COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 54, ALGEBRAIC LOGIC, BUDAPEST (HUNGARY), 1988

Structural Co~pleteness in and Logic

CLIFFORD BERGMAN

A number of authors, particularly those studying "intermediate" intu­ it\onistjc logic, have asked whether a given logic is structurally complete. See [14] and [12] for example. Roughly speaking, a logic is structurally com­ plete if every proper extension must contain new logical axioms (as opposed to nothing but new rules of inference). In particular, structural complete­ ness can be seen as a kind of maximality condition on a logic. Using techniques of algebraic logic, it is possible to study many log­ ics by finding an appropriate class of in which various (algebraic) properties reflect the logical ones of interest. The property corresponding to structural completeness turns out to be quite interesting algebraically, and can be studied independently of any connections to logic. This has been done in a number of papers such as [7], [2] and [9]. In this paper we will briefly consider the notion of structural complete­ ness in its original logical context. We then go on to define and study the algebraic analog, as well as a hereditary version, called deductiveness. Unlike the situation in , there is not yet any general agreement in algebraic logic on the definition of basic concepts (such as a logic!) or even on the motivating examples. Since our intent in section 1 is ' primarily motivational, we do not feel it is necessary to completely specify what we mean by a "logic". Instead we give what we hope is a minimal set of conditions that will hold in any reasonable definition of the term. For a rigorous treatment of the subject, the reader can consult [1], [3], or [5].

Research partially supported by National Science Foundation grant DMS-8701643 60 CLIFFORD BERGMAN

The drawback of this approach is that we will not be able to establish any real connection between the definitions in .section 1 and the theorems ,, in section 2. We leave that to others with more expertise in the field.

1. The Logical Version

Structural completeness arises naturally when considering the "deduc­ tive power" of a logic. Informally, one can think of a logic £, over a set Fm (the set of formulas), as being composed of a set !1 of logical axioms, and a set R of rules of inference. For any set r of formulas, the consequences of r are those formulas derivable from r together with all instances of formulas from !1 by means of the rules in R. In particular, a tautology is a formula derivable directly from !1 via R. We can write ,C = (!1, R) to denote the logic determined by those axioms and rules of inference, and r 1--c

In fact, we identify a logic with its consequence operator, so that two logics ,C and £' are considered equal if Cc = CC'. The only assumptions we make of Cc is that it be an algebraic closure operator on the set Fm. That is, for all r,~ <;;Fm

(cl) r <;;G(r) (c2) G(G(r)) <;;G(r) (c3) r <;;~ =* c(r) <;;c(~) (c4) G(r) <;;LJ { G(ro) : ro is a finite subset of r}.

For the remainder .of this section, one can take (cl)-( c4) as the definition of a logic.1 However, the naive conception of a logic as a pair (!1,R) provides a more intuitive basis for the notions we will discuss.

1 Most modern conceptions of a logic diverge from this definition. For example, [3] requires that every set C .c(r) be closed under substitution. One can easily check that our results remain true under that particular assumption. On the other hand, in [1], condition (c4) defines the special case of compact logics. It is not clear that Proposition 1.2 holds in the failure of (c4). STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 61

We write £, < £', (and say that £,' is an extension of C) if £,' is a stronger logic in the sense that we can "derive more things". Furthermore, write £, =o £,' if £, and £,' have the same tautologies. In terms of the ,, consequence operator: ' c,:::;C' = (\7T <;;;Fm) Cc (r) <;;;Cc, (r) C =o C' = Cc (0) = Cc, (0).

Definition 1. 1. A logic £, is structurally complete if for every logic £,',

C < C' =* C.,to C'.

In other words, £, is structurally complete if it is not possible to extend £, without creating new tautologies. Do structurally complete logics generally exist? The answer is 'yes' as long as we assume that (cl)-(c4) constitutes a definition of a consequence operator.

Proposition 1.2. Every logic C, has a unique, structurally complete ex­ tension C', such that C, =o C'.

Proof. Let x = {GM:£,:::; M & C, =o M }. x can be thought of as a subset of the monoid of self-maps of Sb(Fm) (the ) under the operation of function-composition. Let x be the submonoid generated by x, Define a new operator D by

D(r) = u{ G(r): CE x} for r <;;;Fm.

