SYMMETRIC BOOLEAN ALGEBRAS. Introduction

Transcription

1 Acta Math. Univ. Coenianae Vol. LXXIX, 2(2010), pp SYMMETRIC BOOLEAN ALGEBRAS R. DÍAZ and M. RIVAS Abstract. In order to study Boolean algebras in the category of vector spaces we introduce a prop whose algebras in set are Boolean algebras. A probabilistic logical interpretation for linear Boolean algebras is provided. An advantage of defining Boolean algebras in the linear category is that we are able to study its syetric powers. We give an explicit forulae for products in syetric and cyclic Boolean algebras of various diensions and forulate syetric fors of the inclusionexclusion principle. Introduction Fix k a field of characteristic zero. A fundaental fact is the existence of the functor ( ) : Set Vect, fro the category of sets to the category of k-vector spaces, that sends x into x the free k-vector space generated by x, and sends a ap f : x y to the linear transforation f : x y whose value at i x is f(i). Notice that both Set and Vect are syetric onoidal categories with coproducts and that ( ) is a onoidal functor that respects coproducts. The onoidal structure on Set is the Cartesian product and the coproduct is the disjoint union. The onoidal structure on Vect is the tensor product and the coproduct is the direct su. Notice also that the restricted functor ( ) : set vect fro finite sets to finite diensional vector spaces is such that the diension di(x) of x is equal to the cardinality x of x. Using ( ) one can transfor (cobinatorial) set theoretical notions into (finite diensional) linear algebra notions. For exaple, if x is a onoid, then x carries the structure of an associative algebra. Siilarly, if x is a group, then x carries a structure of a Hopf algebra. Thus associative algebras and Hopf algebras are the linear analogues of onoids and groups, respectively. Our ain goal in this work is to uncover the linear analogue for Boolean algebras, i.e. we propose an answer to the question: what is the natural algebraic structure on B if B is a Boolean algebra? Boolean algebras [5, 17, 20] has been known at least since 1854 and constitute a cornerstone of odern atheatics. Received March 3, 2009; revised October 8, Matheatics Subject Classification. Priary 06E99, 06A06, 03B48. Key words and phrases. Boolean algebras; cobinatorics; probabilistic logic. Dedicated a J. R. Castillo Ariza.

2 182 R. DÍAZ and M. RIVAS For ost atheaticians the word algebra iplies a linear structure, a property that is not present in the traditional definition of Boolean algebras. In this work the the presence or absence of a linear structure is the ost iportant issue, thus we call our objects of study linear Boolean algebras to distinguish the fro proper Boolean algebras. Thus by definition if B is a Boolean algebra, then B is a linear Boolean algebra. Our second goal in this work is to study the syetric powers of linear Boolean algebras. We copute the structural constants of such algebras in various diensions, and show that each syetric function can be used for forulating generalization of the inclusion-exclusion principle for the syetric powers of linear Boolean algebra. Our third goal is to propose a logical interpretation for linear Boolean algebras. This work is organized as follows. In Section 1 we introduce the axios for linear Boolean algebras and show that the linear span of a Boolean algebra is a linear Boolean algebra. The ain difficulty lies in choosing the structural operations present in linear Boolean algebras. In Section 2 we otivate our choice of axios for linear Boolean algebras. What we do is to construct a prop Boole such that Boole-algebras in Vect are linear Boolean algebras. The prop Boole is the linear span of the prop Boole in Set, and one can show that Boole-algebras in Set are precisely Boolean algebras. Once we have a solid definition of linear Boolean algebras we proceed to study soe of the properties of this kind of atheatical entities. In Section 3 we discuss the logical interpretation of linear Boolean algebras. We show that they are naturally related to probabilistic logic. The advantage of working in the linear category is that we can ake use of any powerful techniques available in linear algebra. In Section 4 and 5 we apply Polya functors as defined in [10] to linear Boolean algebras, in particular, we study syetric and cyclic powers of linear Boolean algebras. In Section 6 we close providing an extension of the inclusion-exclusion principle that applies to the syetric products of Boolean algebras. 1. Linear Boolean algebras We recall the definition of Boolean algebras for definiteness and for the reader convenience, so that he or she ay contrast it with the definition of linear Boolean algebras given below. We have chosen axios that ake transparent that Boolean algebras are a particular kind of lattices. Thus a linear analogue for lattices can be readily obtained fro the definition of linear Boolean algebras given below. The reader should notice that while the definition of Boolean algebras involve five structural aps, the definition of linear Boolean algebras involve seven structural aps, including quite unexpectedly, a coproduct. A Boolean algebra is a set B together with the following data: 1. Maps : B B B, : B B B, and c : B B called a union, an intersection and a copleent, respectively. 2. Distinguished eleents e, t B called the epty and total eleents, respectively. This data should satisfy the following identities for a, b, c B:

