◮ Modal algebras form the algebraic semantics of normal modal logics. They are Boolean algebras with a unary operation ✷ s.t. ✷ 1 ≈ 1 and ✷ ( x ∧ y ) ≈ ✷ x ∧ ✷ y . ◮ Modal algebras can be presented also as BAs with a unary Varieties of positive interior algebras: operation ✸ . The two presentations produce term-equivalent structural completeness varieties setting ✷ x := ¬ ✸ ¬ x and ✸ x := ¬ ✷ ¬ x . ◮ Positive modal algebras are �∧ , ∨ , ✷ , ✸ , 0 , 1 � -subreducts of modal algebras. Equivalently... Tommaso Moraschini Definition Institute of Computer Science of the Czech Academy of Sciences An algebra A = � A , ∧ , ∨ , ✷ , ✸ , 0 , 1 � is a positive modal algebra if � A , ∧ , ∨ , 0 , 1 � is a bounded distributive lattice s.t. for all a , b ∈ A , November 14, 2017 ✷ ( a ∧ b ) = ✷ a ∧ ✷ b ✸ ( a ∨ b ) = ✸ a ∨ ✸ b ✷ a ∧ ✸ b ≤ ✸ ( a ∧ b ) ✷ ( a ∨ b ) ≤ ✷ a ∨ ✸ b . ◮ Positive modal algebras form a variety (a.k.a. equational class). 1 / 18 2 / 18 ◮ We focus on two central varieties of (positive) modal algebras: ◮ It is well the congruences of a modal algebra A are in Recall that... one-to-one correspondence with its open filters. ◮ The algebraic semantics of K 4 consists of K4-algebras, i.e. ◮ Fundamental application: This correspondence identifies modal algebras validating ✷ x ≤ ✷✷ x (equiv. ✸✸ x ≤ ✸ x ). subdirectly irreducible modal algebras as those which have a smallest open filter different from { 1 } . ◮ The algebraic semantics of S 4 consists of interior algebras, i.e. ◮ Problem: The correspondence between congruences and open modal algebras validating ✷✷ x ≤ ✷ x ≤ x (equiv. x ≤ ✸ x ≤ ✸✸ x ). filters is lost in the positive setting. A useful description of subdirectly irreducible positive modal algebras is unknown. Definition ◮ A couple of facts are known, e.g. every finitely subdirectly Let A be a positive modal algebra. irreducible positive interior algebra A is well-connected, i.e. 1. A is a positive K4-algebra if it satisfies ✷ x ≤ ✷✷ x and if ✷ a ∨ ✷ b = 1, then a = 1 or b = 1 ✸✸ x ≤ ✸ x . We denote by PK4 the variety they form. if ✸ a ∧ ✸ b = 0, then a = 0 or b = 0 . 2. A is a positive interior algebra if it satisfies ✷✷ x ≤ ✷ x ≤ x and x ≤ ✸ x ≤ ✸✸ x . We denote by PIA the variety they form. However, there are well-connected positive interior algebras, which are not finitely subdirectly irreducible. ◮ Positive K4-algebras (resp. interior algebras) are subreducts of K4-algebras (resp. interior algebras). 3 / 18 4 / 18
◮ Some example of simple positive interior algebras: 1 • ◮ The free one-generated ✸ • algebra of PK4 is infinite: • it contains an infinite • • ✸ descending chain • • • { � ✷ n x � : n ∈ ω } . • • • • • • • • ◮ The free two-generated x • • • • ✸ • ✷ positive interior algebra is ◮ Obs: The positive interior algebras above contain a non-simple infinite. • • • • and non-trivial subalgebra (e.g. any four-element chain). ◮ While the free • • • • one-generated algebra of • • • Corollary PIA is finite. This • • ✷ Positive interior algebras do not have the congruence extension contrasts with the • property (CEP). Thus they do not have EDPC. full-signature case. ✷ • ◮ This contrasts with the full signature case, since interior 0 • algebras have EDPC. 5 / 18 6 / 18 ◮ Playing with these one-generated subdirectly irreducible ◮ From the knowledge of the structure of the free one-generated algebras, we obtain a description of the bottom part of the positive interior algebra we can derive all one-generated subvariety lattice Λ( PIA ) of positive interior algebras. subdirectly irreducible positive interior algebras: Theorem 1 • 1 • 1 • 1. There is a unique minimal subvariety DL of PIA, ✸ • ✸ • • 1 • 1 • 1 • term-equivalent to that of bounded distributive lattices. • ✸ • ✷ • ✷ • • ✷ • 2. The unique covers of DL in Λ( PIA ) are the ones generated by 1 • one of the following algebras D 3 , C a 3 , C b 3 and D 4 : 0 • 0 • 0 • 0 • 0 • 0 • 0 • 1 • 1 • 1 • 1 • 1 • ✸ • 1 • 1 • 1 • ✸ • • ✸ • • • ✸ • ✷ • ✷ • ✷ • ✸ • • • ✷ • • ✸ 0 • 0 • 0 • 0 • • ✷ • • • ✷ 3. If K is a subvariety of PIA such that DL � K, then K includes 0 • 0 • 0 • 0 • one of the covers of DL. 7 / 18 8 / 18
