internalizing equality in boolean algebras

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Algebra univers. 43 (2000) 187 – 195 0002–5240/00/030187 – 09 $ 1.50 + 0.20/0 © Birkh¨ auser Verlag, Basel, 2000 Internalizing equality in Boolean algebras Desmond Fearnley-Sander and T. Stokes Abstract. We show that equality may be internalized in Boolean algebras, in a number of possible ways, as a binary operation satisfying reflexivity and replacement properties. The variety of equality Boolean rings is shown to be equivalent to the variety of modal rings (Boolean rings endowed with a generalised interior operator and important in modal logic). Varying the strength and exact nature of the replacement property corresponds to selecting from a number of natural varieties of modal rings. The work generalises a result of Suszko who considered the S4 case. In propositional logic, the bi-conditional operation internalizes the notion of logical equivalence. In modal logic various notions of “known truth” are captured, including necessity and provability. We focus here on a common generalisation of these ideas in which “known equivalence” is captured via a binary operator that we call an internal equality. We view every Boolean algebra as a Boolean ring and vice versa under the usual cor- respondence. Meet will usually be denoted by juxtaposition. Thus if R is a Boolean algebra/ring, a 0 = a + 1 is the complement of a R; a b = a + b + ab is the join of a and b; a + b = (a b)(a 0 b 0 ), a b = a + ab + 1 and a b = a + b + 1 for all a,b R. R is given the usual partial order: a b if and only if ab = a. Thus any inequality in a Boolean algebra/ring can be expressed as an equation. 1. Equality algebras In Fearnley-Sander and Stokes, [5], as part of an attempt to capture algebraically the most fundamental properties of reasoning involving equality, we introduced the notion of an equality algebra and obtained various representations of them using congruences. An equality algebra (A, L, = i ) is a universal algebra A together with a semilattice (L, ·, 1) and a binary function = i : A × A L such that, for all x,y A, 1. (x = i x) = 1; and 2. (x = i y)f(x) = (x = i y)f(y), for all functions f : A L derived from the operations on A, the semilattice operations and = i , and in which the constants from A and L may be used. We call these respectively, the reflexivity rule and the replacement rule. Presented by Professor Wim Blok. Received May 26, 1998; accepted in final form September 15, 1999. 187

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Algebra univers. 43 (2000) 187 – 1950002–5240/00/030187 – 09 $ 1.50+ 0.20/0© Birkhauser Verlag, Basel, 2000

Internalizing equality in Boolean algebras

Desmond Fearnley-Sander and T. Stokes

Abstract. We show that equality may be internalized in Boolean algebras, in a number of possible ways, as a binaryoperation satisfying reflexivity and replacement properties. The variety of equality Boolean rings is shown to beequivalent to the variety of modal rings (Boolean rings endowed with a generalised interior operator and importantin modal logic). Varying the strength and exact nature of the replacement property corresponds to selecting froma number of natural varieties of modal rings. The work generalises a result of Suszko who considered the S4 case.

In propositional logic, the bi-conditional operation⇔ internalizes the notion of logicalequivalence. In modal logic various notions of “known truth” are captured, includingnecessity and provability. We focus here on a common generalisation of these ideas in which“known equivalence” is captured via a binary operator that we call aninternal equality.

We view every Boolean algebra as a Boolean ring and vice versa under the usual cor-respondence. Meet will usually be denoted by juxtaposition. Thus ifR is a Booleanalgebra/ring,a′ = a + 1 is the complement ofa ∈ R; a ∨ b = a + b + ab is the join ofa andb; a + b = (a ∨ b)(a′ ∨ b′), a ⇒ b = a + ab + 1 anda ⇔ b = a + b + 1 forall a, b ∈ R. R is given the usual partial order:a ≤ b if and only if ab = a. Thus anyinequality in a Boolean algebra/ring can be expressed as an equation.

