ALGEBRAS AND MATRICES FOR ANNOTATED LOGICS

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1 ALGEBRAS AND MATRICES FOR ANNOTATED LOGICS R. A. Lewin, I. F. Mikenberg and M. G. Schwarze Pontificia Universidad Católica de Chile Abstract We study the atrices, reduced atrices and algebras associated to the systes SAL τ of structural annotated logics. In previous papers, these systes were proven algebraizable in the finitary case and the class of atrices analyzed here was proven to be a atrix seantics for the. We prove that the equivalent algebraic seantics associated with the systes SAL τ are proper quasivarieties, we describe the reduced atrices, the subdirectly irreducible algebras and we give a general decoposition theore. As a consequence we obtain a decision procedure for these logics. 1 Introduction Annotated logics were introduced in [14] by V. S. Subrahanian as logical foundations for coputer prograing. Blair and Subrahanian [2] proved that these systes were paraconsistent and showed that they could for the basis of a prograing language for reasoning about data bases that contain inconsistent inforation. Subsequently, nuerous applications in Artificial Intelligence, like inheritance networks, object oriented data bases, etc. have been developed. The first two authors thank the Generalitat de Barcelona for a grant to attend the I WAAL, where an early version of this paper was presented. Research supported by FONDECYT grant

2 A coplete study of these systes, fro the odel theoretical and proof theoretical points of view has been done in [1] and [8]. These papers show that ost classical basic results in odel theory can be adapted to these systes. However, since there are several kinds of axios (soe for coplex forulas, others for atoic forulas and still others for arbitrary forulas), these systes are not structural in the sense that their consequence relation is not closed under substitutions. One of the difficulties with non-structural systes is that they do not have a natural algebraic counterpart. In [11] we introduced a structural version SPτ of annotated logics built with the purpose of obtaining systes as closely related as possible to the original systes Pτ of [8], but to which the techniques of algebraic logic could be applied. The cost of this process was that we had to introduce several unary operation sybols to the language of Pτ and the corresponding axios to control these operations. Then we proved that they are equivalent to the original Pτ systes, in the sense that there is a translation fro the language of Pτ into that of SPτ such that a forula is provable in Pτ if and only if it is provable in SPτ using soe additional preises. For details, see [11]. Finally we studied the algebraizability of these systes using as ain fraework the theory of algebraization of deductive systes developed by Blok and Pigozzi in [3]. The ain result proven in [11] is that these systes are algebraizable. Moreover, we proved that an annotated logic is finitely algebraizable (i.e., it has a finite syste of congruence forulas) if and only if the lattice of annotation constants is finite. All axios and inference rules of SPτ are either (a translation of) an axio or a rule of the systes Pτ, or are necessary to define the new operations or to obtain the above entioned equivalence of the systes. Even so, in the finite case, in SAL τ we had to add a new axio for negations, axio ( 4 ), because one of the weaknesses of Pτ is that the axios and rules pay little or no attention to the peculiarities of the lattice τ or of the particular function that is being used. In order to take these into account in [13] we proposed the systes SAL τ, structural annotated logic based on τ, (for finite τ,) which are axioatic extensions of the SPτ. This iediately iplies that the systes SAL τ are algebraizable. An easy verification shows that an equivalence of translations siilar to that between SPτ and Pτ also holds for SAL τ and Pτ. We then defined a special class of atrices and proved that it is a atrix seantics for SAL τ. In close connection with these systes, [12] proposes a (faily of) se- 2