3 SYMMETRIC BOOLEAN ALGEBRAS 183 a b = b a, a b = b a. a (b c) = (a b) c, a (b c) = (a b) c. a (b c) = (a b) (a c), a (b c) = (a b) (a c). a (a b) = a, a (a b) = a. a e = a, a t = a, a a c = t, a a c = e. To any set x one can associate the Boolean algebra P (x) = {a a x} where the total eleent is x and the epty eleent is, a b = {i x i a or i b}, a c = {i x i / a}, and a b = {i x i a and i b}. Let [n] = {1,..., n} and S n be the group of perutations of [n]. We will always write P [n] instead of P ([n]). Algebras of the for P (x) are essentially the unique odels of finite Boolean algebras according to the following well-known result. Proposition 1. Every finite Boolean algebra is isoorphic to P (x) for a finite set x. Indeed let B be a Boolean algebra. Define a partial order on B by letting a b if a b = a. Let x be the set of priitive eleents or atos of B, that is, we have x = {a A a e and if b a then b = e or b = a}. The ap f : B P (x) given by f(b) = {a x a b} defines the desired isoorphis. Another interesting property of Boolean algebras is the following: if B and C are Boolean algebras, then B C is also a Boolean algebra. Moreover one can show that P (x) is isoorphic to P [1] x. For a k-vector space V we shall use the syetry ap S : V V V V given by S(x y) = y x for x, y V. We denote the identity ap by I : V V. We are ready to define the linear analogue of the notion of Boolean algebras. Definition 2. A linear Boolean algebra is a k-vector space V together with the data: 1. Linear aps : V V V, : V V V, and c : V V called a union, an intersection and a copleent, respectively. 2. Linear aps T : k V, E : k V called the epty ap and the total ap, respectively. 3. Linear ap : V V V called a coproduct. 4. Linear ap ev : V k called the evaluation ap. The axios below ust hold: = S, = S. ( I) = (I ), ( I) = (I ). (I ) = ( ) (I S I) ( I I), (I ) = ( ) (I S I) ( I I). (I ) ( I) = I ev, (I ) ( I) = I ev. (I E) = I, (I T ) = I, (I c) = E ev, (I c) = T ev. ( I) = (I ).

4 184 R. DÍAZ and M. RIVAS S =. Our next result guarantees the existence of infinitely any odels of linear Boolean algebras, naely those naturally associated with Boolean algebras. Proposition 3. If B is a Boolean algebra, then B is a linear Boolean algebra. Proof. The structural aps on B are given as follows. The intersection, the union and the copleent are the linear extensions of the corresponding aps on B. The coproduct is given by: ( ) v a a = v a a a. a B a B The epty and total aps are given for s k by E(s) = se and T (s) = st. Finally, the evaluation ap is given by ev(σ a B v a a) = Σ a B v a. Next result characterizes finite diensional linear Boolean algebras of the for P (x). Proposition 4. If V and W are linear Boolean algebras, then V W is a linear Boolean algebra with the Boolean operations defined coponentwise. Moreover, if x is a finite set then there is a canonical isoorphis of linear Boolean algebras P (x) P [1] x. 2. Boolean prop In this section we provide an explanation for our choice of axios for linear Boolean algebras. We do so by defining a prop Boole over Vect whose algebras are linear Boolean algebras and showing that this prop actually coes fro a prop Boole over Set whose algebras are Boolean algebras. Discovering the prop that defines a given faily of algebras is like unveiling its genetic code [1, 13, 14, 15, 19]. Despite the fact that Boolean algebras have been extensively studied fro a yriad of viewpoints its genetic code has not been study so far. Since the theory of props is not widely known we provide a brief overview. We define props over a syetric onoidal category C, but the reader should bear in ind that in this work C is either Set or Vect. We assue that C is closed and that it adits finite coliits. Definition A prop over C is a syetric onoidal category P enriched over C such that Ob(P ) = N and the onoidal structure is the addition of natural nubers. 2. Let PROP C be the category whose objects are props over C. Morphiss in PROP C are onoidal functors. Explicitly we have that a prop P is given by the following data: Morphiss P (n, ) C P (, k) P (n, k) for n,, k N. Morphiss P (n, ) C P (k, l) P (n + k, + l) for n,, k, l N.