1. Equality algebras

In Fearnley-Sander and Stokes, [5], as part of an attempt to capture algebraically themost fundamental properties of reasoning involving equality, we introduced the notion ofanequality algebraand obtained various representations of them using congruences. Anequality algebra(A, L, =i ) is a universal algebraA together with a semilattice(L, ·, 1) anda binary function=i : A × A → L such that, for allx, y ∈ A,

1. (x =i x) = 1; and2. (x =i y)f (x) = (x =i y)f (y),

for all functionsf : A → L derived from the operations onA, the semilattice operationsand=i , and in which the constants fromA andL may be used. We call these respectively,thereflexivity ruleand thereplacement rule.

Presented by Professor Wim Blok.Received May 26, 1998; accepted in final form September 15, 1999.

187

188 desmond fearnley-sander and t. stokes algebra univers.

Here we consider the important special case in whichL is a Boolean algebra whichis an equality algebra taking values in itself. The case whereL is merely a semilatticeis more easily dealt with, but does not have the interesting connections with modal logicthat we show arise. The ambiguity in the phrase “a Boolean algebra which is an equalityalgebra taking values in itself” allows various distinct systems with equality to be defined,corresponding, as we show, to particular systems of importance in modal logic. Our papermay be thought of as generalising work of Suszko, [11], in which the important S4 Booleancase (the system with the most structure dealt with in what follows) is considered.

The following result, appearing in [5], will be useful.

THEOREM 1.1. Let A be a universal algebra,L a semilattice, and=i : A × A → L.Then(A, L, =i ) is an equality algebra if and only if, for alla, b, c ∈ A,

1. (a =i a) = 1;2. (a =i b) = (b =i a);3. (a =i b)(b =i c) ≤L (a =i c);

and for any basicn-ary operationρ onA and anya1, b1, . . . , an, bn ∈ A,

(a1 =i b1) · · · (an =i bn) ≤L (ρ(a1, . . . , an) =i ρ(b1, . . . , bn)).

2. Equality Boolean rings

The first and most general case we consider is that in which the Boolean ringR hasdefined on it an internal equality=i taking values in the semilattice(R, ·, 1). Examining thedefinition of an equality algebra, this says thatR and=i satisfy the following:(a =i a) = 1for all a ∈ R, and(a =i b)f (a) = (a =i b)f (b), wheref (x) is a derived unary operationonR which is a product of internal equations of the form(g(x) =i h(x)) whereg(x), h(x)

are derived unary Boolean operations onR; so it suffices to require only thatf itself besuch an internal equation.

Thus we say that the Boolean ringR endowed with binary operation=i is anequalityBoolean ring, or anEB-ring, if, for all a, b ∈ R,

1. (a =i a) = 1 and2. (a =i b)(g(a) =i h(a)) = (a =i b)(g(b) =i h(b)) for all derived unariesg, h

defined in terms of the Boolean ring operations onR.

We call =i an internal equalityoperation and an element ofR of the form(a =i b) aninternal equation.

It is immediate from Theorem 1.1 that the Boolean ringR with additional binaryoperation=i is an EB-ring if and only if, for alla, b, c, d ∈ R, (a =i a) = 1, (a =i b) =

Vol. 43, 2000 Internalizing equality in Boolean algebras 189

(b =i a), (a =i b)(b =i c) ≤ (a =i c), (a =i b)(c =i d) ≤ (ab =i cd) and(a =i b)(c =i d) ≤ (a + b =i c + d).

An EB-ringR is type K4if a stronger replacement rule holds: namely, for alla, b ∈ R,

(a =i b)(g(a) =i h(a)) = (a =i b)(g(b) =i h(b))

for all derived unariesg, h defined on the EB-ringR.Again, this is equivalent to saying that(R, +, ·, =i ) is an(R, ·)-valued equality algebra,

and so from Theorem 1.1,R is a type K4 EB-ring if and only if it is an EB-ring in which(a =i b)(c =i d) ≤ ((a =i c) =i (b =i d)) for all a, b, c, d ∈ R. There is a morecompact way to express this additional condition.

PROPOSITION 2.1.R is a type K4 EB-ring if and only if it is an EB-ring in which(a =i b) ≤ ((a =i b) =i 1) for all a, b ∈ R.