3 antically defined annotated logic. This syste differs fro SAL τ in the treatent of the paraconsistent eleents and in the effect of the annotation operators over the atoic forulas. Even though it is an unpublished paper, [12] antecedes the present one, and soe of the ideas, particularly the axios (τ 7 ) and (τ 8 ), are already present there. In this paper we study SAL τ s equivalent algebraic seantics SAL τ. In Section 2 we introduce the syste SAL τ and give soe otivations for the connectives added and for the new axios. In section 3 we define the set of nice atrices, a subset of the class of all SAL τ atrices. In Section 4 we characterize the reduced nice atrices and give two exaples that illustrate the process. Using the reduced nice atrices, in Section 5 we begin the study of SAL τ. The ain results are that SAL τ is a proper quasi variety, that it is generated by a single algebra and a characterization of the subdirectly irreducible algebras is also presented. 2 The Systes SAL τ In this Section we will define structural annotated deductive systes SAL τ that are axioatic extensions of the systes SPτ introduced in [11]. Let τ be a (fixed) finite lattice and : τ τ an arbitrary function. Let, denote the least and the greatest eleents of the lattice. 2.1 The Language The language of SAL τ will consist of the binary logical sybols,, and the unary sybols, and f λ, for each λ τ, a denuerable set P = {p i : i ω} of propositional letters and parentheses. The set F of forulas is defined recursively as usual. For any ω, define n recursively: 0 p = p and n+1 p = ( n p). The forulas of the for kn f λn k n 1 f λn 1 k 1 f λ1 k 0 p i, where for 0 j n, k j ω, λ j τ and i ω, are called hyperliterals. Observe that for n = 0, p i and k p i are hyperliterals. A forula that is not a hyperliteral will be called a coplex forula. The eleents of τ ay be thought of as confidence factors, degrees of belief or degrees of credibility. In this last sense, the forula f λ p stands for the proposition p has credibility greater or equal than λ. They correspond 3

4 to the annotated variables p λ of [8] (or p : λ in other papers on the subject.) Forulas like f µ f λ p ay be considered updatings of the credibility of p. The intended eaning for the forula p is p is well behaved with regard to negations. This unary operation is not unnatural for the intended applications of these logics and was introduced in order to get a structural syste in a sooth way. The function, intended to act as a negation for annotated forulas, is arbitrary. For ω we define recursively in the sae way as. Reark 1. Since the lattice τ has finite cardinality τ, for any λ τ, the set { λ : ω}, is finite. So for any λ, there exists a least integer t λ such that for soe s ω \ {0}, t λ λ = t λ +s λ. Of course, the least such an integer s ω also exists. Call it s λ. Observe that if r t λ and is a ultiple of s λ, then r λ = r+ λ, so defining t = ax{t λ : λ τ }, s = l.c..{2} {s λ : λ τ }, we have, for any λ τ, t λ = t+s λ. In the definition of s above, we have included 2 in the least coon ultiple in order to have a single negation axio 4 both for hyperliterals and for coplex forulas. 2.2 Axios and Inference Rules The axios and inference rules for SAL τ are as follows: 1. Axios for binary connectives: ( 1 ) p (q p), ( 2 ) (p (q r)) ((p q) (p r)), ( 3 ) ((p q) p) p, ( 1 ) (p q) p, ( 2 ) (p q) q, ( 3 ) p (q (p q)), ( 1 ) p (p q), 4

5 ( 2 ) q (p q), ( 3 ) (p r) ((q r) ((p q) r)). 2. Axios for negation: ( 1 ) (p q ) ((p q) ((p q) p)), ( 2 ) p (p ( p q)), ( 3 ) p (p p), ( 4 ) t+s p t p, where t and s are the nubers defined in Reark Axios for : ( 1 ) p ( p), ( 2 ) p (f λ p), ( 3 ) (p ), (p q), (p q), (p q). 4. Axios for annotated forulas: (τ 1 ) (p ) f p, (τ 2 ) (p ) ( k f λ p k 1 f λ p), for k 1, (τ 3 ) f µ p f λ p, for µ λ, (τ 4 ) f µ f λ p f µ p, (τ 5 ) (p ) (f λ p f λ p), (τ 6 ) p (f λ p p). For the next two axios we will need the following definitions. and { fκ p λκ S κ (p) = (f λp f λ p) if κ, (p ) f p if κ = T (λ,κ) (p) = <t+s λ κ p <t+s λκ ( p p). 5

