Varieties of De Morgan Monoids T. Moraschini 1 , J.G. Raftery 2 , and J.J. Wannenburg 2 1 Academy of Sciences of the Czech Republic, Czech Republic 2 University of Pretoria, South Africa SAMSA, November 2016
De Morgan monoids A De Morgan monoid ❆ = � A ; ∨ , ∧ , · , ¬ , t � comprises ◮ a distributive lattice � A ; ∨ , ∧� ◮ a square-increasing ( x ≤ x · x ) commutative monoid � A ; · , t � ◮ x = ¬¬ x ◮ x · y ≤ z iff x · ¬ z ≤ ¬ y ◮ x → y := ¬ ( x · ¬ y ). DM := { all De Morgan monoids } RA := { t -free subreducts of De Morgan monoids } = { Subalgebras of � A , ∨ , ∧ , · , ¬� where ❆ ∈ DM } DM and RA are varieties.
Algebraic Logic Define a logic R t as follows � � γ 1 , . . . , γ n ⊢ R t α iff DM � t ≤ γ 1 & . . . & t ≤ γ n ⇒ t ≤ α. Similarly for RA and the logic R , where we replace every t ≤ α with α → α ≤ α , to which it is equivalent in DM . DM inconsistent s s ❃ ✚ ❩❩❩❩❩ ✚✚✚✚✚ ❩ ⑦ s s R t trivial Subvarieties Axiomatic extensions of R t of DM
DM vs RA Every finitely generated algebra in RA has a unique identity element for · and is therefore a reduct of a De Morgan monoid. RA DM s s V ( ❉ 4 ) s s s ❅ � V ( ❉ 4 ) V ( 2 ) V ( ❈ 4 ) V ( ❙ 3 ) ❅ � ❜❜❜❜ ✧ s s s s ❏ ✡ s ✧ V ( 2 ) V ( ❈ 4 ) V ( ❙ 3 ) ✧ ❏ ✡ ❏ ✧ ✡ s s trivial trivial Subvarieties of RA Subvarieties of DM (´ Swirydowicz 1995)
Important Algebras ❈ 4 ❉ 4 2 ❙ 3 f 2 s f 2 ¬ a f s s s � ❅ � ❅ t f t t = f t s s s s s ❅ s � ❅ � f a ¬ ( f 2 ) ¬ ( f 2 ) s s s f := ¬ t ◮ These are all subdirectly irreducible (which amounts to t having a greatest strict lower bound, say c ) ◮ In fact, they are all simple ( c is the only lower bound of t )
Structural Completeness ◮ Raftery and ´ Swirydowicz (2016) showed recently that the only non-trivial (passively) structurally complete subvariety of RA is the variety of Boolean algebras. ◮ Which subvarieties of DM are structurally complete? ◮ A variety V called structurally complete if every proper subquasivariety of V generates a proper subvariety of V . ◮ V is called passively structurally complete if all the non-trivial algebras in V satisfy the same existential positive sentences.
Passive Structural Completeness in DM Thm. A variety K ⊆ DM is passively structurally complete iff one of the following four (mutually exclusive) conditions hold: 1. K = V ( 2 ) ; 2. K = V ( ❉ 4 ) ; 3. K consists of odd Sugihara monoids; 4. ❈ 4 is a retract of every non-trivial algebra in K . The class { ❆ ∈ DM : ❆ is trivial or ❈ 4 is a retract of ❆ } is a quasivariety but not a variety. For example ❈ 4 is a retract of ❇ × ❈ 4 , but ❇ is a simple homomorphic image of ❇ × ❈ 4 and so can’t map onto ❈ 4 . f 2 s ❇ : f s � ❅ with b · b = f 2 , ¬ b · ¬ b = f 2 and � ❅ ¬ b b s s ❅ s � b · ¬ b = f . ❅ � t ¬ ( f 2 ) s
Exploring condition 4 Thm. There is a largest subvariety M of DM such that ❈ 4 is a retract of all non-trivial members of M . M is axiomatised, relative to DM , by: ◮ t ≤ f , ◮ x ≤ f 2 , [0 := ¬ ( f 2 )] . ◮ (( f → x ) ∨ ( x → t )) → 0 = 0 DM M Odd SM V ( 2 ) V ( ❉ 4 ) V ( ❈ 4 ) V ( ❙ 3 ) trivial
Exploring M Every subdirectly irreducible algebra in M arises by a construction of J. K. Slaney (1993) from a Dunn monoid ❆ [essentially a De Morgan monoid without the involution ¬ ], i.e., a square-increasing distributive lattice-ordered commutative monoid � A ; ∨ , ∧ , · , → , t � that satisfies the law of residuation x ≤ y → z iff x · y ≤ z . Let’s call this construction skew reflection .