Proof. If R is type K4 then lettingb = c = d in (a =i b)(c =i d) ≤ ((a =i c) =i

(b =i d)) gives (a =i b) ≤ ((a =i b) =i 1). Conversely, if this latter condition issatisfied in an EB-ring, then using the replacement rule for EB-rings and the facts that(a =i b) = (a =i b)((a =i b) =i 1) and(c =i d) = (c =i d)((c =i d) =i 1), we have

(a =i b)(c =i d)((a =i c) =i (b =i d))

= (a =i b)(c =i d)((a =i b) =i 1)((c =i d) =i 1)((a =i c) =i (b =i d))

= (a =i b)(c =i d)((a =i b) =i 1)((c =i d) =i 1)

((a =i b)(a =i c) =i (c =i d)(b =i d))

= (a =i b)(c =i d)((a =i b) =i 1)((c =i d) =i 1)

((a =i b)(b =i c) =i (c =i d)(b =i c))

= (a =i b)(c =i d)((a =i b) =i 1)((c =i d) =i 1)((b =i c) =i (b =i c))

= (a =i b)(c =i d),

so(a =i b)(c =i d) ≤ ((a =i c) =i (b =i d)). ¨

Finally, an EB-ringR is type S4if a still stronger replacement rule holds, namely: forall a, b ∈ R,

(a =i b)f (a) = (a =i b)f (b)

for all derived unariesf defined on the EB-ringR.

THEOREM 2.2. The EB-ringR is type S4 if and only if it is type K4 and the epistemicrule a(a =i b) = b(a =i b) holds for alla, b ∈ R.

190 desmond fearnley-sander and t. stokes algebra univers.

Proof. The given rule evidently follows from the strong replacement rule by lettingf (x) = x. Conversely, assuming the epistemic rule holds in the type K4 EB-ringR,let f (x) be any derived unary operation onR as an EB-ring. Thenf (x) can be writtenas a sum of products of the variablex with internal equations and elements ofR, so bydistributivity it suffices to show that replacement holds for any such summand. Thus letf (x) = xαe1(x)e2(x) · · · em(x), where eachej (x) = (gj (x) =i hj (x)), gj (x), hj (x)

derived unaries on the EB-ringR, andα ∈ R. Then for alla, b ∈ R and for eachj , wehave from the fact thatR is type K4 that

(a =i b)ej (a) = (a =i b)(gj (a) =i hj (a)) = (a =i b)(gj (b) =i hj (b))

= (a =i b)ej (b).

Hence

(a =i b)f (a) = (a =i b)aαe1(a)e2(a) · · · em(a)

= (a =i b)bαe1(b)e2(b) · · · em(b) by the epistemic rule= (a =i b)f (b),

as desired. ¨

3. Connection with modal systems

What we shall call amodal ringis a Boolean ring together with a unarymodal operation[ ] satisfying the rules [1]= 1 and [a][b] = [ab] for all a, b ∈ R. These are the Boolean ringversions of the “normal modal algebras” defined by Lemmon in [7], and are among the mostgeneral (models of) modal systems considered, corresponding to the modal logic system K.

In modal logic, [a] is interpreted as the proposition “a is necessarily true”, although otherinterpretations are more appropriate in some circumstances; for instance in Boolos, [1], theinterpretation “a is provable” is the relevant one, a case we return to later.

A modal ringR in which [a] ≤ [[a]] for all a ∈ R is of type K4; see Hughes andCresswell, [6], for instance.

One subvariety of the type K4 modal rings is of particular interest.R is said to be oftype S4if it is of type K4 and satisfies the rule [a] ≤ a for all a ∈ R (theepistemic rulein [7]). It follows that S4 modal rings satisfy [[a]] = [a] for all a ∈ R. In the literature,modal rings of type S4 have attracted great interest, partly because of their link to topology(see McKinsey and Tarski, [8]), but modal rings of type T, satisfying the condition [a] ≤ a

for all a ∈ R but not necessarily the type K4 condition are also a focus of attention. Modalrings of type S4 are calledtopological Boolean ringsby Rasiowa in [10], and are the dualsof closure algebras, first introduced by McKinsey and Tarski in [8] in a topological context.It was shown in [8] that every modal ring is a modal subring of the modal ring of all subsetsof a topological space in which the modal operation is the usual interior operator.

