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. Moraschini1, J.G. Raftery2, and J.J. Wannenburg2

1Academy of Sciences of the Czech Republic, Czech Republic 2University 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 Rt can be characterized as follows γ1, . . . , γn ⊢Rt α iff DM

• t ≤ γ1 & . . . & t ≤ γn
• ⇒ t ≤ α.

LV(DM) :

s s

trivial DM

s s

Rt inconsistent

❩❩❩❩❩ ❩ ⑦ ✚✚✚✚✚ ✚ ❃

Subvarieties

• f DM

Axiomatic extensions

• f Rt

Important algebras

❈4 ❉4 2 ❙♥ (n odd)

s s s s ¬(f 2) t f f 2 s

• s

❅ ❅ s

• ❅

❅ s ¬(f 2) t f f 2 s s f t s q q q s s s q q q s −(n − 1)/2 −1 0 = t = f 1 (n − 1)/2

f := ¬t

◮ 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:

when |i| ≤ |j|, then i · j =

• j

if |i| = |j| i ∧ j

• therwise.

Atoms of LV(DM)

s ❏ ❏ ❏ s ❜❜❜❜ s ✡ ✡ ✡ s ✧ ✧ ✧ ✧ s s

trivial

V(❈4) V(❙3) V(❉4) V(2)

DM Subvarieties of DM We investigate the covers of the atoms in LV(DM).

Covers of V(2) and V(❙3)

s ❏ ❏ ❏ s ❜❜❜❜ s ✡ ✡ ✡ s ✧ ✧ ✧ ✧ s s s

trivial

V(❈4) V(❙3) V(❙5) V(❉4) V(2)

DM Subvarieties of DM

◮ The join of any two atoms is a

cover of both.

◮ The remaining covers are precisely

the join-irreducible (JI) covers. Thm.

◮ V(2) has no JI cover. ◮ The only JI cover of V(❙3) is

V(❙5).

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 ❆.

❉❆♣:

qqq qqq s¬(a2) ❅ ❅ s ap−2

• s¬a

❅ ❅ s ap−1

• ❅

❅ ap = f 2 f s s s¬(ap−2) s a2 ❅ ❅

• s¬(ap−1)
• s

a ❅ ❅

• ❅

❅ ¬(f 2) t s s ◮ For every prime p, the algebra ❉❆♣

generates a cover of V(❉4),

◮ so there are infinitely many covers

• f V(❉4).

A non-finitely generated cover of V(❉4)

❉∞:

❅ ❅ s a3 sa3

f

qqq s¬(a3) ❅ ❅ s ¬(a3

f )

qqq

• s¬(a2)

❅ ❅ s ¬(a2

f )

• s¬a

❅ ❅ s ¬af

• ❅

❅ f f 2 s s

• sa2

f

• s

a2 ❅ ❅

• saf
• s

a ❅ ❅

• ❅

❅ ¬(f 2) t s s

◮ Not all covers of V(❉4) are

finitely generated,

◮ for example, ❉∞ generates

a cover of V(❉4) that is not finitely generated.

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

• f 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 trivial V(2) V(❈4) V(❉4) V(❙3)

Condition 1

• 1. K = V(❆) for some simple 1-generated De Morgan monoid

❆, such that ❈4 embeds into ❆ but is not isomorphic to ❆.

s s s s q q q s s s ¬(f 2) t a a2 ap−1 f = ap f 2

❆♣:

◮ For every prime p, the

algebra ❆♣ generates a cover of V(❈4),

◮ so, there are infinitely

many covers of V(❈4) that satisfy condition 1.

s s s s q q q s s s s ¬(f 2) t a a2 ¬(a2) ¬a f f 2

❆∞:

◮ There are covers of

V(❈4) that are not finitely generated,

◮ for example, ❆∞

generates a cover of V(❈4).

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: ❈5:

❅ ❅ s sf sf 2

• s
• s

t ❅ ❅ s¬(f 2)

❈6:

¬(f 2) s s s

• ❅

❅ f · (t ∧ f ) s ❅ ❅ t s s ❅ ❅

• f

s ¬(f · (t ∧ f )) s s ❅ ❅

• f 2

s

❈7:

s

• s

❅ ❅ ❅ ❅ ❅ s

• s
• s
• s

¬(f 2) f · (t ∧ f ) f t f 2 s ❅ ❅ ❅ ❅ ❅ s s s

❈8:

¬(f 2) s s

• t s
• ❅

❅ s s

• ❅

❅ s ❅ ❅ s s ❅ ❅

• s

s s ❅ ❅

• s
• f

s

• ❅

❅ f 2 s

Condition 3

• 3. K ⊆ M

Every subdirectly irreducible algebra in M arises by a construction

• f 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

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s s s

t a B Dunn monoid

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s s

t′ a′ B′

s s

t a B

Skew Reflection

s

⊥

s s

t′ a′ B′

s s

t a B

s

⊤

Skew Reflection

s

⊥

s s

t′ a′ B′

s s ✑✑✑ ✑

t a B

s

⊤ 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, a < b′ iff t < (a · b)′. Then there is a unique way of turning the structure into a De Morgan monoid S<(❇) = B ∪ B′ ∪ {⊥, ⊤}; ∨, ∧, ·, ¬, t ∈ M,

• f which ❇ is a subreduct, where ¬ extends ′.

In particular, if we specify that a < b′ for all a, b ∈ B, then we get the reflection construction, which is an older idea, see Meyer (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 trivial V(2) V(❈4) V(❉4) V(❙3)

Covers of V(❈4) within M

There are just six of these: R(2):

s s s s ⊥ t′ f ′ ⊤ f t s s

R(❙3):

s s s s s ⊥ −1 1 1′ 0′ −1′ ⊤ s s s

S<(❙3):

s

• s

❅ ❅ ❅ s

• s
• s

⊥ −1 1 1′ 0′ −1′ ⊤ s ❅ ❅ ❅ s s

S<(❈4):

s

• s

❅ ❅ ❅ ❅ ❅ s

• s
• s
• s

⊥ ¬(f 2) t f f 2 (f 2)′ f ′ t′ ¬(f 2)′ ⊤ s ❅ ❅ ❅ ❅ ❅ s s s

S<(❚5):

s s s s

• s

❅ ❅ ❅ s

• s
• ❅

❅ s ⊥ ⊤ t c → t c t′ (c → t)′ c′ s ❅ ❅ ❅ s

• s

s ❅ ❅ s

• S<(❚6):

s s s

• s

❅ ❅ ❅ ❅ ❅ s

• s
• s
• ❅

❅ s ⊥ t c t′ ