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Chapter 10 Certain Classes of Models
CHAPTER 10
CERTAIN CLASSES OF MODELS A structural characterization of quasifree classes of algebras was given in [IX] , $ 5 . Using this result, we state below structural characterizations for universally axiomatizable classes of models and for quasiprimitive classes of algebras (quasiequational classes or quasivarieties). At the same time we resolve the question of an intrinsic, purely algebraic characterization of quasiprimitive classes of algebraic systems - left open in [IV] . Finally, we show that, up to structural equivalence, quasiprimitive classes of algebraic systems are the only elementary (i.e., first-order axiomatizable) classes of models, homomorphically closed in themselves, which admit a theory of defining relations in the sense of [VIII] . In what follows all categories of structures will be assumed to have strong substructures, while direct compositions, in the cases when they exist, will be assumed to coincide with the direct products [VIII] .
$ 1. We agree to say a category % of structures has finitary homomorphisms iff for any two%-structures 3 and 23,no matter what local system {aO1: 01 € r 1 of%-substructures covering % is chosen, any mapping from % into 5!3 that is a homomorphism of %a into an appropriate%-substructure of B for all a€ r is a homomorphism of 2l into B . Corresponding to the usual group-theoretic terminology, a %-structure 2i is called locally finite iff every finite subset of 2i lies in some finite%-substructure. Clearly, all categories of models have finitary homomorphisms; it is also easy to prove the following theorem: Theorem 1: Every category 3c of structures with finitary homomorphisms and locally finite structures is structurally equivalent to an appropriate class
of models. rn A category % of structures is locally compatible iff whenever every finite subset of an arbitrary collection 3 of%-structures, defined on subsets of a given set, is embeddable in some%-structure as a set of%-substructures. then 61
62
Certain classes of models
the whole of 6 can also be simultaneously so embedded in some%-structure. From this definition it follows, in particular, that in a locally compatible category % every increasing chain of %-structures, each embedded in the next, can be embedded in some embracing %-structure. The compactness theorem for first-order predicate logic (FOPL) shows that every elementary class of models is locally compatible. We recall that a class % of models is called universally axiomatizable iff it can be characterized by a collection of universal FOPL sentences, i.e., sentences of the form ( x , ) ... (x,) cp(xl,...,x,), where the expression cp contains no quantifiers. Theorem 2: For a category % of structures to be structurally equivalent to some universally axiomatizable class of models, it is necessary and sufficient that 3c be locally compatible and have finitary homomorphisms, and any subset of a%-structure be a%-substructure. rn Theorem 3: If a universally axiomatizable class of models with fundamental predicates PI', ...,PLk is structurally equivalent to a class of models with fundamental predicates Q i l , ..., Qsl, then throughout the classes there are equivalences of the following form holding: Pl,(xl, ..., xr> ++ qi(x1, ..., x,.~), (i = 1, ..., k ) , Q j ( x l , ..., xsi) * xi@,, ..., xsi) ( j = 1, ..., I ) , where the qi,xi are open formulas constructed with the aid of the equality sign and the predi..., Pk, respectively. cate symbols Q1, ..., Q, and PI, In case the number of fundamental predicates is infinite, Theorem 3 still holds, but infinite expressions must be admissible as the cpi, xi.
5 2. A model 91 with predicates P,,P 2 , ..., whose ranks are n l , 122, ..., respectively, is called an algebraic system of type T = ( I ; n l , n 2 , ...) ,where I is a subset of the index set for the predicates such that for i E I, Pi is the predicate of an operation on the base of 2l (or simply, on a). The class%, of all algebraic systems of type T is bounded, multiplicatively and homomorphically closed in itself, regular, and contains a unit structure. The notions of quasifree and free subclasses of a category 3c of structures were introduced in [IX] . If % is a category of models, then a quasifree or free subclass, distinguishable in % by means of some system of axioms (i.e., first-order axiomatizable relative to%), is called quasiprimitive or primitive in %, respectively. Quasiprimitive (primitive) subclasses of a class %, are called, simply, quasiprimitive (Primitive)classes of algebraic systems of the specified type. From the theorems of Tarski-tog [163,89] and Bing [8] it follows that a subclass f? of a class% of models is quasiprimitive in% iff it can be distinguished in
Certain classes of models
63
'X by axioms of the form (x,) ... &)(I), & ... & I)s I),+,), where the I)i are expressions of the forms Pi(xil,...,xi ) or xk = xl. -+
Suppose the category% : (a) is multiplicatively closed, and (b) contains a unit. Then the intersection of any collection of quasifree (free) subclasses of 3c is again a quasifree (free) subclass. Thus, for every class 3 of%-structures, there is a smallest quasifree (free) subclass TofCX that includes J.The class 7 is called the quasifree (free) closure of d in % and is written 7 = d (7=Jf). It is easy to see that 6 4 consists of all possible%-substructures of direct products of J -structures. In order to obtain an analogous characterization for J f, we lay these additional demands on% : (c) 3c is homomorphGally closed in itself, and (d) if '%, E%, and u is a homomorphism of % ' onto % ,then the pre-image under u of any %-substructure of % is a%-substructure of %. Then the free closure2f of a quasifree subclass .@consists of all possible%-structures which are homomorphic images of2-structures. From this it follows that if % and J are elementary classes of models, then 3 4 and J f are elementary (' ). Furthermore, if a category CK. satisfying (a)-(d) is regular, and 2 is a quasifree subcategory, then every PJ-free %-structure belongs t o 2 . In particular, ifEf contains free structures with any number of 2f-free generators, which are dense under these conditions, then the supply of free structures does not change on passing from 2 to its free closure.