5 SYMMETRIC BOOLEAN ALGEBRAS 185 For n N a group orphis S n P (n, n) such that the following diagra S n S S n+ P (n, n) C P (, ) P (n +, n + ) is coutative. Notice that the ap S n P (n, n) induces a right action of S n on P (n, ) and a left action of S on P (n, ). Let B be the category whose objects are finite sets and whose orphiss are bijections. The actions constructed above can be used to define a functor P : B op B C given by P (a, b) = B(a, [a]) S b P ( a, b ) S b B([b], b), where for a finite set x we define [x] = {1,..., x }. In order to define the free prop generated by a functor G : B op B C we need soe cobinatorial notions. A digraph Γ consists of the following data A pair of finite sets (V, E) called the set of vertices and edges of Γ, respectively. A ap (s, t) : E V V. We call s(e) and t(e) the source and target of e V, respectively. We use the notations in(v) = {e t(e) = v}, i(v) = in(v), out(v) = {e s(e) = v}, and o(v) = out(v). The valence of v V is val(v) = (i(v), o(v)) N 2. Also we introduce the notation V in = {v V i(v) = 0} and V out = {v V o(v) = 0}. An oriented cycle in Γ is a sequence e 1,..., e n of edges in Γ such that t(e i ) = s(e i+1 ) for 1 i n 1 and t(e n ) = s(e 1 ). Digraphs considered in this work do not have oriented cycles. Let a and b be finite sets. An (a, b)-digraph is a triple (Γ, α, β) such that Γ is a digraph; α : a V in is an injective ap; β : b V out is an injective ap. Let DG(a, b) be the groupoid of (a, b)-digraphs. A functor G : B op B C induces a functor G : DG(a, b) C given by G(Γ) = G(in(v), out(v)), v V int where Γ is an object of DG(a, b) and V int = V \ (α(a) β(b)). Definition 6. The prop P G freely generated by G : B op B C is given for n, N by P G (n, ) := li G(Γ) where the coliit is taken over the groupoid DG([n], []), where [0] = and [n] = {1,..., n} for n 1. Copositions in P G are given by gluing of digraphs. To define props via generators and relations we need to know what is the analogue of an ideal in the prop context.

6 186 R. DÍAZ and M. RIVAS Definition 7. Let P be a prop over C. A subcategory I of P is a prop ideal if Ob(I) = Ob(P ) and for n,, k, l N I(n, ) P (, k) I(n, k), P (n, ) I(, k) I(n, k), I(n, ) P (k, l) I(n k, l), P (n, ) I(k, l) I(n k, l). We are ready to define a prop Boole over Set. Boole is a quotient by a prop ideal I B, defined below, of the prop freely generated by vertices representing, respectively, a union, an intersection, a copleent, a coproduct, the epty eleent, the total eleent and the valuation, respectively. V E T The prop ideal I B is generated by the seven relations given below. Each relation corresponds with an axio in the definition of linear Boolean algebras. 1. Coutativity for union and intersection = = 2. Associativity for union and intersection = = 3. Distributivity laws = = 4. Properties of the epty and total eleents C = T V C = E V