Vol. 43, 2000 Internalizing equality in Boolean algebras 191

In [11], Suszko in effect showed that the variety of S4 modal rings is term equivalent tothe variety of type S4 EB-rings, under the correspondences(a =i b) ↔ [a + b + 1] and[a] ↔ (a =i 1). In fact this term equivalence is a special case of a more general equivalencewhich has other special cases that are equally interesting. First we need a preliminary result.

PROPOSITION 3.1.If (R, =i ) is an EB-ring then for alla, b, c ∈ R, the followingthree conditions hold:

1. (a =i a) = 1;2. (a =i 1)(b =i 1) = (ab =i 1) and3. (a =i b) = (a + c =i b + c).

Proof. Suppose that(R, =i ) is an EB-ring. The first condition holds immediately. Forthe second, for alla, b ∈ R, (a =i 1)(b =i 1) ≤ (ab =i 1), and conversely

(ab =i 1) = (ab =i 1)(a =i a) ≤ ((ab) ∨ a =i 1 ∨ a) = (a =i 1),

and similarly(ab =i 1) ≤ (b =i 1), so (ab =i 1) ≤ (a =i 1)(b =i 1), establishingthat (ab =i 1) = (a =i 1)(b =i 1) for all a, b ∈ R. Finally, for all a, b, c ∈ R,(a =i b) = (a =i b)(c =i c) ≤ (a + c =i b + c)(c =i c) ≤ (a =i b), proving the thirdcondition. ¨

THEOREM 3.2. The varieties of modal rings and of EB-rings are term equivalent underthe correspondences(a =i b) ↔ [a + b + 1], [a] ↔ (a =i 1).

Proof. In an EB-ringR, defining [a] = (a =i 1) for all a ∈ R, we have that(a =i b) =[a + b + 1] for all a, b ∈ R, and it is immediate from the previous proposition thatR is amodal ring under the derived modal operation.

Conversely, in a modal ringR, defining (a =i b) = [a + b + 1], we have that[a] = (a =i 1). Furthermore,(a =i a) = [a + a + 1] = 1, (a =i b) = [a + b + 1] =[b + a + 1] = (b =i a), and for alla, b, c, d ∈ R,

(a =i b)(c =i d)(a + b =i c + d)

= [a + b + 1][c + d + 1][a + b + c + d + 1]

= [(a + b + 1)(c + d + 1)(a + b + c + d + 1)]

= [(ac + ad + a + bc + bd + b + c + d + 1)(a + b + c + d + 1)]

= [(ac + ad + a + bc + bd + b + c + d + 1)]

= [(a + b + 1)(c + d + 1)]

= (a =i b)(c =i d)

192 desmond fearnley-sander and t. stokes algebra univers.

as a routine computation shows, and similarly

(a =i b)(c =i d)(ab =i cd)

= [a + b + 1][c + d + 1][ab + cd + 1]

= [(a + b + 1)(c + d + 1)(ab + cd + 1)]

= [(a + b + 1)(c + d + 1)]

= (a =i b)(c =i d).

Hence(a =i b)(c =i d) ≤ (a + b =i c + d) and(a =i b)(c =i d) ≤ (ab =i cd). Thus(R, =i ) is an EB-ring. ¨

Thus modal rings and EB-rings are essentially the same things. Note that the aboveproof shows that the three conditions in Proposition 3.1 characterise EB-rings.

The properties of being type K4 and type S4 correspond to their modal ring namesakes.The type S4 case is essentially the result of Suszko, shown in [11].

THEOREM 3.3. LetR be an EB-ring. Then

1. (R, =i ) is type K4 if and only if(R, [ ]) is of type K4, and2. (R, =i ) is type S4 if and only if(R, [ ]) is of type S4.

Proof. If (R, [ ]) is of type K4 then

(a =i b)(c =i d)((a =i b) =i (c =i d))

= (a =i b)(c =i d)[(a =i b)][(c =i d)][(a =i b) + (c =i d) + 1]

= (a =i b)(c =i d)[(a =i b)(c =i d)((a =i b) + (c =i d) + 1]

= (a =i b)(c =i d)[(a =i b)(c =i d)]

= (a =i b)(c =i d)[(a =i b)][(c =i d)]

= (a =i b)(c =i d)

so(R, =i ) is type K4. Conversely, if(R, =i ) is type K4 then certainly it is type K4 as anEB-ring and so by Proposition 2.1 [a] = (a =i 1) ≤ ((a =i 1) =i 1) = [[a]], so R is oftype K4 as a modal ring.

If (R, =i ) is type S4 then it is type K4, so(R, [ ]) is of type K4, but also,a[a] =a(a =i 1) =1(a =i 1) = [a] for all a ∈ R, so(R, [ ]) is of type S4. If(R, [ ]) is of S4 type,then it is of type K4, so(R, =i ) is type K4. Furthermore, for alla, b ∈ R, a(a =i b) =a[a+b+1] = a(a+b+1)[a+b+1] = (a+ab+a)[a+b+1] = a[ab] = ab[a+b+1],so alsob(a =i b) = ab[ab] and so(R, =i ) is type S4 by Theorem 2.2. ¨

SupposeR is type S4. Then the set of all internal equations is the setPR = {[a]|a ∈ R},and is the set of elements ofR fixed under the modal operation. It is easily checked thatPR

Vol. 43, 2000 Internalizing equality in Boolean algebras 193

is a sublattice ofR viewed as a Boolean algebra. In fact,PR is a Heyting algebra in whichpseudo-implication is given bya → b = [a + ab + 1] for all a, b ∈ PR; moreover everyHeyting algebra is isomorphic toPR for some S4-modal ringR, as is shown in [9].

A further natural class of equality rings is the subclass of the type K4 modal rings whichwe call type Pmodal rings. To see how this class arises, we refer to [1], where a modallogic of provability is considered. The rules used can be phrased for Boolean rings asfollows: [1] = 1, [a] ≤ [[a]], [a ⇒ b] ≤ [a] ⇒ [b] and [[a] ⇒ a] ≤ [a]. Now therule [a][b] = [ab] follows from the third of these (for a proof see [6]), and can be easilyshown to imply it. The fourth, re-statable as [a][[ a] + [a]a + 1] = [a] and called theLobrule in [1], simplifies down to [a] = [[a] + a[a] + 1] upon using the rule [a][b] = [ab].Thus we define a type P modal ring to be a modal ring of type K4 in which additionally[a] = [[a] + a[a] + 1] holds. Such modal rings can be described neatly in terms of theequality operation. Let us say that an EB-ring istype Pif it is type K4 and satisfies the rule(a(a =i b) =i b(a =i b)) = (a =i b).

PROPOSITION 3.4.(R, [ ]) is of type P if and only if(R, =i ) is type P.

Proof. If (R, [ ]) is of type P then(R, =i ) is type K4 and

(a(a =i b) =i b(a =i b)) = [a[a + b + 1] + b[a + b + 1] + 1]= [(a + b + 1)[a + b + 1] + [a + b + 1] + 1]= [a + b + 1]= (a =i b).

Conversely, if(R, =i ) is type K4 and satisfies(a(a =i b) =i b(a =i b)) = (a =i b), then(R, [ ]) is of type K4, but additionally, for alla ∈ R,

[[a] +a[a] + 1] = [(a =i 1) + a(a =i 1) + 1] = (1(a =i 1) =i a(a =i 1))

= (a =i 1) = [a],

so(R, [ ]) is of type P. ¨

Let R be an EB-ring. Note that [a] = a holds for alla ∈ R if and only if (a =i b) =a ⇔ b for all a, b ∈ R. Indeed we have the following

THEOREM 3.5. LetR be an EB-ring. The following are equivalent.

1. [a] = a for all a ∈ R.2. R is type S4 and=i is associative.3. The epistemic rule is satisfied and=i is associative.

194 desmond fearnley-sander and t. stokes algebra univers.