We claim that DE x. The axioms(cl)-(c4) are all easily verified, with the exception of (c2). For this, we make the following two observations. First, every member ofx satisfies axiom (c4). Second, the family { G(r): CE x} is upper-directed by set-theoretic union. Now, if 'Y E D(D(r)), then for some C E x, 'Y E G(D(r)). Therefore, (by the first observation), there is

·1~" a finite subset A <;;;D(r) such that 'Y E C(A), and by upper-directedness, there is C' E x su_ch that A <;;;C'(r). Hence, 'Y E C(C'(r)) <;;;D(r), proving ( c2). Let £,' be the logic with consequence operator D. (In terms of logical axioms and rules of inference, those of £,' are obtained by taking the union 62 CLIFFORD BERGMAN over all of the logics referenced in x.) Then we clearly have £ :S £' =o .C, so that. D E x. To see that .C' is structurally complete, let £' :S M. 1° If £' =o M, then CM E x, from which it follows that M :S £'. The uniqueness is similar. I From the proposition, one sees that the structurally complete logics are precisely the, maximal members of the =a-equivalence classes. It is in this sense that structural completeness is a maximality condition. We now introduce the hereditary version of the same concept.

Definition 1.3. A logic £ is deductive if every (not necessarily proper) extension of £ is structurally complete.

Proposition 1.4. Let £ be a logic. The following are equivalent. (1) £ is deductive; (2) For each M c".£, and r <;;Fm, CM (r) = Cc (r u A), where A=CM(0) Proof. (1) * (2): First note that A<;; Cc (A) <;;CM (A)= A. Now define a consequence operator D by D(r) = Cc (AU r). That D satisfies (cl)-(c4) is straightforward. For any set r,

(*) D(r) = Cc (Au r) <;;CM (Au r) <;; CM (Au CM (r)) = CM (CM (r)) = CM (r).

In other words, if £' denotes the logic with consequence operator D, then £ :S £' :S M. Also, by taking r = 0 in (*), A = D(0) <;; CM (0) = A. Therefore,£' =o M. Now by the deductiveness of£, D = CM. (2) * (1): To show £ is deductive, let £ :S M :S N and assume M =o N. We must show M = N. Since M =o N, A= CM (0) = CN (0). Therefore CN (r) = C;:.(r U A)= CM (F), as desired. I According to the above proposition, dedcutiveness can be interpreted in the following way: Every new rule of inference can be replaced by a set of logical axioms. For if £ = (!1, R) is a deductive logic, and r is a new rule, let M = (!1,RU {r} ). The logic £' constructed in the proposition, is (!1U A, R), and th~ proof of 1.4 demonstrates that£'= M (more precisely, they have the same consequence operator). One of the goals of algebraic logic is to establish a correspondence be­ tween individual logics and classes of algebras. In the best of all possible STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 63

worlds, each logic£ would correspond to a quasivariety, £,Q, Under this correspondence

£ :::;M {cc=} £q =2Mq ( t) £ =o M {cc=} V(£Q) = V(Mq).

In other words, logical axioms correspond to identities ( on the associated algebra~) and rules of inference to quasi-identities. It is on this intuitive idea that the definitions of section 2 are based. Whether or not the relationships in ( t) are realized is a subject of debate in Algebraic Logic. In any case, the algebraic notions turn out to be quite interesting in their own right. Because it is not particularly natural from an algebraic standpoint, it is unlikely that the concept of a structurally ,: complete variety would ever have been considered if not for this intuitive I correspondence between algebras and logics. ' 2. The Algebraic Version

By applying the idealized translation ( t) of section 1 to the notions of structural completeness and deductiveness, we produce corresponding notions for classes of algebraic structures (Definition 2.1, below). In order to proceed, we briefly review the relevant notions from universal algebra. A class ICof algebras is a quasivariety if it is defined by a set of quasi-identities, i.e. universal· sentences of the form 1 rn (f\o-i(x) cs Ti(x)) -->. µ(x) cs v(x) i=O

in which n and tare natural numbers, and each

Q(IC) = SPPu(IC) V(IC) = HSP(IC) 66 CLIFFORD BERGMAN which in turn is equivalent to the quasi-identity

axsoO-> bxsoO where a = (a1, a2, ... , ak, n) [greatest common divisor], and b (a,b'). Finally, this latter formula is equivalent to the-identity

dx so 0 where d = ( ~n ) since b = (d, a). I an a,a , There is a nice characterization of structurally complete varieties in terms of generic algebras.