Theorem 4: Let the regular category %, satisfying (a)-(d), contain a finite structure 9l. Then: (i) %-free structures with differentfinite numbers of free generators are not isomorphic; (ii) in the minimal quasifree subclass { and free subclass { '%}fcontaining a, every structure with a finite generating set is finite;(iii) if the number of non-isomorphic%-structures of finite power is finite, then in { are included only a finite number of minimal quasifree and free subclasses containing more than units. The statements (i) and (iii) are generalizations of theorems of Fujiwara [43] and Scott [ 1481 ,proved for varieties of algebras.
9 3. Let ( r, <) be a partial ordering in which any two elements have a common greater element, and let 3c be a general category. Suppose that with every a E r is associated an object '%aE%, and with every pair ( a ,0)(a< 0; a,0E F) is associated a homomorphism nap: 91a -+ apsuch that a < y < 0 implies nap = na,nYp. We say that and the mappings a + %a and (a,0)-+nap constitute a direct spectrum. An object % of % with specified maps na: !?la -+ % (a E r) is called the limit of the spectrum [28] iff na = n a p p (a< P), and for any !B E% and any system of homomorphisms ua: %a !B -+
64
Certain classes of models
(a f r)satisfying the conditions ua = 7 ~ (a< ~ p), ~thereu is one ~ and only one homomorphism g: (II + % for which ua = nag (aE r).In case % ' is a category of structures, it will be assumed without further mention that r has a least element 0, and the mappings map are homomorphisms of ( I I a onto \up (a< 8). Then it is possible to consider the (a E I?) and %?I = lim 9ia to be defined on $?lo with appropriately chosen equality relations (cf. [V]). If % is the category of all models of a fixed similarity type, then for any direct spectrum under the stated conditions the limit model exists, and its construction is given in [V] . There it is also shown that if a universal or positive sentence holds in all models of the spectrum, then it holds as well in the limit model. Inasmuch as all universally axiomatizable classes of algebraic systems are characterized by positive or universal axioms, every such class contains spectral limits of its systems. The possibility of inverting this gives the following theorem: Theorem 5: A multiplicatively closed class of algebraic systems of type r containing all %,-subsystems of its members is elementaiy i f f it contains limits of all direct spectra of its members, I In particular, a quasifree class of algebraic systems is quasiprimitive iff it contains limits of all spectra of its systems. Taking into account Theorem 5 of [IX] , we get: in order that a category% of structures be structurally equivalent to an elementary class of algebras, multiplicatively closed and containing all subalgebras of its members, it is necessary and sufficient that % be multiplicatively and homomorphically closed in itself, bounded, regular, and additive, and have divisible homomorphisms and limits of direct spectra of its structures. Adding to these conditions the demand for existence of a unit structure, we obtain a structural characterization of quasivarieties of algebras. Let % be an arbitrary category of structures, and let % €%. An equivalence relation 8 defined on the base of (II is called a congruence on 2l (cf. [61]) iff 8 is associated with some homomorphism of 9i onto an appropriate %-structure. An equivalence on (II is called an outer congruence iff for any two homomorphisms p , u of an arbitrary%-structure %?I, generated by {b,: a € r},into % , if bap bau(0) for all a E r, then bp fba(8) for all b € 'x3. Obviously, in order that a quasifree class% of algebras be free, it is necessary and sufficient that every outer congruence in a %-algebra be a congruence. Thus, in order to get an intrinsic structural characterization for varieties of algebras, it suffices to adjoin to the above collection of structural properties characterizing quasivarieties of algebras, the demand that outer congruences be congruences.
