BENIAMINOV ALGEBRAS REVISED: ONE MORE ALGEBRAIC VERSION OF FIRST-ORDER LOGIC

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1 Jānis Cīrulis BENIAMINOV ALGEBRAS REVISED: ONE MORE ALGEBRAIC VERSION OF FIRST-ORDER LOGIC 1. Introduction E.M. Benjaminov introduced in [1] a class of algebras called by him relational algebras. The term comes from database theory, and it generally refers to algebras of a certain kind suggested by E.F. Codd. Beniaminov described another version of the relational data model and also proposed a set of axioms to characterize abstractly his class of concrete relational algebras. However, he did not present any results concerning strength of the axiom system, neither did he compare his algebras with other known algebras of finitary relations. By the way, his axiom set proves to be incomplete in a strong sense. We show here that a slightly restricted class of Beniaminov s relational algebras is interdefinable with that of locally finite cylindric algebras (if the dimensions of the algebras in both classes are concordant) and provides therefore an adequate algebraic version of first-order logic. It is worthwhile to note here that relational algebras are interesting by themselves, as they may be treated as heterogeneoys aanalogues of Craig s trans-boolean algebras [4]. For the history of the problem and earlier results, see Introduction in [2]. The present paper is in fact an extended abstract of the related sections of [2]. 2. Relational algebras Assume we are given a set of sorts Σ. A sorted set is a set each element of which is correlated with some sort. Where X and Y are two sorted sets, an agreement of X with Y is a sort preserving mapping ϕ: X Y (see [1]). 0 Supported by Latvian Council Grant No 93/

2 Beniaminov begins with the category of all finite sorted sets and agreements which we denote by Σ. The category is, however, too big: as the collection of all finite sets is a proper class, it is difficult to compare relational algebras in the original Beniaminov s sense with algebras of relations traditionally arising in algebraic logic. With this in mind, we shall restrict the category Σ to its small subcategory whose objects are subsets of some fixed set. Thus, let V be any sorted set which is assumed to be fixed throughout the rest. We assume that V contains infinitely many elements (sometimes called variables) of each sort. A relation type is a finite subset of V. Let RT stands for the set of all such types. We shall base the concept of a Beniaminov algebra on the category Σ V of all relational types and agreements over V rather than on Σ. Given two relation types X and Y, we denote by Agr(X, Y ) the set of agreements of X with Y. Agr means the set of all agreements from Σ V. Now let us assume that we are given a heterogeneous algebra A := (A X, s α, t α ) X,Y RT, α Agr, where (a) every A X is a Boolean algebra, (b) for α Agr(X, Y ), s α and t α are operations of types A X A Y and A Y A X, respectively, and (c) the following conditions are satisfied: r1: s α ( a) = s α a, r2: s α a s α (a a ), r3: t α b t α (b b ), r4: b s α t α b, r5: t α s α a a, r6: s ε a = a if ε is an identity agreement (of a relation type with itself), r7: s β s α a = s βα a if β and α are composable. These conditions are contained in the first four of Beniaminov s original axioms in [1] and are, in fact, equivalent to them. (In particular, every s α is a Boolean homomorphism, and every t α is additive.) His fifth (and the last) axiom can be stated in the present notation as follows: b: if Z (X Y ) = and α Agr(X, Y ), then t α (s ι2 a s ι3 b) = s ι1 t α a s ι3 b, where ι 1, ι 2, ι 3 are the embeddings X X Z, Y Y Z and Z Y Z, 141

3 respectively, while α is the agreement X Z Y Z that acts as α on X and as the identity map on Z. However, a bit stronger axiom is really needed. Definition 1. We call A a relational algebra (of dimension type V ), briefly a RelA V, if it satisfies (along with r1 r7) the following condition: r8: if Z (X Y ) =, Y Z U and α Agr(X, Y ), then t α s ι2 a = s ι1 t α a, where ι 1 and ι 2 are embeddings X X Z and Y U, respectively, while α is the agreement X Z U that acts as α on X and as the identity map on Z. We shall use the abbreviature RelA V also as the name of the respective class of algebras. The point is that b can be derived from r1 r7 and a particular case of r8 with U = Y Z. Curiously, this particular case (we call it r9) is also an instance of b obtained by substituting 1 for b; so r9 and b are equivalent. It turns out that, in a RelA V, every operation s ι with ι: X Y an embedding is injective (this cannot be proved in full extent if r9 is assumed rather than r8; therefore, the above axiom system of a RelA V is indeed stronger than that proposed by Beniaminov). We shall say that a RelA V is flat if A X A Y for X Y and each s ι is the respective embedding of A X into A Y. Proposition 2. Any relational algebra is isomorphic to a flat algebra. 3. Transformation algebras with conjugates From the viewpoint of their structure, relational algebras are too far from cylindric algebras to be handily compared immediately. The nearest analogues of RelA s among the classes considered in the literature on algebraic logic in some detail are the trans-boolean algebras of Craig [4]. Since we are only interested in locally finite algebras, the original definition of [4] is weakened as follows (see [3]). By a transformation we mean any sort-preserving self-map of V. Let T r ω stand for the set of finite transformations, i.e. transformations α 142