7 5. Absorption Laws SYMMETRIC BOOLEAN ALGEBRAS 187 = = 6. Coassociativity and cocoutativity = = Given an object x of a category C we let End C x be the prop given for n, N by End C x (n, ) = C(x n, x ). Let P be a prop over C. A P -algebra in C is a pair (x, r) where r : P End C x is a prop orphis and x is an object of C. In practice a P -algebra x is given by a faily of orphiss in C r : P (n, ) C(x n, x ) satisfying the natural copatibility conditions. Theore 8. B is a Boole-algebra in Set if and only if B is a Boolean algebra. Proof. Assue that (B, r) is a Boole-algebra in Set where r : Boole End Set B is a prop orphis. The iages under r of the generators of Boole give operations,, ( ) c, t, e,, ev, respectively. For exaple t : {1} B and e : {1} B are identified with eleents of B. ev : B {1} is the constant ap and plays no essential part in this story. We also get a ap : B B B which does see to fit into the definition of Boolean algebra. Assue that is given by (a) = (f(a), g(a)) for a B. We use the relations in Boole. The cocoutativity graph iplies that f = g. The coassociativity graph iplies that f 2 = f. One of the absorption graphs iplies the identity f(a) (f(a) b) = a for a, b B. Thus we obtain f(a) = f 2 (a) (f 2 (a) b) = f(a) (f(a) b) = a. Thus (a) = (a, a) and it is a siple check that all other relations in Boole turn B into a Boolean algebra. Assue that B is a Boolean algebra with operations,, ( ) c, and distinguished eleents t and e that ay be thought as aps fro {1} to B. Take ev to be the constant ap fro B to {1} and let be given by (a) = (a, a). Let r be the ap assigning to each generator of the Boole prop the corresponding ap fro the list above. The fact that B is a Boolean algebra guarantees that all the relations defining Boole are satisfied and r is extending to a prop orphis r : Boole End Set B.

8 188 R. DÍAZ and M. RIVAS Notice that the functor ( ) : Set Vect induces a functor ( ) : PROP Set PROP Vect sending P into P given by P (n, ) = P (n, ) and with the induced copositions. The following result follows fro the fact that each generator of the prop Boole corresponds with an operation on linear Boolean algebras and each relation in the set of generator of the prop ideal I B corresponds with an axio in the definition of linear Boolean algebras. Theore 9. V is a Boole-algebra in Vect if and only if V is a linear Boolean algebra. 3. Probabilistic logic and linear Boolean algebras It is hard to do any work on Boolean algebras and not to ention its relation with classical propositional logic at all. Indeed the otivation of Boole hiself was to describe the algebraic structures underlying the laws of thought. Propositional logic deals with the deduction relation aong sets of sentences of propositions S constructed recursively fro a finite set of propositions connected by a fixed set of connecting sybols. There are any ways [18] to describe a syste of propositional logic but in any of the one can iagine that there exists a sort of logical agent capable of perforing the following tasks: Recognize when a graatical construction is an eleent of S. The agent is able to translate in S expressions of the for s t, s t into sentences, and s for sentences s and t in S. Decide wether or not a sequence of sets of sentences c 1,..., c n is a deduction. A sentence s is said to iply a sentence t if there exists a deduction c 1,..., c n such that c 1 = {s} and c n = {t}. Assign a truth-value to sentences in S when provided with an assignent of truth values for propositions in P, i.e. construct an eleent of {0, 1} S given an eleent in {0, 1} P. The logical agent is said to be sound and coplete if in addition the following property holds: A sentence s iplies a sentence t if for any assignents of truth values to propositions in P the truth value of t is 1 if the truth value of s is 1. It is not hard to show the existence of sound and coplete logical agents, for exaple, see [18]. Boolean algebras appear within the context of propositional logic as follows. We call sentences s and t in S equivalent if s iplies t and t iplies s. Let B(S) be the quotient of S by that equivalence relation. B(S) coes equipped with a natural structure of Boolean algebra with operations defined by [s] [t] = [s t], [s] [t] = [s t], and [s] c = [ s]. The total eleent is [s s] and the epty eleent is [s s]. The Boolean algebra B(S) is isoorphic to the Boolean algebra P ({0, 1} C ) via the ap : B(S) P ({0, 1} P )