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A s a s Dunn monoid
Skew Reflection t s s A s a s Dunn monoid
Skew Reflection t s s A s a s Dunn monoid
Skew Reflection t s s A s a s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s s A a s s Dunn monoid
Skew Reflection t s a ′ s A A ′ a s t ′ s
Skew Reflection a ′ t s s A ′ A t ′ a s s
Skew Reflection A ′ a ′ s t s t ′ s a A s
Skew Reflection A ′ a ′ s t s t ′ s a s A
Skew Reflection A ′ a ′ s t ′ s t s a s A
Skew Reflection A ′ a ′ s t ′ s t s a s A
Skew Reflection A ′ a ′ s t ′ s t s s a A
Skew Reflection A ′ a ′ s t ′ s t s a s A
Skew Reflection s 1 A ′ a ′ s t ′ s t s a s A 0 s
Skew Reflection Declare that a < b ′ for certain a , b ∈ A in such a way that � A ∪ A ′ ∪ { 0 , 1 } ; ≤� is a distributive lattice, t < t ′ and for all a , b , c ∈ A , s 1 a · b < c ′ iff a < ( b · c ) ′ . A ′ Then there is a unique way of turning the struc- a ′ s ture into a De Morgan monoid t ′ ✑ s ✑✑✑ S < ( ❆ ) = � A ∪ A ′ ∪ { 0 , 1 } ; ∨ , ∧ , · , ¬ , t � ∈ M , t s a s of which ❆ is a subreduct, where ¬ extends ′ . In particular if we specify that a < b ′ for all A a , b ∈ A , then we get the reflection construc- 0 tion, which is an older idea, see Meyer (1973) s and Galatos and Raftery (2004). In this case we write R ( ❆ ).
Recall Q: Which subvarieties of M are structurally complete? The map W �→ V { R ( ❆ ) : ❆ ∈ W } , from varieties of Dunn monoids to subvarieties of M , preserves structural incompleteness . Therefore some subvarieties of M are not structurally complete e.g. V { R ( ❆ ) : ❆ a Brouwerian algebra }
Covers of V ( ❈ 4 ) Let K be a cover of V ( ❈ 4 ) within M . Then K = V ( ❆ ) Thm. for some skew reflection ❆ of a subdirectly irreducible Dunn monoid ❇ , where 0 is meet-irreducible in ❆ , and ❆ is generated by the greatest strict lower bound of t in ❇ .
Constructing R ( ❙ 3 ) s s ¬ a ❙ 3 : s s t s s a
Constructing R ( ❙ 3 ) s s ¬ a s ❙ 3 : s t s s a
Constructing R ( ❙ 3 ) s s ¬ a s ❙ 3 : s t s s a
Constructing R ( ❙ 3 ) s s ¬ a s s ❙ 3 : s t s a
Constructing R ( ❙ 3 ) s s s s ¬ a ❙ 3 : s t s a
Constructing R ( ❙ 3 ) s s s s ¬ a ❙ 3 : s t s a
Constructing R ( ❙ 3 ) s s s s ¬ a ❙ 3 : s t s a
Constructing R ( ❙ 3 ) s s s s ¬ a ❙ 3 : s t s a
Constructing R ( ❙ 3 ) s s s s ¬ a ❙ 3 : s t s a
Constructing R ( ❙ 3 ) s s s s ¬ a ❙ 3 : s t s a
Constructing R ( ❙ 3 ) 1 s a ′ s t ′ s s ¬ a ′ s ¬ a s t s a s 0
Constructing R ( ❙ 3 ) 1 s a ′ s t ′ s s ¬ a ′ R ( ❙ 3 ): s ¬ a s t s a s 0
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : t s s a s s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : t s s a s s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : t s s a s s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : t s s a s s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : t s s a s s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : s t s s a s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : s t s s a s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : s t s s a s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : s t s s a s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : s t s s a s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : s t s s a s
Constructing S < ( ❙ 3 ) ¬ a s s ❙ 3 : t s s a s s
Constructing S < ( ❙ 3 ) ¬ a s s a ′ t s s t ′ a s s ¬ a ′
Constructing S < ( ❙ 3 ) a ′ s s ¬ a t ′ s t s ¬ a ′ s a s
Constructing S < ( ❙ 3 ) a ′ s s ¬ a t ′ s t s ¬ a ′ s a s
Constructing S < ( ❙ 3 ) a ′ s s ¬ a t ′ s s t ¬ a ′ s a s
Constructing S < ( ❙ 3 ) a ′ s s ¬ a t ′ s s t ¬ a ′ s a s
Constructing S < ( ❙ 3 ) a ′ s t ′ s ¬ a s ¬ a ′ s t s a s
Constructing S < ( ❙ 3 ) a ′ s t ′ s s ¬ a ¬ a ′ s s t a s
Constructing S < ( ❙ 3 ) a ′ s t ′ s s ¬ a ¬ a ′ s s t s a
Constructing S < ( ❙ 3 ) a ′ ✓ s ✓ t ′ ✓ s ✓ ✓ ¬ a s ¬ a ′ s ✓ ✓ s ✓ t ✓ s a
Constructing S < ( ❙ 3 ) 1 s S < ( ❙ 3 ): a ′ ✓ s ✓ t ′ ✓ s ✓ ✓ ¬ a s ¬ a ′ s ✓ ✓ s ✓ t ✓ s a 0 s
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