4 whose effective domain edα := {x V : αx x} is finite. Clearly, T r ω is a monoid: it contains the identity map ε and is closed under composition. Definition 3. A transformation algebra of type V with conjugates, briefly a TAC V, is an algebra A := (A, s α, t α ) α T rω, where A is a Boolean algebra and s α, t α are operations on A satisfying the following axioms (visually like to r1 r7): t1: s α ( a) = s α a, t2: s α a s α (a a ), t3: t α a t α (a a ), t4: b s α t α b, t5: t α s α a a, t6: s ε a = a, t7: s α s β a = s αβ a. A TAC V is said to be locally finite (if, for every a A, there is a finite subset X V such that s α a = a whenever α agrees with ε on X. The term transformation algebra with conjugates is patterned after Pinter s substitution algebra with conjugates [5]. The axioms t1 t5, t7 correspond to Craig s axioms TB1 TB6 in [4], and t6 is derivaable by the help of his TB9. The following theorem reveals the intimate connections between RelA s and locally finite TAC s (LfTAC V s, for short). Theorem 4. The class frela V to LfTAC V. of flat RelA V s is definitionally equivalent Moreover, the classes frela V and LfTAC V are isomorphic as categories. Together with Proposition 2, this means that the category of all RelA V s is equivalent to LfTAC V. 4. Cylindric algebras over Σ V The principal result of [4] shows that trans-boolean algebras and polyadic algebras are essentially the same structures. In [3], we have anounced a theorem that asserts the same for locally finite TAC s and locally finite polyadic 143

5 algebras. A version of it, with cylindric algebras instead of polyadic ones, is proved in [2]. Strictly speaking, arbitrary (not necessary finite) transformations were admitted in [3]; it is not of importance, however, under the assumption of local finiteness. To state the theorem here, we adopt the standard definition of a cylindric algebra in the following form. Definition 5. By a cylindric algebra of type V, or briefly a CA V, we mean an algebra A := (A, c x, d xy ) x V, (x,y) V 2, where A is a Boolean algebra, every c x is a unary operation on A, every d xy is an element of A, V 2 is the subset of V 2 consisting of homogenous pairs both components of which are of the same sort, and the following axioms are satisfied: c1: c x 0 = 0, c2: a c x a, c3: c x (a c x a ) = c x a c x a, c4: c x c y a = c y c x a, c5: d xx = 1, c6: d xz = c y (d xy d yz ) if y x, z, c7: c x (a d xy ) c x ( a d xy ) = 0 if x y. Theorem 6. The classes LfTAC V and of LfCA V are definitionally equivalent; any TAC-homomorphism is also a CA-homomorphism and vice versa. It follows that the categories LfTAC V and LFCA V are isomorphic. We conclude that the category RelA V is equivalent to LfCA V. This is the equivalence result mentioned in Introduction. References [1] E. M. Beniaminov, The algebraic structure of relational models of databases (in Russian). Nauchno tekhnicheskaya Informacija (1980), ser. 2, no 9. pp [2] J. Cīrulis, Abstract algebras of finitary relations: several nontraditional axiomatizations (submitted to Acta Universitatis Latviensis). 144

6 [3] Ja. P. Cirulis, Two results in algebraic logic (in Russian). Abstracts 8-th All-Union Conf. Math. Logic, Moscow, 1986, p [4] W. Craig, Unification and abstraction in algebraic logic. Studies in Algebraic Logic. Math. Assoc. Amer., Washington, 1974, pp [5] C. Pinter, Cylindric algebras and algebras of substitutions. Trans. AMS 175 (1973), pp Department of Computer Science University of Latvia Raina b. 19 Rigia, LV 1586 Latvia dmpk@mii.lu.lv 145