9 SYMMETRIC BOOLEAN ALGEBRAS 189 that sends [s] B(S) into the set of its odels: ([s]) = {v {0, 1} P the truth value of s according to v is 1}. Suarizing sentences in S describes subsets of {0, 1} P and two sentences describe the sae subset if and only if they are equivalent. The expressive power of a logical description of P ({0, 1} P ) lies in the possibility of describing the sae set in a variety of different ways. For exaple, the logical agent ay be said that a subset of {0, 1} P is described by a sentence s, another subset of {0, 1} P is described by a sentence t and be asked to provide a sentence which describes the union of those sets. It will readily answer that s t is the sought sentence. It is natural to wonder if any logical eaning can be ascribed to the linear Boolean algebra B(S). We venture a possible answer: assue the logical agent is said that a sentence s i describes an unknown subset of {0, 1} C with probability p i for 1 i n and a sentence t j describes another unknown subset of {0, 1} C with probability q j for 1 j. If asked to find a sentence that describes the union of those subsets the logical agent will answer: the sentence s i t j describes the union of the unknown sets with probability p i q j. This is the only consistent answer with the product rules on B(S) which is given by ( n ) n, p i [s i ] q j [t j ] = p i q j [s i t j ]. j=1,j=1 This probabilistic interpretation applies as well to the linear Boolean algebra B. Let v and w be a couple of vectors in B given by v = a B v aa and w = b B v bb. Assue that the coefficients of v and w, respectively, are positive and add to one. This allows us to think that v a represents the probability that the unknown subset v of x is equal to a. Siilarly w b represents the probability that w is equal to b. Under this conditions we have that The probability that v w is equal to c is given by (v w) c = v a w b. a b=c The probability that v w is equal to c is given by (v w) c = v a w b. The probability that v c is equal to a is v a c. a b=c We invite the reader to take a look at the structural coefficients of the algebras Sy 2 P [1] and P [1] 3 /Z 3 given in Section 4 and Section 5 below, and check that they are indeed consistent with the probabilistic interpretation just outlined. 4. Syetric powers of Boolean algebras The following ideas introduced in [10] and further applied in [7, 8, 11] are useful for studying the syetric powers of algebras. After a brief review of the general theory we shall apply it to study the syetric powers of linear Boolean algebras. Suppose that a group G acts by autoorphiss on the k-algebra A. The space of co-invariants A/G = A/{ga a g G and a A}

10 190 R. DÍAZ and M. RIVAS is a k-algebra with the product given by ab = 1 a(gb). G g G For each subgroup K S n the Polya functor P K : k-alg k-alg fro the category of associative k-algebras into itself is defined as follows. If A is a k-algebra then P K A denotes the k-algebra whose underlying vector space is P K A = A n / a 1 a n a σ 1 (1) a σ 1 (n) : a i A, σ K. The rule for the product of eleents in P K A is provided by our next result. Theore 10. For any {a ij },n,j=1 A the following identity holds in P KA: ( n n ) K 1 =. j=1 a ij σ {id} K 1 j=1 a iσ 1 i (j) In particular for each algebra A and each positive integer the Polya functor P Sn yields an algebra P Sn A which we denote by S n A. Recall that P [k] denotes the k-vector space generated by the subsets of [k]. The structural aps,, and ( ) c for P [k] are the linear extensions of the union, intersection, and copleent on P [k]. Definition 11. We call S P [k] the syetric Boolean algebra of type (, k). The structural operations on S P [k] are induced fro the corresponding operations on P [k]. The group S x acts by autoorphiss on P (x) for any finite set x. The next result gives a characterization of the algebra of co-invariants P (x)/s x. Proposition 12. We have that P (x)/s x S x P [1] and therefore di(p (x)/s x ) = x + 1. A basis for P [k]/s k is given by 0,..., k where i denotes the equivalence class of [i] [k]. Let us study the operation of union, intersection, and copleents on the space P [k]/s k in details. Below we use the notation P (x, k) = {c P (x) c = k} for any set x. Theore 13. For 0 a, b, k, the following identities hold in P [k]/s k. Let = in(k a, b), then a b = ( 1 ( )( ) a k a ) (a + l). k b l l l=0 b Let = in(a, b), then a b = ( 1 ) k b l=0 ( ) a l. l

11 Also we have that a c = k a. SYMMETRIC BOOLEAN ALGEBRAS 191 Proof. For the first identity we have that a b = 1 [a] σ[b] = ( 1 ) k! k σ S k b = ( 1 ) [a] c = ( 1 ) k k c 0 P ([k]\[a],l) b c 1 P ([a],b l) b = ( 1 ) k b l=0 ( a b l )( k a l c P ([k],b) ) (a + l). [a] c c 0 P ([k]\[a],l) c 1 P ([a],b l) [a] c 0 The second identity follows fro the fact that the nuber of perutations σ S k such [a] σ[b] = l is given by ( a l )( b l ) l! ( k a b l ) (b l)!(k b)! The third identity is obvious. Let π = {b 1,..., b k } be a partition of x and S π S x the Young subgroup consisting of block preserving perutations of x. Our next result characterizes algebras of the for P (x)/s π. Proposition 14. There is an isoorphis P (x)/s π k S bi P [1], thus we have that di(p (x)/s π ) = k ( b i + 1). 5. Cyclic Boolean Algebras In this section we consider another application of Polya functors in the context of linear Boolean algebra, naely, we consider the cyclic powers of the linear Boolean algebra P [1], i.e. the algebras P [1] /Z. For = 2 one gets the space P [1] 2 /Z 2 which has a basis given by (0, 0), (1, 0) and (1, 1).