Proof. The first condition implies the second because(R, ⇔) is certainly type S4 by thesecond part of Theorem 3.3, and⇔ is associative. Clearly the second condition implies thethird. Finally, if =i is associative, then

((a =i 1) =i a) = ((a =i a) =i 1) = (1 =i 1) = 1

so if R satisfies the epistemic rulea(a =i b) = b(a =i b)), then

a = a((a =i 1) =i a) = (a =i 1)((a =i 1) =i a) = (a =i 1) = [a]

for all a ∈ R; thus the third condition implies the first. ¨

4. Conclusion

We should now say something about our original motivation. We had believed thatthe notion of provability could be captured using the type S4 replacement rule, and weindependently re-discovered Suszko’s correspondence between S4 modal rings and whatwe once called replacement rings (here called type S4 EB-rings). But then we read [1] andhad to re-think: as Boolos makes clear, S4 modal logic, and hence the strongest possiblereplacement rule, do not model provability at all. But it turns out that a weaker version ofreplacement does hold in the “logic of provability” (and the epistemic rulea(a =i b) =b(a =i b) is replaced by the Lob rule(a =i b) = (a(a =i b) =i b(a =i b))). Moregenerally, a form of replacement rule underlies some of the most important modal systems.

We believe (see Bulmer, Fearnley-Sander and Stokes, [2], Fearnley-Sander, [3] andFearnley-Sander, [4]) that internalization of equality has significance that goes beyond itsintrinsic interest in algebra and modal logic. It allows the development of a formal theoryof reasoning, quite different from classical logic, in which equality is a fundamental oper-ation on a par with conjunction. In particular it offers immediate support for computation,since fundamental algorithms such as the Knuth-Bendix, Gauss-Jordan and Grobner basealgorithms may be incorporated at an algebraic level: they are simply reductions to normalform of conjunctions of internal equations. Moreover, it is compatible with a typing disci-pline and enables reasoning about cases (or branching programs) via what we have calledBoolean affine combinations(see [2]).

Acknowledgements

The work reported here was supported by Australian Research Council Large GrantsA49132001 and A49331346.

REFERENCES

[1] Boolos, G., ‘The logic of provability’, Am. Math. Monthly91 (1984), 470–479.[2] Bulmer, M., Fearnley-Sander, D. andStokes, T., ‘Towards a Calculus of Algorithms’, Bull. Austral.

Math. Soc.50 (1994), 81–89.

Vol. 43, 2000 Internalizing equality in Boolean algebras 195

[3] Fearnley-Sander, D., ‘The idea of a diagram’, inResolution of Equations in Algebraic Structures, ed.Hassan Ait-Kaci and Maurice Nivat, Academic Press, 1989, 127–150.

[4] Fearnley-Sander, D., ‘Definitional reasoning and optimization’, inProceedings of the InternationalWorkshop on Analysis and its Applications, ed. C. V. Stanojevic and O. Hadzic, Institute of Mathematics,Novi Sad, 1992, 365–379.

[5] Fearnley-Sander, D. andStokes, T., ‘Equality Algebras’, Bull. Aust. Math. Soc.56 (1997), 177–191.[6] Hughes, G. E. andCresswell, M. J., An Introduction to Modal Logic, Methuen, London, 1968.[7] Lemmon, J., ‘Algebraic semantics for modal logics I’, J. Symbolic Logic31 (1966), 46–65.[8] McKinsey, J. C. C. andTarski, A., ‘Algebra of topology’, Ann. Math.45 (1944), 141–191.[9] McKinsey, J. C. C. andTarski, A., ‘Some theorems about the sentential calculi of Lewis and Heyting’, J.

Symbolic Logic13 (1948), 1–15.[10] Rasiowa, H., An Algebraic Approach to Non-Classical Logics, North-Holland, 1974.[11] Suszko, R., ‘Abolition of the Fregean axiom’, Springer Lecture Notes in Mathematics453(1974), 169–239.

Department of MathematicsUniversity of TasmaniaGPO Box 252–37 HobartTasmania 7001Australiae-mail: [email protected]

Department of Mathematics and StatisticsMurdoch UniversityMurdochW.A. 6150Australiae-mail:[email protected]