6 (τ 7 ) (p ) κ τ S κ(p), (τ 8 ) S κ (p) λ τ T (λ,κ)(p). 5. Inference Rules (R1) Modus Ponens p, p q q (R2) p f λ q, p f µ q p f λ µ q. Reark The first three axios for negations are tautologies of classical propositional calculus relativized to well behaved forulas. Axio ( 4 ) codes into the syste the fact that all functions over a finite set are eventually periodic. Together with (τ 2 ), (τ 5 ), (τ 7 ) and (τ 8 ) it handles the negation for annotated forulas. 2. The axios for control this new unary operation. The third one states that coplex forulas are well behaved. The other two state that the behaviour of a forula is not changed by negations or by the operations f λ. 3. The first three groups of axios guarantee that the set of coplex forulas behave classically with respect to the Boolean operations. 4. The first three axios for annotated forulas are relativized translations of those of Pτ. The next three were introduced to control the action of the unary operations f λ. Essentially, they say that the degrees of credibility apply only to the core p of the hyperliterals. 5. For each lattice τ, SAL τ is an axioatic extension of the corresponding syste SPτ of [11] obtained by adding axios τ 7 and τ 8. As we said in the introduction, in the axioatization of SPτ there is only a crude consideration of the specific lattice τ and the function that we are using. Since τ and are arbitrary, these axios take a rather cubersoe for, nevertheless they are not coplicated in spirit. 6

7 6. Intuitively, the forula S κ (p) is true only if the axiu degree of credibility of the forula p is κ. Axio τ 7 states that every hyperliteral has a certain axiu degree of credibility. 7. The forula T (λ,κ) (p) codes soe of the finer aspects of the behavior of the negations of the degree of credibility λ with respect to a given degree of credibility κ. Axio τ 8 states that if a forula has axiu credibility κ, then its negations will behave like soe given degree of credibility λ with respect to κ. 3 Nice Annotated Matrices In this Section a special class M τ of SAL τ atrices is defined; these atrices are a slight siplification of the ones that were introduced in [13], where we proved that they for a atrix seantics for the systes SAL τ. We first observe that since τ is finite, all its ideals are principal, that is, if I is an ideal, then for soe κ τ, I = {λ τ : λ κ}. This ideal will be denoted by I κ. For I ω and 0, 1 / τ define L = (I τ ) {0, 1}, and for any I indexed faily κ i : i I of eleents of τ, let D = i I ({i} I κi ) {1}. The set M τ of nice atrices is M τ = {M = L, D : I ω, κ i : i I }, where L and D are defined as above and the operations of L are defined as follows. { 1 if a D or b D, a b = 0 otherwise. { 1 if a D and b D, a b = 0 otherwise. 7

8 { 1 if a / D or b D, a b = 0 otherwise. a = f λ a = { 0 if a I τ, 1 if a {0, 1}. { i, λ if a = i, µ, a if a {0, 1}. 0 if a = 1, a = 1 if a = 0, i, µ if a = i, µ. A special case of these atrices is when I = {i} is a singleton. In this case we siply drop the ordered pairs and identify {i} τ with τ. We call these eleentary atrices. The ain theore proven in [13] is the following copleteness theore. Theore 1. Γ A if and only if Γ M τ A. 4 Reduced Nice Matrices Reduced atrices play an iportant role in the study of the classes of algebras that arise in the algebraization process. A well known result is that for any algebraizable syste S whose class of reduced S atrices is M, the equivalent quasivariety seantics Q is Q = {A : A, F M, for soe S filter F }, that is, the algebras in the equivalent algebraic seantics are the algebra reducts of the reduced atrices. There is one further interesting result that we will use in this section, Theore I.14 in [6]. Using the sae notations of the previous paragraph this theore ay be stated as follows: if S is algebraizable and K M is a 8