12 192 R. DÍAZ and M. RIVAS The union ap : S 2 P [1] S 2 P [1] S 2 P [1] is given by (0, 0) (1, 0) (1, 1) (0, 0) 1,0,0 0,1,0 0,0,1 (1, 0) 0,1,0 0, 1 2, 1 2 0,0,1 (1, 1) 0,0,1 0,0,1 0,0,1 The intersection : S 2 P [1] S 2 P [1] S 2 P [1] is given by (0, 0) (1, 0) (1, 1) (0, 0) 1,0,0 1,0,0 1,0,0 (1, 0) 1,0,0 1 2, 1 2, 0 0,1,0 (1, 1) 1,0,0 0,1,0 0,0,1 The copleent ( ) c : S 2 P [1] S 2 P [1] is given by (0, 0) (1, 0) (1, 1) 0,0,1 0,1,0 1,0,0 Although the algebra S 2 P [1] does not satisfy all the axios required to ake it into a linear Boolean algebra (the absorption laws fail!) it does share any properties of linear Boolean algebras, and in any case it is a atheatical object of great interest. For = 3 the space P [1] 3 /Z 3 has the basis (0, 0, 0), (1, 0, 0), (1, 1, 0) and (1, 1, 1). The union ap : P [3] 3 /Z 3 P [3] 3 /Z 3 P [3] 3 /Z 3 is given by (0, 0, 0) (1, 0, 0) (1, 1, 0) (1, 1, 1) (0, 0, 0) 1,0,0,0 0,1,0,0 0,0,1,0, 0,0,0,1 (1, 0, 0) 0,1,0,0 0, 1 3, 2 3, 0 0, 0, 2 3, 1 3 0,0,0,1 (1, 1, 0) 0,0,1,0 0, 0, 2 3, 1 3 0, 0, 1 3, 2 3 0,0,0,1 (1, 1, 1) 0,0,0,1 0,0,0,1 0,0,0,1 0,0,0,1

13 SYMMETRIC BOOLEAN ALGEBRAS 193 The intersection ap : P [3] 3 /Z 3 P [3] 3 /Z 3 P [3] 3 /Z 3 is given by (0, 0, 0) (1, 0, 0) (1, 1, 0) (1, 1, 1) (0, 0, 0) 1,0,0,0 1,0,0,0 1,0,0,0 1,0,0,0 2 (1, 0, 0) 1,0,0,0 3, 1 3, 0, 0 1 3, 2 3, 0, 0 0,1,0,0 1 (1, 1, 0) 1,0,0,0 3, 2 3, 0, 0 0, 2 3, 1 3, 0 0,0,1,0 (1, 1, 1) 1,0,0,0 0,1,0,0 0,0,1,0 0,0,0,1 The copleent ap ( ) c : P [3] 3 /Z 3 P [3] 3 /Z 3 is given by ( ) c (0, 0, 0) (1, 0, 0) (1, 1, 0) (1, 1, 1) 0,0,0,1 0,0,1,0 0,1,0,0 1,0,0,0 6. Syetric inclusion-exclusion principles Perhaps the ost fundaental eleentary result concerning Boolean algebras is the inclusion-exclusion principle. In this section we consider the extensions of this principle for linear Boolean algebras. The reader will find interesting inforation on the inclusion-exclusion principle and its generalizations in several works by Rota and his collaborators [16]. We use the inclusion-exclusion principle in the following for: Proposition 15. Let a 1,..., a n P (x), then n a i = ( 1) I +1 a i. I [n] In this section we consider vector spaces over the coplex nubers and we write {a 1,..., a } for the basis eleent i I a 1 a S P [k] = P [k] /S. The following result follows fro Theore 10. Theore 16. Let {a i 1,..., a i } be in the basis of S P [k] for 1 i n. The union ap on S P [k] is given by n {a i 1,..., a i 1 { n n } = a i σ i(1),..., a i σ i() }. For exaple for, n = 2, one gets σ {1} S (n 1) {a, b} {c, d} = 1 2 {a c, b d} + 1 {a d, b c}. 2 Recall that a easure on a finite set x is a ap µ : P (x) C such that µ(a b) = µ(a) + µ(b)