Certain classes of models
65
54. A class % of models is said to have local embeddability iff whenever every finite submodel of an arbitrary model %l of appropriate type is isomorphically embeddable in some%-model, then %l itself is embeddable in some%-model.
Lemma 1:Let the class % of models with local embeddability contain %-free models with free dense generating sets of arbitrary finite powers. Then for each Goperation anar (cf. [IX] , §3), a formula of the form 'P,JXl,
...,xn)= Xn+1
-
(axn+*>... ( 3xs)qnpncu(xl ...,xs) 7
(1)
(where qnais an appropriate conjunction of formulas of the form xi = xl,
E;(xil,...,xi,)) is valid in all %-models. rn 'Lehma 2: If a class % of models with local embeddability is homomorph-
ically rbsed in itself and contains%-free models with %-free,finitely %-dense subsets of every cardinality, then by augmenting the findamental predicates with the @-operations,we turn % into a structurally equivalent universally axiomatizable class of algebraic systems. rn On the basis of these lemmas the following can be proved:
Theorem 6: Suppose % is an elementary, homomorphically closed in itself class of models which is also R-complete in the class of all models of the type of %. Then 3c is StructuralIy equivalent to a quasiprimitive class of algebraic systems. Indeed, assuming% is all of the above and non-trivial, we find that Rcompleteness implies % contains models with %-free %-dense generating sets of arbitrary cardinality [VIII] . Since % is homomorphically closed it follows from Theorem 1 of [IX] that 7C is regular, and the non-empty intersection of %-submodels of a %-model is a %-submodel. By virtue of the basic result of [VII] ,% is additive, in view of its first-order axiomatizability. Theorems 1 and 2 of [IX] now show that every free generating subset of a %-model is finitely dense. Local embeddability follows immediately from axiomatizability by way of the compactness theorem for FOPL. Finally, by Lemma 2 , the expansion of %-models by the @-operationsas defined in (1) yields a quasiprimitive class of algebraic systems structurally equivalent to %. rn
NOTE (') J q is not necessarily elementary; cf. [XXXI] ,second corollary
CERTAIN CLASSES OF MODELS A structural characterization of quasifree classes of algebras was given in [IX] , $ 5 . Using this result, we state below structural characterizations for universally axiomatizable classes of models and for quasiprimitive classes of algebras (quasiequational classes or quasivarieties). At the same time we resolve the question of an intrinsic, purely algebraic characterization of quasiprimitive classes of algebraic systems - left open in [IV] . Finally, we show that, up to structural equivalence, quasiprimitive classes of algebraic systems are the only elementary (i.e., first-order axiomatizable) classes of models, homomorphically closed in themselves, which admit a theory of defining relations in the sense of [VIII] . In what follows all categories of structures will be assumed to have strong substructures, while direct compositions, in the cases when they exist, will be assumed to coincide with the direct products [VIII] .
$ 1. We agree to say a category % of structures has finitary homomorphisms iff for any two%-structures 3 and 23,no matter what local system {aO1: 01 € r 1 of%-substructures covering % is chosen, any mapping from % into 5!3 that is a homomorphism of %a into an appropriate%-substructure of B for all a€ r is a homomorphism of 2l into B . Corresponding to the usual group-theoretic terminology, a %-structure 2i is called locally finite iff every finite subset of 2i lies in some finite%-substructure. Clearly, all categories of models have finitary homomorphisms; it is also easy to prove the following theorem: Theorem 1: Every category 3c of structures with finitary homomorphisms and locally finite structures is structurally equivalent to an appropriate class
of models. rn A category % of structures is locally compatible iff whenever every finite subset of an arbitrary collection 3 of%-structures, defined on subsets of a given set, is embeddable in some%-structure as a set of%-substructures. then 61
62
Certain classes of models
the whole of 6 can also be simultaneously so embedded in some%-structure. From this definition it follows, in particular, that in a locally compatible category % every increasing chain of %-structures, each embedded in the next, can be embedded in some embracing %-structure. The compactness theorem for first-order predicate logic (FOPL) shows that every elementary class of models is locally compatible. We recall that a class % of models is called universally axiomatizable iff it can be characterized by a collection of universal FOPL sentences, i.e., sentences of the form ( x , ) ... (x,) cp(xl,...,x,), where the expression cp contains no quantifiers. Theorem 2: For a category % of structures to be structurally equivalent to some universally axiomatizable class of models, it is necessary and sufficient that 3c be locally compatible and have finitary homomorphisms, and any subset of a%-structure be a%-substructure. rn Theorem 3: If a universally axiomatizable class of models with fundamental predicates PI', ...,PLk is structurally equivalent to a class of models with fundamental predicates Q i l , ..., Qsl, then throughout the classes there are equivalences of the following form holding: Pl,(xl, ..., xr> ++ qi(x1, ..., x,.~), (i = 1, ..., k ) , Q j ( x l , ..., xsi) * xi@,, ..., xsi) ( j = 1, ..., I ) , where the qi,xi are open formulas constructed with the aid of the equality sign and the predi..., Pk, respectively. cate symbols Q1, ..., Q, and PI, In case the number of fundamental predicates is infinite, Theorem 3 still holds, but infinite expressions must be admissible as the cpi, xi.