9 atrix seantics for S, (the wording in [6] is K is strongly adequate for S,) then M = SPP U (K), where S, P and P U are the usual subatrix, direct product and ultraproduct atrix theoretic operators. This theore states that M is the sallest atrix quasivariety containing K. 4.1 The Leibniz Operator One of the ain tools in algebraic logic is the so called Leibniz operator, which is extensively studied in [3, 5, 9, 10], and other places. We give here the ain definitions and properties for future reference. For any algebra A and D A, we define the Leibniz congruence relation on A over D Ω A (D) = { a, b : ϕ A (a, c 1,..., c n ) D iff ϕ A (b, c 1,..., c n ) D, for any forula ϕ(x, y 1,..., y n ) F and c 1,..., c n A}. The function Ω A whose doain is P(A) is called the Leibniz operator on A. The ost useful characterization of the Leibniz operator is probably Theore 1.5 in [3]. We say that the congruence θ on an algebra A is copatible with a subset F of A, if a θ b and b F, then a F. Theore 2. The Leibniz operator Ω A on A assigns to each X A the largest congruence Ω A (X) of A that is copatible with X. 4.2 Reduced Matrices A atrix M = A, F is reduced if Ω A (F ) is the identity relation on A. Lea 3. Let M = L, D M τ, where D = I κ {1}, for soe κ τ, be an eleentary atrix. Then the following are equivalent, for all µ, ν τ. 1. For all < s + t, µ κ if and only if ν κ, 2. T L (µ,κ) (ν) = 1, 3. T L (ν,κ) (µ) = 1. 9

10 Proof. If for all < s + t, µ κ if and only if ν κ, then T (µ,κ) (p) = T (ν,κ) (p). Also, for all < s + t, µ D if and only if ν D, so T(µ,κ)(ν) L = 1 = T(ν,κ)(µ). L This proves that 1 iplies 2 and 3. On the other hand, assuing T L (µ,κ)(ν) = 1, the first part of the forula T (µ,κ) (p) iplies that for all < s + t, if µ κ, then ν D, so ν κ. The second part of the forula T (µ,κ) (p) iplies that for all < s + t, if µ κ, then ( ν ν) D and since ν ν is either 0 or 1, this is equivalent to ν ν = 0, or equivalently, ν / D or ν κ. This proves that 2 iplies 1. The equivalence of 1 and 3 is proven siilarly. Theore 4. Let M = L, D M τ, where D = i I ({i} I κ i ) {1}. Then a = b or a Ω L (D) b iff a = i, µ, b = j, ν, κ i = κ j = κ for soe κ and T L (µ,κ)(ν) = 1 = T L (ν,κ)(µ). Proof. It is straightforward that the relation Θ defined on the right hand side is an equivalence relation that preserves the binary connectives and the unary connectives f λ and. The preservation of negations is obtained by the condition T L (µ,κ)(ν) = 1 = T L (ν,κ)(µ), which states that the negations of µ and ν have the sae behavior with respect to κ. See Reark 2., 7. The condition κ i = κ j = κ iplies I κi = I κj which insures copatibility with D, since by Lea 3 and the definition of D i, µ D iff µ I κ iff ν I κ iff j, ν D. To prove that the congruence Θ defined on the right hand side above is the largest congruence copatible with D, let θ be any other congruence copatible with D. If i, µ θ j, ν, then i, κ i = f κi i, µ θ f κi j, ν = j, κ i 10

11 and since i, κ i D, by copatibility, j, κ i D, so κ i κ j. Siilarly κ j κ i, so κ i = κ j, and thus I κi = I κj = I κ for soe κ. Finally since i, µ θ j, ν, we get i, µ θ j, ν, so by copatibility, µ I κ if and only if ν I κ, which by Lea 3 is equivalent to T L (µ,κ) (ν) = 1 = T L (ν,κ) (µ). This proves that θ Θ, so the latter is axial copatible with D and thus equals Ω L (D). In order to siplify notations, we will first build the eleentary reduced atrices. So let us consider the atrix M = A, D κ = 2 τ, I κ {1}. If we let [a] κ be the class of a odulo Ω A (D κ ) and τ κ the set of equivalence classes, then [0] κ = {0}, [1] κ = {1}, [λ] κ = {µ τ : T A λ,κ(µ) = 1}. τ κ should always be understood as associated to the obvious reduced ideal, naely, I κ = {[λ] κ : λ κ} and D κ = I κ {[1] κ }. So the reduction M of atrix M is (isoorphic to) M = A, D κ = 2 τ κ, D κ. We let 2 κ be the algebra reduct of M. Again for notational convenience, we will assue that for different κ, the corresponding τ κ are disjoint. The following theore follows easily fro Theore 4 if we observe that if a atrix contains repeated copies of the sae ideal of τ, then these copies are identified under the Leibniz congruence. Theore 5. Let L, D M τ and let I κ1,..., I κ be all the distinct ideals that appear in D. Then the reduced atrix L, D = L/Ω L (D), D/Ω L (D) 11