14 194 R. DÍAZ and M. RIVAS for a, b x disjoint. Let us fix a easure µ on [k]. An eleent {a 1,..., a } in the basis of S P [k] deterines a vector (µ(a 1 ),..., µ(a )) C /S. Functions on C /S are known as syetric functions. There are any interesting exaples of polynoial syetric functions such as the power functions, the eleentary syetric functions, the hoogeneous functions, the Schur functions and so on. For exaple, the polynoial x l x l is S -invariant. Each syetric function can be used to obtain a syetric for of the inclusion-exclusion principle. We consider explicitly the syetric inclusion-exclusion principles derived fro the power, eleentary, and hoogeneous syetric functions; other syetric functions ay be considered as well but we shall not do so here. Notice that Gessel [12] uses the nae syetric inclusion-exclusion to refer to a different atheatical gadget. The power function p l : S P [k] C is given on the basis eleents by: p l ({a 1,..., a }) = µ(a i ) l. We use the power functions p l to get a syetric for of the inclusion-exclusion principle. Theore 17. Let {a i 1,..., a i } be in the basis of S P [k] for 1 i n. Then n p l ( {a i 1,..., a i 1 }) = Proof. σ {1} S (n 1) j {1,...,} Σc I =l n p l ( {a i 1,..., a i 1 }) = σ {1} S (n 1) j {1,...,} 1 = σ {1} S (n 1) j {1,...,} 1 = σ {1} S (n 1) j {1,...,} 1 = σ {1} S (n 1) j {1,...,} Σc I =l ( ) l {c I } I [n]( 1) ( I +1)cI µ( a i σ i(j) )c I. i I n µ( a i σ i(j) ) l ( I [n]( 1) I +1 µ( i I ) l a i σ i(j) ) ( ( ) l ) {c I } Σc I =l I [n][( 1) I +1 µ( a i σ i(j) )] c I i I ( ) l {c I } I [n]( 1) ( I +1)cI µ( a i σ i(j) )c I. i I

15 For exaple, for l = 1, one gets n p 1 ( {a i 1,..., a i 1 }) = For l = 1, n = 2, we get p 1 ({a 1 1,..., a 1 } {a 1 1,..., a 2 }) = 1! SYMMETRIC BOOLEAN ALGEBRAS 195 σ {1} S (n 1) j {1,...,} I [n] ( 1) I +1 µ( i I a i σ i(j) ). {µ(a 1 j) + µ(a 2 σ(j) ) µ(a1 j a 2 σ( j) )}. σ S j [] Next we consider a generalized inclusion-exclusion principle using the eleentary syetric functions e l (x 1,..., x ) = 1 t 1<t 2< <t l j=1 l x tj. Theore 18. Let {a i 1,..., a i } be in the basis of S P [k] for 1 i n. Then ( n e l {a i 1,..., a i } ) 1 = σ {1} S n 1 1 t 1<t 2< <t l f:[l] P ([n]) l ( 1) f(j) +1 µ ( j=1 i f(j) a i σ i(t j)). For n = 2, = 2 and l = 2, the ap e 2 : S 2 P [k] R is given by e 2 ({a, b}) = µ(a)µ(b) and Theore 18 iplies that 2e 2 ({a, b} {c, d}) = 2µ(a)µ(b) + 2µ(c)µ(d) + µ(a)µ(d) + µ(c)µ(b) + µ(a)µ(c) + µ(d)µ(b) µ(a)µ(b d) + µ(c)µ(b d) + µ(b)µ(a c) + µ(d)µ(a c) + µ(a)µ(b c) + µ(d)µ(b c) + µ(b)µ(a d) + µ(c)µ(a d). Next we describe the generalization of the inclusion-exclusion principle using the hoogenous syetric functions h l (x 1,..., x ) = 1 t 1 t 2 <t l j=1 l x tj. Theore 19. Let {a i 1,..., a i } be in the basis of S P [k] for 1 i n. Then ( n h l {a i 1,..., a i } ) 1 = σ {1} S n 1 1 t 1 t 2 t l f:[l] P ([n]) l ( 1) f(j) +1 µ ( a i σ i(t j)). j=1 i f(j)