5 2. A model 91 with predicates P,,P 2 , ..., whose ranks are n l , 122, ..., respectively, is called an algebraic system of type T = ( I ; n l , n 2 , ...) ,where I is a subset of the index set for the predicates such that for i E I, Pi is the predicate of an operation on the base of 2l (or simply, on a). The class%, of all algebraic systems of type T is bounded, multiplicatively and homomorphically closed in itself, regular, and contains a unit structure. The notions of quasifree and free subclasses of a category 3c of structures were introduced in [IX] . If % is a category of models, then a quasifree or free subclass, distinguishable in % by means of some system of axioms (i.e., first-order axiomatizable relative to%), is called quasiprimitive or primitive in %, respectively. Quasiprimitive (primitive) subclasses of a class %, are called, simply, quasiprimitive (Primitive)classes of algebraic systems of the specified type. From the theorems of Tarski-tog [163,89] and Bing [8] it follows that a subclass f? of a class% of models is quasiprimitive in% iff it can be distinguished in
Certain classes of models
63
'X by axioms of the form (x,) ... &)(I), & ... & I)s I),+,), where the I)i are expressions of the forms Pi(xil,...,xi ) or xk = xl. -+
Suppose the category% : (a) is multiplicatively closed, and (b) contains a unit. Then the intersection of any collection of quasifree (free) subclasses of 3c is again a quasifree (free) subclass. Thus, for every class 3 of%-structures, there is a smallest quasifree (free) subclass TofCX that includes J.The class 7 is called the quasifree (free) closure of d in % and is written 7 = d (7=Jf). It is easy to see that 6 4 consists of all possible%-substructures of direct products of J -structures. In order to obtain an analogous characterization for J f, we lay these additional demands on% : (c) 3c is homomorphGally closed in itself, and (d) if '%, E%, and u is a homomorphism of % ' onto % ,then the pre-image under u of any %-substructure of % is a%-substructure of %. Then the free closure2f of a quasifree subclass .@consists of all possible%-structures which are homomorphic images of2-structures. From this it follows that if % and J are elementary classes of models, then 3 4 and J f are elementary (' ). Furthermore, if a category CK. satisfying (a)-(d) is regular, and 2 is a quasifree subcategory, then every PJ-free %-structure belongs t o 2 . In particular, ifEf contains free structures with any number of 2f-free generators, which are dense under these conditions, then the supply of free structures does not change on passing from 2 to its free closure.
Theorem 4: Let the regular category %, satisfying (a)-(d), contain a finite structure 9l. Then: (i) %-free structures with differentfinite numbers of free generators are not isomorphic; (ii) in the minimal quasifree subclass { and free subclass { '%}fcontaining a, every structure with a finite generating set is finite;(iii) if the number of non-isomorphic%-structures of finite power is finite, then in { are included only a finite number of minimal quasifree and free subclasses containing more than units. The statements (i) and (iii) are generalizations of theorems of Fujiwara [43] and Scott [ 1481 ,proved for varieties of algebras.
