semisimple varieties of modal algebras

T.KowalskiM.Kracht

Semisimple Varieties of ModalAlgebras

Dedicated to the memory of Willem Johannes Blok

Abstract. In this paper we show that a variety of modal algebras of finite type is

semisimple iff it is discriminator iff it is both weakly transitive and cyclic. This fact

has been claimed already in [4] (based on joint work by the two authors) but the proof

was fatally flawed.

Keywords: modal logic, semisimple varieties

1. Introduction

A variety of algebras is said to be semisimple if every subdirectly irreduciblemember is simple. In this paper we show that if the signature is finite,semisimple varieties of modal algebras can be characterised by the conjunc-tion of two properties: they are weakly transitive and cyclic. The proofoffered in [4] contains a fatal mistake. We shall also indicate why the resultis false if the signature is infinite.1

2. Modal Logic

The set of formulae, Fmκ (otherwise referred to as terms over the given setof variables), is the smallest set such that

∗ pi : i ∈ ω ⊆ Fmκ

∗ ⊥ ∈ Fmκ.

∗ If ϕ, χ ∈ Fmκ then ¬ϕ,ϕ ∧ χ ∈ Fmκ.

∗ If ϕ ∈ Fmκ and i < κ then iϕ ∈ Fmκ.

Throughout this paper κ is assumed to be finite. ⊥, 3iϕ, ϕ ∨ χ and ϕ → χare abbreviations. A subset L ⊆ Fmκ is a (normal) modal logic if

1This paper was prepared while the second author was visiting the Research Schoolof Information Sciences and Engineering (RSISE) at ANU in Canberra. We both greatlyappreciate the hospitality of the RSISE. Additionally, we thank Tadeusz Litak and ananonymous referee for helping us to improve the paper.

Special issue of Studia Logica in memory of Willem Johannes BlokEdited by Joel Berman, W. Dziobiak, Don Pigozzi, and James RafteryReceived December 2, 2004, Accepted February 16, 2006

c©Springer 2006DOI: 10.1007/s11225-006-8308-2Studia Logica (2006) 83: 351–363

T. Kowalski and M. Kracht

∗ L contains all tautologies of classical logic.

∗ L is closed under substitution.

∗ For every i < κ: i(p0 → p1) → (ip0 → ip1) ∈ L

∗ For every i < κ: if ϕ ∈ L then iϕ ∈ L.

The smallest normal modal logic in κ operators is called Kκ. In what is tofollow we shall write

ϕ := ϕ ∧∧

i<κ

iϕ (i)

This is an example of a compound modality, where by a compound modal-ity we understand a term δ(p) over one variable that does not contain ¬ or⊥. Examples are p ∧ 1(p ∧ 0 2 p), non-examples are 0(p ∧ q) and1(1p ∨ 0p). This is slightly less general than the definition given in[4], but that difference is inessential here. Note that p itself also is a com-pound modality. Also, the distinction between a formula (an algebraic term)and the modality it denotes (the corresponding term-operation) is often—harmlessly—blurred. We shall denote compound modalities by £, ¢, whichmake them look like modal operators. Indeed, if £ is a compound modality,then for any normal modal logic L the following holds:

∗ £(p → q) → (£p → £q) ∈ L

∗ If ϕ ∈ L then £ϕ ∈ L

So, a compound modality has all properties of a normal modal operator.The methods we use here work just as well for polyadic operators. Recall

that the following is required for a polyadic operator of arity n. For everyi < n, put

Oi(q) := O(p0, · · · , pi−1, q, pi+1, · · · , pn−1) (ii)

Then

∗ For every i < n: L contains Oi(>).

∗ For every i < n: L contains Oi(q → r) → (Oi(q) → Oi(r)).

