â be as Example 53, then Â ⋆ Â = Â if ∅ ∈ A, and Â ⋆ Â = ∅ otherwise. Indeed, a ∈ P[n, ñ] belongs to Â ⋆ Â if a1 ⊔ b̃ ∈ Â, i.e. a1 = b ∈ A, and (a1 + b) ⊔ ˜(a2 + b) ∈ Â, i.e. ∅ ∈ A and a1 = a2 ∈ A. Example 55. For A ∈ PP[n] set Ã = {a ∈ P[n, ñ] \| a1 = a2 ∈ A}. Then Ã ⋆ Ã = Ã if [n] ∈ A, and Â ⋆ Â = ∅ if [n] 6∈ A. Indeed, a ∈ P[n, ñ] belongs to Ã ⋆ Ã if a1 ⊔ b̃ ∈ Ã, i.e. a1 = b ∈ A, and (a1 + b) ⊔ ˜(a2 + b) ∈ Â, i.e. [n] ∈ A and a2 = ∅ + b = b = a1. Finally we introduce the product ∗ on PP[n, ñ]. Theorem 56. There are maps ∗ : PP[n, ñ] × PP[n, ñ] → PP[n, ñ] and ∗ : PP[n, ñ] × PP[n] → PP[n], turning PP[n, ñ] into a Z2-algebra and PP[n] into a module over PP[n, ñ], such that the pair (PP[n, ñ], PP[n]) is isomorphic to ( EndZ2(M(Zn 2 ,Z2)), M(Zn 2 ,Z2) ) via the maps A → ∑ a∈A xa1sa2 and F → ∑ a∈F xa. R. Díaz 141 Proof. From Theorem 31 and Proposition 22 we see that the desired products ⋆ are constructed as follows. For A, B ∈ PP[n, ñ], the product A ∗ B ∈ PP[n, ñ] is given by { a ∈ P[n, ñ] ∣∣O{b, c, d, e ∈ P[n] \|e ⊆ c ∩ d, b ∪ d \ e = a1, b ⊔ c̃ ∈ A, d ⊔ ˜(c + a2) ∈ B} } . Let A ∈ PP[n, ñ] and F ∈ PP[n], then A ∗ F ∈ PP[n] is given by A ∗ F = {a∈P[n] \| O{b, c, d∈P[n], e∈F \|c ⊆ d∩e, b ∪ e\c=a, b ⊔ d̃∈A}}. We provide a few applications of Theorem 56. Example 57. In PP[3, 3̃] we have {{1, 2}} ∗ {{2, 3, 1, 2}} = {{1, 3, 1}, {1, 2, 3, 1}}. Indeed we must have b = {1}, c = {2}, d = {2, 3}, and a2 = {2}+{1, 2} = {1}. Since e ⊆ {2} ∩ {2, 3} = {2}, there are two options, either e = ∅ and then a1 = {1, 2, 3} and a = {1, 2, 3, 1}, or e = {2} and then a1 = {1, 3} and a = {1, 3, 1}. Example 58. For A ∈ PP[n] set A′ = π−1({∅} × A). Let F ∈ PP[n] then A′ ∗ F = {a ∈ P[n] \| O{c ∈ P[n], d ∈ A, e ∈ F \| c ⊆ d ∩ e, e \ c = a}}. Therefore (∑ a∈A sa ) ◦ (∑ b∈F xb ) = ∑ a∈A′∗F xa. Example 59. For A ∈ PP[n] let Â = {a ∈ P[n, ñ] \| a1 = a2 ∈ A}. Let F ∈ PP[n], then Â ∗ F = { a ∈ P[n] ∣∣∣O{b ∈ A, c ∈ P[n], e ∈ F \| c ⊆ b ∩ e, b ∪ e \ c = a} } . Therefore (∑ a∈A xasa ) ◦ (∑ b∈F xb ) = ∑ a∈Â∗F xa. 8. Final remarks Our work leaves several open problems and questions for future rese- arch: 1) We considered the structural aspects of quantization in characte- ristic 2. Dynamical aspects will be considered elsewhere. 2) We studied 142 Quantum Boolean algebras the analogue for the Weyl algebra in characteristic 2. Recently, there have been several attempts to construct a suitable category that could play the role of the category of modules over the non-existing field with characte- ristic one [8, 9, 21]. It is natural to ponder whether it is possible to build analogues for the Weyl algebra in such new contexts. 3) Categorification of the Weyl algebra has been considered in [11]; categorification of the Boole-Weyl and shifted Boole-Weyl algebras remain to be addressed. 