Typical application: V ( D 4 ) has no join-irreducible cover in Λ( PIA ) . ◮ To climb higher (from bottom to top) in the lattice Λ( PIA ) , ◮ Consider the positive interior algebras D 3 , C a 3 , C b 3 and D 4 : we need the following: 1 • Definition ✸ • 1 • 1 • 1 • Let K be a variety. A subdirectly irreducible A ∈ K is a splitting algebra in K if there is a largest subvariety of K excluding A . • ✸ • ✷ • ✷ • 0 • 0 • 0 • 0 • Lemma (McKenzie) ◮ D 3 is splitting in PK4 with splitting identity If a congruence distributive variety is generated by its finite ✸ x ∧ ✷✸ x ≤ x ∨ ✷ x ∨ ✸✷ x . (1) members, then its splitting algebras are finite. ◮ C a 3 and C b 3 are splitting in PIA, respectively with splitting Corollary identities Splitting algebras in PK4 and in PIA are finite. ✸✷✸ x ≈ ✸ x and ✷✸✷ x ≈ ✷ x . (2) ◮ V ( D 4 ) is axiomatized by ◮ Problem: Which finite s.i. algebras are splitting in PK4 and in ✷✸ x ≈ ✷ x and ✸✷ x = ✸ x . (3) PIA? In the full-signature case the answer is all (by EDPC). In the positive case it is unknown. ◮ Since (3) is a consequence of (1, 2), we are done. 9 / 18 10 / 18 ◮ The next figure shows the join-irreducible covers of the varieties generated by D 3 , C a 3 , C b 3 and D 4 in Λ( PIA ) : Figure: Join-irreducible varieties of depth ≤ 4 in Λ( PIA ) . 11 / 18 12 / 18
Definition Definition Let K be a variety. Let K be a variety and consider a quasi-equation 1. K is actively structurally complete (ASC) if every active Φ := ϕ 1 ≈ ψ 1 & . . . & ϕ n ≈ ψ n → ϕ ≈ ψ. admissible quasi-equation is derivable. 2. K is passively structurally complete (PSC) if every passive 1. Φ is active in K when there is a substitution σ such that (admissible) quasi-equation is derivable. K � σϕ i ≈ σψ i for every i ≤ n . 3. K is structurally complete (SC) if every admissible 2. Φ is passive in K if it is not active in K. quasi-equation is derivable. 3. Φ is admissible in K if for every substitution σ : 4. K is hereditarily structurally complete (SHC) if every if K � σϕ i ≈ σψ i for every i ≤ n , then K � σϕ ≈ σψ. subvariety of K is SC. 4. Φ is derivable in K if K � Φ . ◮ Clearly, (ASC) + (PSC) = (SC), and (SHC) implies (SC). ◮ We aim to understand the various structural completeness in ◮ Observe that passive quasi-equations are vacuously admissible. subvarieties of PK4. 13 / 18 14 / 18 Theorem Let K be a non-trivial variety of positive interior algebras. TFAE: Theorem 1. K is actively structurally complete. Let K be a SC variety of positive modal algebras. Either 2. K excludes D 3 , C a 3 and C b 3 . K = V ( B 2 ) or there are n , m ≥ 1 such that 3. K = DL or K = V ( D 4 ) . K � ✷ x ∧ · · · ∧ ✷ n x ≤ x and K � x ≤ ✸ x ∨ · · · ∨ ✸ m x . 4. K is hereditarily structurally complete. 5. K is structurally complete. Corollary 6. K satisfies the equations ✷✸ x ≈ ✷ x and ✸✷ x ≈ ✸ x . Let K be a SC variety of positive K4-algebras. Either K = V ( B 2 ) or Corollary K ⊆ PIA. Let K be a non-trivial variety of positive K4-algebras. TFAE: ◮ Hence, while trying to spot SC subvarieties of PK4, we can 1. K is structurally complete. restrict to subvarieties of PIA. 2. K = V ( B 2 ) or K = DL or K = V ( D 4 ) . 3. K is hereditarily structurally complete. ◮ There are only 3 non-trivial SC subvarieties of PK4. 15 / 18 16 / 18
Finally... ◮ Problem: What about ASC and PSC subvarieties of PK4? ◮ For ASC the answer is unknown. ◮ Let C 2 be the two-element positive interior algebra. For PSC we have: Theorem Let K be a non-trivial variety of positive K4-algebras. TAFE: 1. K is passively structurally complete. ...thank you for coming! 2. Either K = V ( B 2 ) or ( Fm K ( 0 ) = C 2 and C 2 is the unique simple member of K). 3. Either K = V ( B 2 ) or ( Fm K ( 0 ) = C 2 and K excludes D 3 ). 4. Either K = V ( B 2 ) or K � ✸ 1 ≈ 1 , ✷ 0 ≈ 0 , ✸ x ∧ ✷✸ x ≤ x ∨ ✷ x ∨ ✸✷ x . ◮ There are infinitely many PSC subvarieties of PIA. 17 / 18 18 / 18
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