Hence for arity > 1 the derived unary operators Oi(p) are normal operators.Define

[O]p :=∧

i<n

Oi(p) (iii)

Now, let 〈A, 〈Oj : j < κ〉〉 be a polyadic BAO. Put

A¦ := 〈A, 〈[Oj ] : j < κ〉〉 (iv)

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Semisimple Varieties of Modal Algebras

Then it turns out that Con(A¦) = Con(A), where Con(A) denotes the lat-tice of congruences of the algebra A. This reduces the polyadic signatureto a monadic signature. For it is the case that (a) A is subdirectly irre-ducible (simple) iff A¦ is subdirectly irreducible (simple), (b) a variety V

of polyadic BAOs is semisimple iff all subdirectly irreducible algebras inV¦ := A¦ : A ∈ V are simple, and (c) the notions of weak transitivityand cyclicity are translated from the monadic case via (iii). The class V¦,although not a variety, is closed under direct products and homomorphic im-ages and this is enough to carry out the relevant constructions and proofs.Hence, we can always pretend that we are working in a monadic signature.The polyadic case will be given a rather short shrift here also because acommon generalisation of results from the present paper and from [2] isunder way.

A modal algebra is an algebra 〈A, 1,−,∩, 〈¥i : i < κ〉〉 such that〈A, 1,−,∩〉 is a boolean algebra and for every i < κ: ¥i1 = 1 and ¥i(a∩b) =¥ia ∩¥ib. An example is the algebra obtained by Fmκ /ΘL, where ϕ ΘL χiff ϕ ↔ χ ∈ L. It is more convenient to regard a modal algebra as a pair〈A, 〈¥i : i < κ〉〉, where A = 〈A, 1,−,∩〉 is the underlying boolean algebra.A filter is called open if from a ∈ F follows for all i < κ that ¥ia ∈ F . Anelement a is open if for all i < κ: ¥ia ≥ a iff the filter it generates (in theboolean sense) is open. The map γ 7→ 1/γ is a lattice isomorphism betweencongruences and open filters (see [4], Section 1.7). If κ is finite, put

¥a := a ∩⋂

i<κ

¥ia (v)

Furthermore, for a polymodal algebra A, let A := 〈A, ¥〉. This is a mono-modal algebra. For a variety V we put V := A : A ∈ V. It is not hard toshow the following.

Lemma 1. ConA = ConA. It follows that A is subdirectly irreducible iffA is, and that A is simple iff A is.

This is because a congruence of A is a congruence of A and conversely. Itdoes not hold, however, that every subalgebra of A is a subalgebra of A. So,V is not a variety, but SV is, because by the congruence extension propertyfor BAOs we have HSPV = SHPV and (−) commutes with products andhomomorphic images. The following is reformulation of a theorem foundin [5].

Lemma 2 (Rautenberg). A is subdirectly irreducible iff there is a c < 1 suchthat for every a < 1 there is an n ∈ ω such that

¥na ≤ c (vi)

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(Notice that ¥a ≤ a, by definition.) For c one may take any elementthat generates the unique smallest open filter 6= 1 if it exists. In a simplealgebra, this filter is A itself.

Proposition 3. A is simple iff A is simple iff for each a ∈ A with a < 1there is an n ∈ ω such that ¥na = 0.

3. Weakly Transitive Logics

Consider the rules MP: ϕ,ϕ → χ/χ and MN: ϕ/ i ϕ. Let L be a modallogic. We define the following consequence relations:

∗ The local consequence relation of L: ∆ `L ϕ iff ϕ can be provedfrom ∆ ∪ L using only MP.

∗ The global consequence relation of L: ∆ °L ϕ iff ϕ can be derivedfrom ∆ ∪ L using only MP and MN.

∆ °L ϕ iff for all modal algebras A such that A ² L and all valuations β: ifβ(δ) = 1 for all δ ∈ ∆ then β(ϕ) = 1. A monomodal logic L is transitive ifit contains p → p. Weak transitivity generalises this notion as follows.We say that £ contains ¢ in L if `L £p → ¢p. A modality is a mastermodality for L if it contains all compound modalities in L. Notice that if£ is a master modality then `L £p → £ £ p, since ££ is a compoundmodality and so £ contains it in L. Also notice that if £ and ¢ both aremaster modalities, then £p ↔ ¢p ∈ L, so they are semantically equivalentin L.

Definition 4. L is weakly transitive if it has a master modality.

In finite signatures we can characterise ¢ as follows.

Proposition 5. A polymodal logic L of finite signature is weakly transitiveiff there is an n such that `L np ↔ n+1p iff there is an n such that n

is a master modality.

(Recall that V ² ¥x ≤ x, so that V ² ¥n+1x ≤ ¥nx is guaranteed bydefinition.) Notice that a variety V of polymodal algebras of finite type isweakly transitive iff there is an n such that

V ² ¥nx = ¥n+1x (vii)

Also, if L is weakly transitive then the least open filter generated by x is thefilter generated by ¥nx. Thus, a principal open filter is also principal in theboolean reduct.

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4. Cyclic Varieties

Let be a modal operator. is called a dual conjugate of in L ifp → ¬ ¬p ∈ L. Intuitively, this means that for every x in a Kripke–frame, if x¢y then y¢−x, where ¢− is the relation corresponding to . (So,¢− contains the converse of ¢.) V is called cyclic if every modal operatorhas a dual conjugate compound modality (thus a dual conjugate need notbe a basic modality). Recall the definition of from (i).

Proposition 6. The following are equivalent.

➀ Every compound modality has a dual conjugate.

➁ Every basic modality has a dual conjugate.

➂ has a dual conjugate.

It follows that V is cyclic iff V is cyclic. Obviously, V is cyclic iff wehave p → ¬n ¬p for some n. (In which case V is called n-cyclic.)

Proposition 7. If V is cyclic then every finite subdirectly irreducible algebrais simple.

(This follows from Theorem 9 below since the theory of a finite algebrais weakly transitive.) This cannot be generalised to infinite algebras. TakeF := 〈Z, ¢〉, where x ¢ y iff |x − y| = 1. Then the algebra Z of finite andcofinite subsets is subdirectly irreducible. The subset of cofinite sets formsan open filter. It is not hard to see that ConZ ∼= 3. The variety generatedby Z is cyclic.

Suppose that V is weakly transitive and cyclic. Then there is a mastermodality. Let £ be the corresponding operator in the algebras. Since V iscyclic, V ² x ≤ £−£− x. For £ has a dual conjugate, and it is easy to seethat £ is a dual conjugate for any modality. Say that a compound modalityis a universal modality for L if it is a master modality and satisfies thelaws of S5. Then we have

Proposition 8. L is weakly transitive and cyclic iff it has a universalmodality.

Proposition 9. Suppose V is weakly transitive and cyclic. Then everysubdirectly irreducible algebra is simple.

Proof. Assume that A ∈ V is subdirectly irreducible. Let F be the leastopen filter 6= 1. Since V is weakly transitive, and F is principal as an

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open filter, it is also open as a boolean filter, hence F = y : y ≥ x forsome x such that x = £x. We claim that −x is also open. For notice that£− x = £−£x = £−£−−x ≥ −x, since £ is a dual conjugate of £. Onthe other hand, £−x ≤ −x, so the two are equal. Now, the least open filtercontaining G and F is A. Also, F is the smallest nontrivial open filter. ButF ⊆ G cannot hold for then x ∈ G and so x ≥ −x, from which x = 1. SoG = 1, showing −x = 1, and x = 0. Thus A is simple.

5. Discriminator Varieties

A discriminator term for A is a term t(x, y, z) such that for all elementsa, b, c: t(a, b, c) = c if a = b and a otherwise. An algebra with a discriminatorterm is simple. A variety is a discriminator variety if it is generated fromalgebras with the same discriminator term. Put u(a) := −t(1, a, 0). Thenu(a) = 1 if a = 1, and u(a) = 0 else.

Proposition 10. A modal algebra has a discriminator term if there is aterm such u that u(a) = 0 iff a < 1, and u(1) = 1.

Such a u is often called a unary discriminator. Let V be m-transitiveand cyclic. Then τ(p) := mp is a (unary) discriminator for every subdi-rectly irreducible member. Now we show that the converse also holds.

Proposition 11. V is a discriminator variety iff it is weakly transitive andcyclic.

Proof. Assume u is a discriminator term for V. Then if V is not weaklytransitive, for every n there is An and an < 1 such that ¥nan > ¥n+1an.Let n be the modal degree of u. Then An+1 ² u(an+1) ↔ u(1) > 0. Contra-diction. So V is weakly transitive and has a master modality, to which theoperator £ corresponds. We show that V ² x ≤ £ − £ − x. If x = 0, thisis clearly true. If x > 0, however, we have −x < 1 and so £ − x = 0, fromwhich £(−£−x) = £1 = 1, as promised.

6. The Main Theorem

A variety is semisimple if every subdirectly irreducible member is simple.