12 is such that and L = {0, 1} i=1 τ κi D = Iκ i {1}. i=1 We let 2 κ1,...,κ be the algebra reduct of L, D. If = τ, that is, all possible ideals appear in D, we let 2 τ = 2 κ1,...,κ τ. Corollary 6. There are 2 τ reduced atrices in M τ. Proof. Since all ideals are principal, there are as any ideals as eleents in the lattice, so there are as any reduced atrices as subsets of τ. The last corollary provides us with a decision procedure for annotated logics: we just have to check 2 τ finite atrices. As a atter of fact, we will see in the next theore that we only need to check a single atrix. Let us consider the reduced atrix M whose algebra reduct is 2 τ. The following lea is straightforward. Lea 7. Every reduced atrix in M τ is a subatrix of M. Theore 8. The class {M} is a atrix seantics for syste SAL τ, that is, Γ A if and only if Γ M A. 4.3 Two Exaples We will illustrate the process deterining the reduced nice atrices for two classes of annotated logics which appear in the literature. Exaple 1. Let τ = {0, 1, 2,..., 1, 1} with the usual order and let be defined by a = 1 a, that is, τ is the ( + 1) eleent chain with Lukasiewicz negation. There are two cases, the first of which is when is even, that is τ is odd. In this case 1 τ. Then there are four possibilities κ = 1. Then for all λ τ, [λ] 1 = τ, so τ 1 = {[0] 1 } = I 1. 12

13 2. κ = j > 1. Then there are three equivalence classes. 2 so [0] j [ j ] j [1] j = {0, 1,..., j 1 }, = { j,..., j } = [1 2 ] j, = { j + 1,..., 1}, τ j = {[0] j, [ 1 2 ] j, [1] j } and I j = {[0] j, [ 1 2 ] j }. 3. κ = 1 2. so [0] 1 2 [ 1 2 ] 1 2 [1] 1 2 = {0, 1,..., 2 2 }, = { 1 2 }, = { + 2 2,..., 1}. τ 1 2 = {[0] 1, [ ] 1, [1] 1 } and I = {[0] j, [ 1 2 ] j }. 4. κ = j < 1 2. so [0] j [ j + 1 ] j [1] j = {0, 1,..., j }, = { j + 1,..., j 1 = { j,..., 1}, } = [ 1 2 ] j, τ j = {[0] j, [ 1 2 ] j, [1] j } and I j = {[0] j }. 13

14 One should observe that for κ 1, even though as ordered sets the τ κ s are isoorphic, the operations in particular the f λ s are not defined in the sae way. For exaple, if > 2, for any x f 1 [x] 1 2 = [ 1 ] 1 2 = [0] 1 2 while f 1 [x] 0 = [ 1 ] 0 = [ 1 2 ] 0. The second case is when is odd. Then 1 / τ and there are two central 2 points, naely, 1 +1 and. 2 2 If κ > 1 1 or κ <, τ 2 2 κ is like in the first case. The only difference arises when κ = 1 for then 2 so [0] 1 2 [1] 1 2 = {0, 1,..., 1 2 }, = { + 1 2,..., 1}, τ 1 2 = {[0] 1 2, [1] 1 } and I = {[0] 1 }. 2 In the first case 2 τ = {0, 1} τ j = = 3( + 1), so the algebra 2 τ has 3( + 1) eleents. In the second case the algebra 2 τ has eleents. In case = 2, then τ = {0, 1, 1} and 2 2 τ has 9 eleents, naely {1, 0} {0 1, 0 a, 0 0, a a, a 0, 1 a, 1 0 } and D = {1, 0 1, 0 a, 0 0, a a }, where in order to siplify notation we use a instead of 1 and we drop the brackets. The 2 following is a table of the iplication and of the unary operations. The first five eleents belong to D. 14