Theorem 2.6. A variety V is structurally complete if and only if

(\IF EV) V = V(F) ===>V = Q(F).

Proof. If V is structurally complete, then V ( Q(F)) = V implies that Q(F) = V. Conversely, suppose the condition holds. Let Q be a sub­ quasivariety, and suppose that V(Q) = V. Setting F = FQ(w) and ap­ plying Lemma 2.2, F = Fv(w). Therefore, V = V(F), so by assumption, V = Q(F) ~ Q ~ V. Thus Vis structurally complete. I As we remarked above, structurally complete varieties have some pow­ erful properties. The following Theorem and Corollary give an example of. this. For any class K of algebras, we denote by K,- the class of all nontrivial members of K.

Theorem 2. 7. Let V be a structurally complete variety, and let

Proof. Suppose A E V and A ,I"

(by Theorem 2.6) = SPP0 (FxB). It is easy to check that every ultraproduct of F x B is isomorphic to a product of an ultrapower of F with an ultrapower of B. Therefore, (using the non-triviality of B) there is a homomorphism h from B to an ultrapower F* of F. Since F and F* satisfy the same first­ order sentences, F* I= ,cp. Since

Corollary 2.8. Let V be a structurally complete variety of finite type. '-' (1) Either every member ofV has an idempotent, or no nontrivial member of V has an idempotent.

,1 (2) Let A be a finite, simple member of V. If some member of V fails to have an idempotent, then every nontrivial member of V extends A. {So A is unique.} (3) IfV is semi-simple and residually finite, then either every mem- ber of V has an idempotent, or V is equationally complete. Proof. By an idempotent of an algebra, we mean an element e, such that { e} is a subalgebra of the algebra. Since V is of finite type, there is a first order, positive, existential sentence whose content is: "I have an idempo- . tent". By Theorem 2.7, either every member of V satisfies this sentence (i.e. has an idempotent), or no nontrivial member does. If A is any finite algebra ( of finite type), then there is a sentence

A E V = Q(F) = SPP 0 (F), by Theorem 2.6. Since F is finite, Pu(F) = 68 CLIFFORD BERGMAN I(F), and since A is subdirectly irreducible, A E SP(F) =- A E S(F). I We now turn our attention to deductive varieties. In order to derive a characterization, we need the notion of a primitive algebra. " Definition 2.10. Let P be a subdirectly irreducible algebra in a variety V. • Pis primitive in V if P satisfies (VA EV) PE H(A) =-PE S(A); • Pis weakly primitive in V if P satisfies (VA EV) PE H(A) =-P E SPu(A).

Obviously, every primitive algebra is weakly primitive. The two notions coincide in an important special case.

Lemma 2.11. Let V be a residually finite variety of finite type. Then every weakly primitive algebra is primitive.

Proof. Let P be weakly primitive in V. Since P is subdirectly irreducible and V is residually finite, P is finite. Since our language is of finite type, there is a first order sentence

Theorem 2.12. A variety V is deductive if and only if every subdirectly irreducible algebra is weakly primitive.

Proof. Let V be deductive, B E V,;, and suppose that B is a homomorphic image of an algebra A in V. By deductiveness, B E V(A) = Q(A) = SPPu(A). Since B is subdirectly irreducible, B E SPu(A), that is, B is weakly primitive. Conversely, suppose that the condition holds, and let Q be a subquasi­ variety of V. We wish to show that Q is.closed under homomorphic images. Let C E Q and A E H(C). A is a subdirect product of subdirect irre­ ducibles from V. Since every quasivariety is closed under subdirect prod­ ucts, it suffices to show that a subdirectly irreducible homomorphic image of A lies in Q. But if B is such an algebra, then B is assumed to be weakly primitive. Therefore BE H(A) i;; H(C) =-BE SPu(C) i;; Q. I STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 69

Theorem 2.13. Let V be a locally finite vMiety. The following Me equiv­ alent. (1) Vis deductive; (2) Every finite, subdirectly irreducible member of V is weakly primitive; (3) Every finite, subdirectly irreducible member of V is primitive.