Theorem 12. Let the signature be finite. Then the following are equivalent:

➀ V is semisimple.➁ V is discriminator.➂ V is weakly transitive and cyclic.

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Proof. By Proposition 11, ➁⇔➂. ➂⇒➀ is Proposition 9. By Theorem 13below, ➀ implies weak transitivity. Cyclicity follows right away. For supposethat the variety is n-transitive but not cyclic. Then there is a subdirectlyirreducible A and a such that a ¥¨na. So, ¨na 6= 1. Its complement −atherefore generates an open filter F = x : x ≥ ¥n−a 6= A. Contradiction.

The work therefore consists in showing

Theorem 13. In a finite signature, all semisimple varieties of modal alge-bras are weakly transitive.

The rest of this section is devoted to the proof of this theorem. Theproblem can be simplified at the outset by looking at V (or SV for thatmatter). So, from now on we assume that V is a variety of monomodalalgebras satisfying x ≤ ¨x. Then V is n-transitive iff V ² ¨n+1x = ¨nx.In what is to follow we assume that V is a nontrivial semisimple variety ofmodal algebras.

Let A be an algebra in V such that for a nonzero element a ∈ A we have¨na < 1 for every n ∈ ω. Such an algebra obviously exists: for instancethe free algebra FrV(x) on one generator x must be such, as otherwise V

would satisfy ¨nx = 1 for some n ∈ ω, and therefore also ¨n0 = 1, andthus 0 = 1, forcing V to be trivial, contrary to the initial assumption. Putα := CgA(a, 0), where CgA(x, y) denotes the least congruence of A containingthe pair (x, y). By assumption on A, 0 < α < 1. As α is principal, theremust be a congruence β with β ≺ α, that is, β is a lower cover of α.

Lemma 14. For every congruence β with β ≺ α there is an m ∈ ω such that

➀ ¨m+1a ≡β ¨ma, and➁ −¨ma ≡β ¨− ¨ma.

Proof. Consider the set Γ := θ ∈ ConA : θ ≥ β, θ 6≥ α. Observe thatif Γ = β then A/β is si but not simple. So there is a θ ∈ Γ such thatθ 6= β. By congruence distributivity, γ :=

∨Γ is a member of Γ. Therefore,

A/γ is si hence simple. From this and congruence permutability it followsthat α γ = 1. Thus, (0, 1) ∈ α γ, and there must be a c ∈ A with(0, c) ∈ α and (c, 1) ∈ γ; hence also (−c, 0) ∈ γ. Now, (0, c) ∈ α iff for somem ∈ ω we have ¨ma ≥ c. Thus, −¨ma ≤ −c and therefore (−¨ma, 0) ∈ γ.We can then assume c = ¨ma. By definition we have α ∩ γ = β, i.e.,0/α ∩ 0/γ = 0/β. Now, to prove ➀, consider ¨m+1a ∩ −¨ma. This elementbelongs to 0/α ∩ 0/γ = 0/β and thus we obtain ¨m+1a ≡β ¨ma. Then, for➁, consider ¨ma∩¨−¨ma. This element also belongs to 0/α∩ 0/γ = 0/β;therefore −¨ma ≡β ¨− ¨ma.

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Now, consider the following condition on V.

(?) For every k ∈ ω there are r, ` ∈ ω such that V ² x ≤ ¨`¥k¨rx.

Notice that (?) is a weakened form of cyclicity.

Lemma 15. All p-cyclic varieties of modal algebras satisfy (?) for r = kpand ` = 0.

Proof. p-cyclic varieties satisfy x ≤ ¥¨px. We will show that this forcesx ≤ ¥k¨kpx, for every k ∈ ω. For k = 0, 1 this is trivial. Suppose it holdsfor k.

¥k+1¨(k+1)px = ¥(¥k¨kp(¨px)) (viii)

= ¥(¥k¨kp(¨px))≥ ¥¨px

≥ x

Thus, x ≤ ¨0¥k+1¨(k+1)px, as required.

Suppose now that V falsifies (?). Then there is a k ∈ ω such that for allr, ` ∈ ω our variety V falsifies x ≤ ¨`¥k¨rx. Let K be the smallest such k;note that for all k′ ≥ K the variety V also falsifies x ≤ ¨`¥k′¨rx.