15 a 0 0 a a a 0 1 a f 0 f a f a a a 0 a a a 1 a a a a a a 0 a a a 1 a 0 a a a a a a 0 a a a 1 a a Exaple 2. Let τ be the bilattice F OUR = {, f, t, }. F OUR is ordered as the four eleent Boolean algebra with greatest eleent and least eleent. The negation is defined by =, f = t, t = f, =. 1. κ =. Then for all λ τ, [λ] = τ, so τ = {[ ] } = I κ = t or κ = f. Then for all λ τ, [λ] t = {λ} = [λ] f, so τ t = {[ ] t, [t] t, [f] t, [ ] t }, and I t = {[ ] t, [t] t }. and τ f = {[ ] f, [t] f, [f] f, [ ] f }. and I f = {[ ] f, [f] f }. Observe that in these two cases the lattices τ κ with their associated ideals I κ are isoorphic to τ with the ideal I κ. 3. κ =. Then so [ ] = { }, [ ] = {, t, f}, τ = {[ ], [ ] } and I = {[ ] }. In this exaple the algebra 2 τ has 13 eleents. 15

16 5 The Quasi variety SAL τ of SAL τ algebras In this Section we will describe the quasivariety of SAL τ algebras and find soe of its properties. The theory of algebraization of deductive systes developed in [3] provides us with a standard axioatization for the quasivariety that arises in the process. This is the content of Theore 2.17 in that onograph. In order to apply these results in our case, we observe that SAL τ is an axioatic extension of the algebraizable syste SPτ, so it is algebraizable too. Moreover, the sae defining equations and equivalence forulas used in [11] for the algebraization of systes SPτ can be used for SAL τ. These are the following. The single defining equation is p p p p and the equivalence forulas are (p, q) = p q, k (p, q) = k p k q, for 0 k < t + s, λ (p, q) = f λ p f λ q, for all λ τ. If we let J = { } {0, 1,..., t + s 1} τ, we write p q as an abbreviation of the conjunction j J j(p, q) of the equivalence forulas. We will introduce two new constant sybols to the language, 1 to stand for the class of all SAL τ theores and 0 to stand for the class of all negations of a SAL τ theore. Theore 9. The class of all SAL τ algebras is the quasivariety of all algebras A = A,,,,,, f λ, 0, 1 λ τ axioatized by the following identities σ 1 (1) for every axio σ of SAL τ, together with the following quasi identities p p 1 & p q 1 q q 1 (2) p f λ q 1 & p f µ q 1 x f λ µ q 1 (3) p q 1 p q. (4) Proof. It should be noted that this axioatization is an obvious siplification of the one obtained by direct application of [3], Theore 2.17 taking into account that if σ is coplex then (σ σ) σ. 16

17 Given an algebra A SAL τ, it is iediate that if we let B = {b A : b = 1}, B is a Boolean algebra with unary operations defined by f λ (b) = b and b = 1, for all b B and all λ τ. B will be referred to as the Boolean part of A. We will let H = {b A : b 1} and call it the hyperliteral part of A, thus any SAL τ algebra is of the for A = B H,,,,,, f λ, 0, 1 λ τ. We will see that there is a close connection between the Boolean and hyperliteral parts. The next theore is the ost iportant application of the theory to the reduced nice atrices of the previous Section. It is a consequence of the general theoretical rearks at the beginning of Section 4 and in Theore 8. Theore 10. The class SAL τ 5.1 SAL τ is not a Variety of SAL τ algebras is given by SAL τ = SP({2 τ }). Theore 11. The class of all SAL τ algebras is not a variety. Proof. Let us consider the reduced algebra 2 whose universe is 2 {a}, for which the operations are defined as follows: 0 a a a a a a f λ a 0 a Let B = {α, };,,,,, f λ,, with operations defined by α α Finally define α α α α f : 2 B 0 a α 1 17 f λ α α