Proof. (1) =;,- (2) by the previous theorem. To see (2) =;,- (3), let B be finite, subdirectly irreducible, and B E H(A). Since B is finite, there is a finitely generated subalgebra A' of A such that B E H(A'). By local finiteness, A' is finite. By weak primitivity, B E SPu(A') = S(A') since A' is finite. Therefore, B E ~(A), so B is primitive. Now assume (3) holds. We show V is deductive. By Theorem 2.12 it suffices to show that every subdirectly irreducible of V is weakly primitive. So let C be subdirectly irreducible, and suppose that C E H (A). We need to show that C E SPu(A). Any algebra can be embedded into an ultraproduct of its finitely generated subalgebras (see [4, V.2.14]). Let C' be a finitely generated subalgebra of C. It suffices to show that C' E PsS(A), for then C E SPuPsS(A) ~ Q(A) = SPPu(A), from which it follows that C E SPu (A) since C is subdirectly irreducible. Now C' is finitely generated, and V is locally finite, so arguing as we did above, there is a finite subalgebra A' of A such that C1 E H(A'). In particular, C' is finite. Therefore, C' is a subdirect product of finite, subdirectly irreducible algebras, which are all primitive, by assumption. Let B be such a subdirect factor of C'. Then B E H(C') ~ H(A') ===} B E S(A') ~ S(A). It follows that C' E PsS(A) as desired. I Theorem 2.13 is quite powerful in practice. It has been used to identify a number of deductive varieties. Example 2.5 follows easily from this theorem. We present a few more examples.

Example 2.14.1. The vMiety of Boolean algebras is deductive since the only subdirectly irreducible algebra, the two-element chain is projective, hence primitive.

Example 2.14.2. Let n be a positive integer, and Va variety'ofn-dimen­ sional cylindric algebras. Then Vis deductive if and only ifV is equationally complete (or trivial). If V is equationally complete, then as in the previous example, V has a unique subdirectly irreducible algebra, which is projec­ tive, hence primitive, so Theorem 2.13 applies. Conversely, suppose that V 70 CLIFFORD BERGMAN

is deductive. · Note that no nontrivial cylindric algebra has an idempotent element. Every finitely generated subvariety of V is semi-simple and residu­ ally finite, so by Corollary 2.8(3) (and deductiveness) must be equationally complete. From this it follows that V itself is equationally complete.

Example 2.14.3. (Igosin [9]) V(Mw) is deductive, where Mw is the modu­ lar lattice of height 2 with a countably infinite set of atoms. The subdirectly irreducibles in this variety are the lattices Mn, for n any cardinal not ·equal to 1 or 2. Note that this class is closed under ultraproducts. Let Q be a proper subquasivariety. If Q contained Mw, then Q would contain every one of the subdirectly irreducibles, and would not be proper. Therefore, there is a natural number n such that Q,i <;;;S(Mn), hence Y(Q) = V(Mn), But one can show that MJ is primitive in this latter variety, for all j :,; n. Therefore Q = V(Mn),

We also note that the variety Y(Ns) is deductive, since the only sub­ directly irreducibles are projective, hence primitive.

------! I j Example 2.14.4. (This example was discovered during a problem session held in conjunction with the conference.) Let F denote the Fano plane 3 (that is the lattice of subspaces of the (Z2 ) over Z2). Then V = V(F) is a structurally complete variety that is not deductive.

By [8] V,i = {2, M3 , M3,3 , F}, where 2 is the two-element chain, M3 is as in the previous example, and M3,3 is shown in the figure below. It is easy to check that each of these subdirectly irreducible (actually simple) algebras is a subalgebra of F. Now, M3,3 is not primitive, since it is a homomorphic image of, but not embeddable in the lattice B. (B is a subdirect product of Ma,3 x 2, so B E V.) Therefore, by Theorem 2.13, Vis not deductive. However it follows from (10, Theorem 6.2], that F is projective in V. Therefore, if Q is any quasivariety such that Y(Q) = V, then F E H(Q) implies F E Q. But then V,i <;;;S(F) <;;;Q implying that V = Q. So V is structurally complete.

Example 2.14.5. (Tsitkin [14]) There exist five finite Heyting algebras, A1, ... , As such that: (1) Y(Ai) is not structurally complete, for i = 1, ... , 5. (2) IfV is a variety of Heyting algebras excluding A 1, ... , As, then V is structurally complete. (3) A variety Vis deductive if and only ifV excludes A1, ... , As, STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 71

B (1) follows easily from the fact that Ai is not primitive in the variety it generates (imitate the proof of Theorem 2.13). (3) follows easily from (1) and (2). Unfortunately, a proof of (2) was not included in [14].