In FrV(x), the algebra freely generated by x, we have x 6≤ ¨`¥K¨rx, forall r, ` ∈ ω. For each r ∈ ω define θr := CgFrV(x)(¥K¨rx, 0). Since ¥K¨rx ≤¥K¨r+1x, the family of congruences θr : r ∈ ω forms an increasing chain.Put α := CgFrV(x)(x, 0) and Θ :=

∨r∈ω θr.

Lemma 16. 0 < Θ < α.

Proof. If Θ = 0, then, since the reasoning takes place in the one-generatedfree algebra in V, we get that V ² ¥K¨rx = 0, for all r ∈ ω. In particular,substituting 1 for x we obtain V ² 1 = ¥K¨r1 = 0. This means V is trivial,contradicting the initial assumptions.

That Θ ≤ α is clear from the definitions. Since α is principal, Θ = α iffthere is a finite set S ⊆ θr : r ∈ ω with

∨S = α. However, as θr : r ∈ ω

is an increasing chain there is an r ∈ ω such that θr = α. Now, both θr andα are principal, and thus θr = α iff there is an ` ∈ ω with x ≤ ¨`¥K¨rx. Aswe reason in the free algebra, it means that V satisfies x ≤ ¨`¥K¨rx. Thiscontradicts our assumption that V falsifies (?).

Lemma 17. All semisimple varieties of modal algebras satisfy (?).

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Proof. Since our fixed variety V has been chosen arbitrarily, it suffices toprove that it satisfies (?). We proceed by reductio. Suppose V falsifies (?)and let K be the smallest number witnessing that, precisely as in Lemma 16.In view of that lemma, we can choose a congruence β such that Θ ≤ β ≺ α.Then, by Lemma 14, we obtain ¨mx ≡β ¥¨mx, for some m ∈ ω. Sinceθm ≤ Θ ≤ β, this gives us ¨mx ≡β ¥K¨mx ≡β 0. Therefore, x ≡β 0 andthus β ≥ α contradicting the choice of β.

From now on we assume that V satisfies (?). Define a function r : ω → ω,by taking r(i) to be the smallest number such that there exists an ` ∈ ωwith V ² ¨`¥i¨r(i)x ≤ x.

Lemma 18. The function r is nondecreasing.

Proof. Suppose the contrary. Then, for a certain i ∈ ω, we have r(i) >r(i + 1). By definition of r there is an ` ∈ ω such that

x ≤ ¨`¥i+1¨r(i+1)x ≤ ¨`¥i¨r(i+1)x (ix)

This is a contradiction, since r(i) is by definition the smallest number forwhich a suitable ` exists, yet r(i + 1) is strictly smaller.

Now let us define another function ` : ω → ω by taking `(i) to be thesmallest number such that V ² ¨`(i)¥i¨r(i)x ≤ x. Thus, ` depends on i viar(i).

Lemma 19. Let V be semisimple. If V is not weakly transitive then for eachi ∈ ω there is a simple algebra Ai in V and an element ai ∈ Ai such that¨r(i)ai < 1 but ¨r(i)+1ai = 1.

Proof. Suppose otherwise. Then there is an i ∈ ω such that for eachsimple algebra A ∈ V and each element a ∈ A we have: ¨r(i)+1a = 1 implies¨r(i)a = 1. By Proposition 3 this gives ¨r(i)a = 1 if a > 0. This in turnforces V to satisfy ¨n+1x = ¨nx for n = r(i), contrary to the assumption.

Now suppose V is not weakly transitive and let Ai and ai ∈ Ai be as inthe lemma above, for each i ∈ ω. Then ai > 0. Put bi := −¨r(i)ai and fix ak ∈ ω.

Lemma 20. For every i ≥ k we have: ¨kbi < 1 and ¨`(k)+r(k)+1 − ¨kbi = 1.

Proof. To prove the first statement of the claim, suppose the contrary, andnote that ¨kbi = 1 forces ¨ibi = 1, since i ≥ k. By the definition of bi, we

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have ¨i − ¨r(i)ai = 1, thus ¥i¨r(i)ai = 0. Therefore, ¨`¥i¨r(i)ai = 0, forany ` ∈ ω. This falsifies ¨`(i)¥i¨r(i)ai ≥ ai.

