Varieties of De Morgan Monoids II: Covers of Atoms T. Moraschini 1 , - PowerPoint PPT Presentation

Varieties of De Morgan Monoids II: Covers of Atoms 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 TACL, June 2017

De Morgan monoids A De Morgan monoid ❆ = � A ; ∨ , ∧ , · , ¬ , t � comprises ◮ a distributive lattice � A ; ∨ , ∧� , ◮ a square-increasing ( x ≤ x · x ) commutative monoid � A ; · , t � , ◮ satisfying x = ¬¬ x ◮ and x · y ≤ z iff x · ¬ z ≤ ¬ y . ◮ x → y := ¬ ( x · ¬ y ) DM denotes the variety of all De Morgan monoids.

Algebraic logic The logic R t can be characterized as follows � � γ 1 , . . . , γ n ⊢ R t α iff DM � t ≤ γ 1 & . . . & t ≤ γ n ⇒ t ≤ α. inconsistent DM s s ❃ ✚ ❩❩❩❩❩ ✚✚✚✚✚ L V ( DM ) : ❩ ⑦ s s R t trivial Subvarieties Axiomatic extensions of R t of DM

Important algebras 2 ❙ ♥ ( n odd) ❈ 4 ❉ 4 ( n − 1) / 2 s f 2 s q f 2 q s f s q � ❅ � ❅ 1 s t f t s t s s s ❅ s � ❅ � 0 = t = f s f ¬ ( f 2 ) ¬ ( f 2 ) s s − 1 s q q f := ¬ t q − ( n − 1) / 2 s ◮ The first three are exactly the simple 0-generated De Morgan monoids, see Slaney (1989). ◮ For any positive odd number n , the · of ❙ ♥ is as follows: � j if | i | � = | j | when | i | ≤ | j | , then i · j = i ∧ j otherwise.

Atoms of L V ( DM ) DM s V ( ❉ 4 ) V ( 2 ) ❜❜❜❜ ✧ s s s s ❏ ✡ ✧ V ( ❈ 4 ) V ( ❙ 3 ) ✧ ❏ ✡ ❏ ✧ ✡ s trivial Subvarieties of DM We investigate the covers of the atoms in L V ( DM ).

Covers of V ( 2 ) and V ( ❙ 3 ) DM s ◮ The join of any two atoms is a cover of both. ◮ The remaining covers are precisely the join-irreducible (JI) covers. V ( ❙ 5 ) Thm. s V ( ❉ 4 ) V ( 2 ) ◮ V ( 2 ) has no JI cover. ❜❜❜❜ ✧ s s s s ❏ ✡ ✧ V ( ❈ 4 ) V ( ❙ 3 ) ✧ ❏ ✡ ◮ The only JI cover of V ( ❙ 3 ) is ❏ ✡ ✧ s V ( ❙ 5 ) . trivial Subvarieties of DM

Covers of V ( ❉ 4 ) Thm. Every join-irreducible cover of V ( ❉ 4 ) has the form V ( ❆ ) for some simple 1-generated De Morgan monoid ❆ , where ❉ 4 embeds into ❆ but is not isomorphic to ❆ . a p = f 2 s � ❅ � ❅ ❉❆ ♣ : a p − 1 s s f � ❅ � ◮ For every prime p , the algebra ❉❆ ♣ � ❅ � a p − 2 s ¬ a s ❅ � generates a cover of V ( ❉ 4 ), ❅ � qqq s ¬ ( a 2 ) a 2 qqq ◮ so there are infinitely many covers s � ❅ � ❅ a s ¬ ( a p − 2 ) s � ❅ � of V ( ❉ 4 ). � ❅ � t s ¬ ( a p − 1 ) s ❅ � ❅ ¬ ( f 2 ) � s

A non-finitely generated cover of V ( ❉ 4 ) f 2 s � ❅ � ❅ f ❉ ∞ : ¬ a f s s � ❅ � ¬ ( a 2 � ❅ � f ) s s ¬ a ◮ Not all covers of V ( ❉ 4 ) are � ❅ � ¬ ( a 3 � ❅ � f ) s ¬ ( a 2 ) s finitely generated, ❅ � qqq ❅ � a 3 qqq s ¬ ( a 3 ) s ◮ for example, ❉ ∞ generates � ❅ � ❅ a 2 s a 3 s a cover of V ( ❉ 4 ) that is not � ❅ � f � ❅ � a s a 2 s finitely generated. � ❅ � f � ❅ � t s a f s ❅ � ❅ ¬ ( f 2 ) � s

Covers of V ( ❈ 4 ) More cases, as ❈ 4 has diverse homomorphic pre-images. In fact: Thm. (Slaney) If h : ❆ → ❇ is a homomorphism from a finitely subdirectly irreducible De Morgan monoid into a 0-generated De Morgan monoid, then h is an isomorphism or ❇ ∼ = ❈ 4 . ◮ There is a largest subvariety U of DM such that every non-trivial member of U has ❈ 4 as a homomorphic image. ◮ U is finitely axiomatized. ◮ There is a largest subvariety M of DM such that ❈ 4 is a retract of all non-trivial members of M . ◮ M is axiomatized, relative to U , by t ≤ f .