18 Then f is a hooorphis, so B is a hooorphic iage of 2 SAL τ but it is not a SAL τ algebra since it fails to verify axio (4). 5.2 Subdirectly Irreducibles in SAL τ Theore 12. A SAL τ algebra is subdirectly irreducible if and only if its Boolean part is 2. Proof. Let A be a subdirectly irreducible SAL τ -algebra. One ust observe that if θ is a Boolean congruence on the Boolean algebra B, then it is also an absolute SAL τ congruence on B considered as a SAL τ algebra. Moreover, if we define ˆθ = θ A, where A is the identity on the universe A of A, then ˆθ is an absolute SAL τ congruence on A. Now suppose that the Boolean part B 2. Then there are two non trivial Boolean congruences θ 1 and θ 2 such that θ 1 θ 2 = B. But then ˆθ 1 ˆθ 2 = A, so A is not subdirectly irreducible, a contradiction, so B = 2. Suppose B = 2. We will prove that the SAL τ congruence Θ = B B A is a onolith for the lattice of congruences of A. Let θ be any non-trivial SAL τ congruence and let a θ b, with a b. If both a, b B, then Θ θ. If a B and b H, since 1 = a θ b = 0, again Θ θ. If both a, b H and we assue that a b = 0, then 1 = a a θ a b = 0 and thus Θ θ. A siilar arguent shows that if we assue that either f λ a f λ b = 0, for soe λ τ or that k a k b = 0, for soe k ω, then Θ θ. So the only case left is when for all λ τ, f λ a f λ b = 1 and for all k ω, k a k b = 1. Since for a, b H we also have a b = 1, we ay conclude that a b = 1, so by axio (4), we have that a = b, a contradiction, so this case is not possible and Θ is a onolith for the lattice of congruences of A. Let A SAL τ have universe A = B H. Given any eleent a A, a a B. As a atter of fact, even though B could be larger, at least it contains the Boolean algebra generated by {a a : a H}. In this sense, the Boolean part of A cannot be too sall with respect to H. The iplications of this fact can be ost easily seen in algebras whose Boolean part is 2. 18

19 Theore 13. Let A be a SAL τ algebra whose Boolean part is 2. Then A is isoorphic to a subalgebra of a reduced SAL τ algebra 2 κ1,...,κ. Proof. Theore 10 states that SAL τ = SP({2 τ }), so the Boolean eleents of A ay always be thought of as tuples of zeros and ones and the hyperliteral eleents of A as tuples of zeros, ones and eleents of τ. Let A be a subalgebra of Π i I 2 τ whose Boolean part B is 2. If for soe i, j I, a(i) τ κ for soe κ and a(j) {0, 1}, then a (i) = 0 and a (j) = 1. This contradicts the fact that a B = {0, 1}. So either for all i I a(i) {0, 1}, or for all i I a(i) τ κ for soe κ. Assue that for i, j I, a(i) τ κi and a(j) τ κj, with κ j κ i, say κ j κ i. Then f κj a(i) / Iκ i but f κj a(j) Iκ j, and thus f κj a(i) f κj a(i) = 0 and f κj a(j) f κj a(j) = 1, and thus f κj a f κj a / B, which is a contradiction. So for any a H, for all i I, a(i) τ κ for a unique κ. This eans that H is contained in a union of powers of τ κ s. Assue now that for i, j I, a(i) a(j). Then for soe < t + s, a(i) I κ, but a(j) / I κ, so a(i) a(i) = 1 and a(j) a(j) = 0, and thus a a / B, which again is a contradiction. So if we let then A is a subalgebra of diag(τ I κ) = {a : for all i, j I, a(i) = a(j)}, B = 2 κ F diag(τ I κ), where F τ. To finish our proof, taking F = {κ 1,..., κ }, we siply have to point out that fixing one i I, the apping is a onoorphis. f : B 2 κ1,...,κ a a(i), for a H, 19

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