9 3. Let ( r, <) be a partial ordering in which any two elements have a common greater element, and let 3c be a general category. Suppose that with every a E r is associated an object '%aE%, and with every pair ( a ,0)(a< 0; a,0E F) is associated a homomorphism nap: 91a -+ apsuch that a < y < 0 implies nap = na,nYp. We say that and the mappings a + %a and (a,0)-+nap constitute a direct spectrum. An object % of % with specified maps na: !?la -+ % (a E r) is called the limit of the spectrum [28] iff na = n a p p (a< P), and for any !B E% and any system of homomorphisms ua: %a !B -+
64
Certain classes of models
(a f r)satisfying the conditions ua = 7 ~ (a< ~ p), ~thereu is one ~ and only one homomorphism g: (II + % for which ua = nag (aE r).In case % ' is a category of structures, it will be assumed without further mention that r has a least element 0, and the mappings map are homomorphisms of ( I I a onto \up (a< 8). Then it is possible to consider the (a E I?) and %?I = lim 9ia to be defined on $?lo with appropriately chosen equality relations (cf. [V]). If % is the category of all models of a fixed similarity type, then for any direct spectrum under the stated conditions the limit model exists, and its construction is given in [V] . There it is also shown that if a universal or positive sentence holds in all models of the spectrum, then it holds as well in the limit model. Inasmuch as all universally axiomatizable classes of algebraic systems are characterized by positive or universal axioms, every such class contains spectral limits of its systems. The possibility of inverting this gives the following theorem: Theorem 5: A multiplicatively closed class of algebraic systems of type r containing all %,-subsystems of its members is elementaiy i f f it contains limits of all direct spectra of its members, I In particular, a quasifree class of algebraic systems is quasiprimitive iff it contains limits of all spectra of its systems. Taking into account Theorem 5 of [IX] , we get: in order that a category% of structures be structurally equivalent to an elementary class of algebras, multiplicatively closed and containing all subalgebras of its members, it is necessary and sufficient that % be multiplicatively and homomorphically closed in itself, bounded, regular, and additive, and have divisible homomorphisms and limits of direct spectra of its structures. Adding to these conditions the demand for existence of a unit structure, we obtain a structural characterization of quasivarieties of algebras. Let % be an arbitrary category of structures, and let % €%. An equivalence relation 8 defined on the base of (II is called a congruence on 2l (cf. [61]) iff 8 is associated with some homomorphism of 9i onto an appropriate %-structure. An equivalence on (II is called an outer congruence iff for any two homomorphisms p , u of an arbitrary%-structure %?I, generated by {b,: a € r},into % , if bap bau(0) for all a E r, then bp fba(8) for all b € 'x3. Obviously, in order that a quasifree class% of algebras be free, it is necessary and sufficient that every outer congruence in a %-algebra be a congruence. Thus, in order to get an intrinsic structural characterization for varieties of algebras, it suffices to adjoin to the above collection of structural properties characterizing quasivarieties of algebras, the demand that outer congruences be congruences.
Certain classes of models
65
54. A class % of models is said to have local embeddability iff whenever every finite submodel of an arbitrary model %l of appropriate type is isomorphically embeddable in some%-model, then %l itself is embeddable in some%-model.
Lemma 1:Let the class % of models with local embeddability contain %-free models with free dense generating sets of arbitrary finite powers. Then for each Goperation anar (cf. [IX] , §3), a formula of the form 'P,JXl,
...,xn)= Xn+1
-
(axn+*>... ( 3xs)qnpncu(xl ...,xs) 7
(1)
(where qnais an appropriate conjunction of formulas of the form xi = xl,
E;(xil,...,xi,)) is valid in all %-models. rn 'Lehma 2: If a class % of models with local embeddability is homomorph-
ically rbsed in itself and contains%-free models with %-free,finitely %-dense subsets of every cardinality, then by augmenting the findamental predicates with the @-operations,we turn % into a structurally equivalent universally axiomatizable class of algebraic systems. rn On the basis of these lemmas the following can be proved:
Theorem 6: Suppose % is an elementary, homomorphically closed in itself class of models which is also R-complete in the class of all models of the type of %. Then 3c is StructuralIy equivalent to a quasiprimitive class of algebraic systems. Indeed, assuming% is all of the above and non-trivial, we find that Rcompleteness implies % contains models with %-free %-dense generating sets of arbitrary cardinality [VIII] . Since % is homomorphically closed it follows from Theorem 1 of [IX] that 7C is regular, and the non-empty intersection of %-submodels of a %-model is a %-submodel. By virtue of the basic result of [VII] ,% is additive, in view of its first-order axiomatizability. Theorems 1 and 2 of [IX] now show that every free generating subset of a %-model is finitely dense. Local embeddability follows immediately from axiomatizability by way of the compactness theorem for FOPL. Finally, by Lemma 2 , the expansion of %-models by the @-operationsas defined in (1) yields a quasiprimitive class of algebraic systems structurally equivalent to %. rn
NOTE (') J q is not necessarily elementary; cf. [XXXI] ,second corollary