Covers of V ( ❈ 4 ) Thm. If K is a join-irreducible cover of V ( ❈ 4 ) , then exactly one of the following holds. 1. K = V ( ❆ ) for some simple 1-generated De Morgan monoid ❆ , such that ❈ 4 embeds into ❆ but is not isomorphic to ❆ . 2. K = V ( ❆ ) for some (finite) 0-generated subdirectly irreducible De Morgan monoid ❆ ∈ U \ M . 3. K ⊆ M . DM U M V ( 2 ) V ( ❉ 4 ) V ( ❈ 4 ) V ( ❙ 3 ) trivial

Condition 1 1. K = V ( ❆ ) for some simple 1-generated De Morgan monoid ❆ , such that ❈ 4 embeds into ❆ but is not isomorphic to ❆ . ❆ ♣ : ❆ ∞ : f 2 f 2 s s f = a p ◮ For every prime p , the f s s ◮ There are covers of a p − 1 algebra ❆ ♣ generates ¬ a s s V ( ❈ 4 ) that are not q a cover of V ( ❈ 4 ), ¬ ( a 2 ) q s finitely generated, q q ◮ so, there are infinitely a 2 s q ◮ for example, ❆ ∞ q many covers of a 2 a s s generates a cover of V ( ❈ 4 ) that satisfy t a V ( ❈ 4 ). s s condition 1. ¬ ( f 2 ) t s s ¬ ( f 2 ) s

Condition 2 2. K = V ( ❆ ) for some (finite) 0-generated subdirectly irreducible De Morgan monoid ❆ ∈ U \ M . Slaney (1989) characterized all the 0-generated subdirectly irreducible De Morgan monoids. They are all finite, and apart from the simple ones, they are: ❈ 7 : ❈ 8 : ❈ 5 : ❈ 6 : f 2 f 2 s s f 2 s s f 2 s s � ❅ � ❅ s � � ❅ f � ❅ ¬ ( f · ( t ∧ f )) ❅ � ❅ f f s s s s ❅ � � ❅ � s � ❅ s s s � ❅ � ❅ � ❅ � ❅ � � ❅ ❅ t � ❅ � ❅ f · ( t ∧ f ) s s s s s � ❅ ❅ � ❅ � s f s s s s ❅ � ❅ � ❅ � ❅ ❅ � ❅ � t t f · ( t ∧ f ) ❅ � s s s s s ❅ � ❅ � t s � ❅ � s ❅ � � ❅ � s s s ❅ s � ¬ ( f 2 ) s ❅ � ¬ ( f 2 ) s ¬ ( f 2 ) s ¬ ( f 2 ) s

Condition 3 3. K ⊆ M Every subdirectly irreducible algebra in M arises by a construction of Slaney (1993) from a Dunn monoid ❇ [essentially a De Morgan monoid without the involution ¬ ], i.e., a square-increasing distributive lattice-ordered commutative monoid � B ; ∨ , ∧ , · , → , 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 B a s s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s s B s a s Dunn monoid

Skew Reflection t s s B s a s Dunn monoid

Skew Reflection t s s B s a s Dunn monoid

Skew Reflection t s s B s a s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s s B a s s Dunn monoid

Skew Reflection t s a ′ s B B ′ a s t ′ s

Skew Reflection a ′ t s s B ′ B t ′ a s s

Skew Reflection B ′ a ′ s t s t ′ s a B s

Skew Reflection B ′ a ′ s t s t ′ s a s B

Skew Reflection B ′ a ′ s t ′ s t s a s B

Skew Reflection B ′ a ′ s t ′ s t s a s B

Skew Reflection B ′ a ′ s t ′ s t s s a B

Skew Reflection B ′ a ′ s t ′ s t s a s B

Skew Reflection s ⊤ B ′ a ′ s t ′ s t s a s B ⊥ s

Skew Reflection Declare that a < b ′ for certain a , b ∈ B in such a way that � B ∪ B ′ ∪ {⊥ , ⊤} ; ≤� is a distributive lattice, t < t ′ and for all a , b ∈ B , s a < b ′ iff t < ( a · b ) ′ . ⊤ B ′ Then there is a unique way of turning the a ′ structure into a De Morgan monoid s t ′ ✑ s ✑✑✑ S < ( ❇ ) = � B ∪ B ′ ∪ {⊥ , ⊤} ; ∨ , ∧ , · , ¬ , t � ∈ M , t s a of which ❇ is a subreduct, where ¬ extends ′ . s In particular, if we specify that a < b ′ for all B a , b ∈ B , then we get the reflection ⊥ construction, which is an older idea, see Meyer s (1973) and Galatos and Raftery (2004). In this case we write R ( ❇ ) for S < ( ❇ ).

Covers of V ( ❈ 4 ) within M Thm. Let K be a cover of V ( ❈ 4 ) within M . Then K = V ( ❆ ) for some finite skew reflection ❆ of a subdirectly irreducible Dunn monoid ❇ , where ⊥ is meet-irreducible in ❆ , and ❆ is generated by the greatest strict lower bound of t in ❇ . DM U M V ( 2 ) V ( ❉ 4 ) V ( ❈ 4 ) V ( ❙ 3 ) trivial