For the second part, applying the definition of bi we get

¨`(k)+r(k)+1 − ¨kbi = ¨`(k)+r(k)+1 − ¨k − ¨r(i)ai (x)

= ¨`(k)+r(k)+1¥k¨r(i)ai

= ¨`(k)+r(k)+1¥k¨r(k)(¨r(i)−r(k)ai)

Then, from the inequality ¨`(k)¥k¨r(k)x ≥ x, which holds in V, we get¨`(k)+r(k)+1¥k¨r(k)x ≥ ¨r(k)+1x, by applying ¨ repeatedly r(k)+1 times toboth sides. Now a substitution yields:

¨`(k)+r(k)+1¥k¨r(k)(¨r(i)−r(k)ai) ≥ ¨r(k)+1(¨r(i)−r(k)ai) (xi)

Rewriting both sides back, we obtain

¨`(k)+r(k)+1¥k¨r(i)ai ≥ ¨r(i)+1ai = 1 (xii)

This is as required.

Now take the ultraproduct B :=∏

i∈ω Ai/U over a nonprincipal ultrafil-ter U on ω. Put b := 〈bi : i ∈ ω〉/U .

Lemma 21. For every k ∈ ω we have ¨kb < 1 and ¨`(k)+r(k)+1 − ¨kb = 1.

Proof. Let k be given. By Lemma 20, ¨kbi < 1 holds in Ai for every i;therefore, it holds in B. Also, the set of indices i for which the equation¨`(k)+r(k)+1 − ¨kbi = 1 is true in Ai is cofinite (it contains i : i ≥ k).Therefore, since U is nonprincipal, ¨`(k)+r(k)+1 − ¨kbi = 1 is true in B.

Lemma 22. If V satisfies (?) it is weakly transitive.

Proof. We will show that assuming otherwise leads to a contradiction.Suppose V falsifies ¨n+1x = ¨nx for all n ∈ ω. Given that, the precedingthree lemmas show how to produce an algebra B ∈ V and an element b ∈ Bwhich for all k ∈ ω satisfies ¨kb < 1 and ¨`(k)+r(k)+1 − ¨kb = 1. Then,putting α := CgB(b, 0), and taking β as in Lemma 14, we get

−¨mb ≡β ¨− ¨mb ≡β ¨`(m)+r(m)+1 − ¨mb = 1 (xiii)

Thus, ¨mb ≡β 0 and therefore b ≡β 0. It follows that β ≥ α, contradictingthe choice of β as a lower cover of α.

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By Lemma 17 all semisimple varieties satisfy (?). This concludes theproof of Theorem 13.

Throughout the proof we made use of ¥, whose definition, (v), relieson the fact that the signature is finite. Indeed, this condition is essential.For suppose that κ is infinite. Then let Hi = 〈H, 〈¥i : i < κ〉〉 be thefollowing algebra. H is the 4-element boolean algebra; ¥j is the identityif j 6= i and the unary discriminator otherwise. Then Hi is simple for eachi < κ. Now, let P = Hκ

i /U be an ultraproduct over a nonprincipal ultrafilterU on κ. Then ¥i~x/U = ~x/U iff j < κ : ¥ix(j) = x(j) ∈ U . Butj < κ : ¥ix(j) = x(j) = κ\i and thus belongs to U . Hence, P is the fourelement algebra in which x = ¥ix holds for all x and each i < κ. So, P is notsimple. In fact it is not difficult to see that any ultraproduct of the algebrasfrom Hi : i < κ is either isomorphic to one of Hi or to P. Consider thevariety W generated by Hi : i < κ. By Jonsson’s Theorem all subdirectlyirreducible algebras from W are in HSPU(Hi : i < κ). By previous remarksPU(Hi : i < κ) = Hi : i < κ∪P. Thus, HSPU(Hi : i < κ) consists ofalgebras with at most four elements. Every two element algebra is simple,hence there is only one nonsimple algebra in this class: P, which is notsubdirectly irreducible. So the variety W is semisimple. It is however not adiscriminator variety otherwise ultraproducts of simple algebras would haveto be simple.