4) The symmetric powers of Weyl algebras and linear Boolean algebras in characteristic zero were studied in [10] and [12], respectively. The analogue problems for the quantum Boolean algebras and the linear quantum Bool- ean algebras are open. 5) Our logical interpretation of quantum Boolean algebras was based on a specific choice of connectives. It remains to study other connectives, perhaps with a more direct logical meaning. References [1] F. Bayen, M.Flato, C. Fronsdal, A. Lichnerowicz, D. Sternheimer, Quantum Mechanics as a Deformation of Classical Mechanics, Lett. Math. Phys. 1 (1977) 521-530. [2] F. Bergeron, G. Labelle, P. Leroux, Combinatorial Species and Tree-like Structures, Cambridge Univ. Press, Cambridge 1998. [3] G. Birkhoff, J. von Neumann, The Logic of Quantum Mechanics, Ann. Math. 37 (1936) 823-843. [4] J. Boardman, R. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math. 347, Springer-Verlag, Berlin 1973. [5] G. Boole, An Investigation of the Laws of Thought, Dover Publications, New York 1958. [6] F. Brown, Boolean Reasoning, Dover Publications, New York 2003. [7] A. Connes, Noncommutative Geometry, Academic Press, San Diego 1990. [8] A. Connes, C. Consani, M. Marcolli, Fun with F1, J. Number Theory 129 (2009) 1532-1561. [9] A. Deitmar, Schemes over F1, in G. van der Geer, B. Moonen, R. Schoof (Eds.), Number Fields and Function Fields - Two Parallel Worlds, Progress in Mathematics 239, Birkhäuser, Basel 2005, pp. 87-100. [10] R. Díaz, E. Pariguan, Quantum Symmetric Functions, Comm. Alg. 33 (2005) 1947-1978. [11] R. Díaz, E. Pariguan, Super, Quantum and Non-Commutative Species, Afr. Dia- spora J. Math. 8 (2009) 90-130. [12] R. Díaz, M. Rivas, Symmetric Boolean Algebras, Acta Math. Univ. Comenianae LXXIX (2010) 181-197. [13] V. Ginzburg, M. Kapranov, Kozul duality for operads, Duke Math. J. 76 (1994) 203-272. R. Díaz 143 [14] J. Harris, Algebraic Geometry, Springer-Verlag, Berlin 1992. [15] M. Markl, Operads and PROPs, Handbook of Algebra 5 (2008) 87-140. [16] M. Markl, S. Shnider, J. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc., Providence 2002. [17] I. Reed, A class of multiple error-correcting codes and decoding scheme, IRE Trans. Inf. Theory 4 (1954) 38-49. [18] G.-C. Rota, Gian-Carlo Rota on Combinatorics, J. Kung (Ed.), Birkhäuser, Boston and Basel, 1995. [19] I. Shafarevich, Basic Algebraic Geometry 1, Springer-Verlag, Berlin 1994. [20] M. Stone, The Theory of Representations for Boolean Algebras, Trans. Amer. Math. Soc. 40 (1936) 37-111. [21] C. Soulé, Les variétés sur le corps à un élément, Moscow Math. J. 4 (2004) 217-244. [22] J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princenton 1955. [23] I. Zhegalkin, On the Technique of Calculating Propositions in Symbolic Logic, Mat. Sb. 43 (1927) 9-28. Contact information R. Díaz Universidad Nacional de Colombia - Sede Medellín, Facultad de Ciencias, Escuela de Matemáticas, Medellín, Colombia E-Mail(s): ragadiaz@gmail.com Received by the editors: 15.10.2015 and in final form 05.12.2015.

Quantum Boolean algebras

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