16 196 R. DÍAZ and M. RIVAS For n = 2, = 2 and l = 2, the ap h 2 : S 2 P [k] R is given by Theore 19 iplies that h 2 ({a, b}) = µ(a) 2 + µ(a)µ(b) + µ(b) 2. 2h 2 ({a, b} {c, d}) = [µ(a) + µ(c) µ(a c)] 2 + [µ(b) + µ(d) µ(b d)] 2 + [µ(a) + µ(d) µ(a d)] 2 + [µ(b) + µ(c) µ(b c)] 2 + 2µ(a)µ(b)+2µ(c)µ(a)+µ(a)µ(d)+µ(c)µ(b)+µ(a)µ(c) + µ(d)µ(a) µ(a)µ(b d) + µ(c)µ(b d) + µ(b)µ(a c) + µ(d)µ(a c)+µ(a)µ(b c)+µ(d)µ(b c)+µ(b)µ(a d) + µ(c)µ(a d). Notice that the structural constants of the syetric and cyclic powers of Boolean algebras are rational nubers. It would be interesting to study the cobinatorics of those nubers along the lines of [2, 3, 4, 9]. Acknowledgeent. This work is dedicated to the eory of Professor J. R. Castillo Ariza who always had a deep interest in Boolean Algebras and wrote an introductory book on the subject [6]. Our thanks to Mauricio Angel, Edundo Castillo and Eddy Pariguan. Part of this work was done while the first author was visiting ICTP, Italy, and the Universidad of Sonora, México. References 1. Adas J., Infinite Loop spaces, Princeton Univ. Press, Princenton Blandín H. and Díaz R., Copositional Bernoulli nubers, Afr. Diaspora J. Math. 7 (2008), , On the cobinatorics of hypergeoetric functions, Adv. Stud. Contep. Math. 14 (2007), , Rational cobinatorics, Adv. Appl. Math. 40 (2008), Boole G., An Investigation of the Laws of Thought, Dover Publications, New York Castillo J., Copendio Eleental de Lógica Mateática, Editorial Nueva Expresión, Caracas Castillo E. and Díaz R., Hoological atrices, in S. Paycha, B. Uribe (Eds.), Geoetric and Topological Methods for Quantu Field Theory, Contep. Math. 432, Aer. Math. Soc., Providence 2007, pp Díaz R. and Pariguan E., On the q-eroorphic Weyl algebra, Sao Paulo J. Math. Sci. 3 (2009), R. Díaz, E. Pariguan, Quantu syetric functions, Co. Alg. 33 (2005), , Super, Quantu and Non-Coutative Species, Afr. Diaspora J. Math. 8 (2009), , Syetric quantu Weyl algebra, Ann. Math. Blaise Pascal 11 (2004), Gessel I., Syetric inclusion-exclusion, Se. Lothar. Cobin. 54 (2005/06), Art. B Mac Lane S., Categorical algebra, Bull. Aer. Math. Soc. 71 (1965), Markl M., Merkulov S. and Shadrin S., Wheeled PROPS, graph coplexes and the aster equation, arxiv:ath.qa/ Merkulov S., PROP profile of Poisson geoetry, Co. Math. Phys. 262 (2006), Rota G.-C., Gian-Carlo Rota on Cobinatorics, Joseph Kung (Ed.), Birkhäusser, Boston and Basel Sikorski R., Boolean Algebras, Springer-Verlag, Berlin 1964.

17 SYMMETRIC BOOLEAN ALGEBRAS Sullyan R., First-order logic, Dover Pub., New York Voronov A., Notes on universal algebra, in: Graphs and Patterns, Matheatics and Theoretical Physics, Aer. Math. Soc., Proc. Syp. Pure Math 73, Riverside, 2005, pp Whitesitt J., Boolean Algebra and its applications, Addison-Wesley Publishing, Massachusetts R. Díaz, Instituto de Mateáticas y sus Aplicaciones, Universidad Sergio Arboleda, Bogotá, Colobia, e-ail: ragadiaz@gail.co M. Rivas, Departaento de Mateáticas, Universidad Sión Bolívar, Caracas, Venezuela, e-ail: ariolysrivas07@gail.co