____ J Let V be an equationally complete variety (i.e. a minimal variety). Then by Proposition 2.4 the notions of deductiveness and structural completeness coincide, and are equivalent to V being minimal as a quasivariety. This suggests the following Problem 2.15. When is an equationally complete variety minimal as a quasiwµ·iety? This was, to a large degree, answered by the following two theorems. Theorem 2.16. (Bergman-McKenzie, [21)Let V be a locally finite, con­ gruence modular variety. If V is minimal as a variety, then V is minimal as a quasivariety. Observe that Examples 2.14.1-2 foHow from this theorem. Theorem 2.17. (1) There is a 3-element algebra generating a variety that is equa­ • tionally complete, but not structurally complete. (2) (Andreka-Nemeti) Let YY (w3) denote the full cylindric set al­ gebra on a 3-element set, of dimension w. Then V(YY (w3)) is equationally complete, but not structurally complete. Note that the variety in (1) will be locally finite, and the one in (2) is congruence modular (in fact, arithmetical). The proof of 2.17 appears in 72 CLIFFORD BERGMAN

[2]. Part (2) relies heavily on results in [11]. In fact, in (2) we can replace w by any infinite cardinal. r. It is an open problem (see for example [6, problem 10]) to characterize those finite algebras that generate an equationally complete variety. Modulo t that problem, can one determine which of those finite algebras generates a structurally complete variety? We conclude by mentioning one more theorem characterizing deductive­ ness among a class of varieties. A ring is said to be left- (right-) uniserial if the lattice ofleft (right) ideals forms a finite chain. A variety V of algebras is directly representable if it has only finitely many finite, directly irreducible members. For such a variety, Vabdenotes the subvariety of abelian algebras, and R(Vab) the ring associated with Vab·

Theorem 2.18. (Hogben-Bergman, [7]) ( 1) Let R be an Artinian ring, V the variety of left R-modules.

Then V is deductive if and only if R = $7=1 Mn, (Ri), where R,; is a left- and right uniserial ring for all i = 1, ... , k. We say "R is a deductive ring". (2) Let V be a directly representable variety of finite type. Then V is deductive if and only if either (a) V is equationally complete or (b) Every member ofV has an idempotent, and R(Vab) is a deductive ring.

Final Remarks. Many of the results in this section can be interpreted as statements abont logic. For example, Problem 2.15 can be interpreted as asking whether a logic with a maximal (proper) set of tautologies is in fact a maximal logic. In the terminology of [1], the former condition is called weak maximality. Thus Theorem 2.16 asserts that among those logics .C such that .CQ is locally finite and congruence modular, weak maximality implies maximality.

Example 2.14.1 is of course equivalent to the assertion that the clas­ u sical propositional calculus is a maximal logic. 2.14.5 characterizes those intuitionistic propositional logics that are deductive, and gives conditions I: for structural completeness. Example 2.14.2 together with Theorem 2.17(2) asserts that in the va­ riety CA" (a-dimensional cylindric algebras), every equationally complete variety is structurally complete if and only if a < w. CAa corresponds to STRUCTURAL COMPLETENESS IN ALGEBRA AND LOGIC 73

first-order predicate logic with a individual variables. The connections be-- • tween predicate logics and cylindric algebras is extensively investigated in [5] and [13].

References

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11. I. Nemeti 1 On varieties of cylindric algebras with applications to logic, Annals of Pure and Applied Logic 36 (1987), 235-277. 12. T. Prucnal, Structural completeness of Medvedev's propositional calculus, Reports on Mathematical Logic 6 (1976), 103-105. 13. A. Tarski and S. Givant, "A Formalization of Set Theory Without Variables," Col­ , loquium Publications 41, Amer. Math. Soc., Providence, R. I., 1987. 14. A. I. Tsitkin, On structurally complete superintuitionistic logics, Dokl. Akad. N auk SSSR 241 (1978), 40-43. English Transl., Soviet Math. Dok!. 19 (1978), 816-819.

,. Clifford Bergman Dept artment of Mathematics Iowa State University Ames, Iowa, 50011 USA :{