7. Relevance of the Notions

We shall illustrate the relevance of the concepts discussed in this paper. Thenotion of weak transitivity is due to Wim Blok and generalises the notion oftransitivity in the right way.2 Algebraically, it amounts to the property ofequationally definable principal congruences (EDPC). In modal logic, weaktransitivity is equivalent to the definability of a master modality. Manyresults in modal logic do not depend on transitivity, they only depend onweak transitivity. Examples are the existence of a deduction theorem forthe global frame consequence, finite equivalentiality, and the existence ofmany splitting algebras. A variety is cyclic if every operator is included ina converse. Cyclicity generalises the notion of symmetry. If a logic is bothweakly transitive and cyclic then a universal modality is definable.

2We have been unable to determine exactly where this notion first appeared in print.Wim Blok and Wolfgang Rautenberg both started to use it in the mid 70s, but withoutany reference. The second author recalls a conversation with Wolfgang Rautenberg inwhich he said that he had met Wim Blok in Amsterdam while Wim was working on hisPhD and that Wim had introduced him to splittings and weak transitivity.

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°L has a deduction theorem iff there is a formula τ(p, q) such that χ °L ϕiff °L τ(χ, ϕ). In general, if °L has a deduction theorem one can chooseτ(p, q) := £(p → q) for some master modality £. Hence L is weakly tran-sitive iff °L has a deduction theorem. `L is equivalential if there is a set∆(p, q) of terms in two variables satisfying

➀ `L ∆(p, p)

➁ ∆(p, q) `L ∆(q, p)

➂ ∆(p, q), ∆(q, r) `L ∆(p, r)

➃ p, ∆(p, q) `L q

➄⋃

i<Ω(f) ∆(pi, qi) `L ∆(f(~p), f(~q))

Similarly one defines equivalentiality for °L. If `L is equivalential in L, so is°L and conversely. `L is finitely equivalential if ∆ can be chosen finite.Every modal logic is equivalential via ∆(p, q) := p ↔ q. K is not finitelyequivalential. We remark that °L is always finitely equivalential; for we maysimply choose ∆(p, q) := p ↔ q. The following is from [1].

Theorem 23 (Blok, Pigozzi). For a modal logic L the following are equiv-alent:

1. L is weakly transitive.

2. °L has a deduction theorem.

3. `L is finitely equivalential.

4. Alg L has equationally definable principal congruences.

5. Alg L has definable principal open filters.

Given a variety V, an algebra A is called finitely presentable if thereis a finite set E of equations such that

FrV(var(E))/Θ(E) ∼= A (xiv)

where var(E) is the set of variables occurring in E and Θ(E) is the congru-ence generated by E. (In particular, finite algebras are finitely presentable.)A is splitting if there is a logic L such that for every logic L′ ⊇ Th V: eitherL′ ⊆ ThA or L′ ⊇ L, but not both. The following is from [3].

Theorem 24. Suppose that V is weakly transitive. Then every subdirectlyirreducible A which is finitely presentable is splitting.

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8. Conclusion

We mention a few applications of Theorem 12. It follows that semisim-ple varieties have a deduction theorem, and that every finitely presentablesubdirectly irreducible algebra is splitting. This has useful applications intense logic. By definition, tense logics are cyclic. K4.2t also is 2-transitive.It follows that its variety is semisimple. A fortiori, tense logics of lineartransitive structures define semisimple varieties. By contrast, the variety ofK4t-algebras is not weakly transitive and therefore not semisimple.

References

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able principal congruences, I’, Algebra Universalis, 15:195–227, 1982.

[2] Kowalski, Tomasz, ‘Semisimplicity, EDPC and Discriminator Varieties of Residu-

ated Lattices’, Studia Logica, 77:255–265, 2004.

[3] Kracht, Marcus, ‘An almost general splitting theorem for modal logic’, Studia Log-

ica, 49:455 – 470, 1990.

[4] Kracht, Marcus, Tools and Techniques in Modal Logic, Number 142 in Studies in

Logic, Elsevier, Amsterdam, 1999.

[5] Rautenberg, Wolfgang, ‘Splitting lattices of logics’, Archiv fur Mathematische

Logik, 20:155–159, 1980.

Tomasz KowalskiResearch School of Information Sciences and EngineeringAustralian National UniversityCanberra, ACT [email protected]

Marcus KrachtDepartment of LinguisticsUCLA3125 Campbell Hall405 Hilgard AvenueLos Angeles, CA 90095–[email protected]

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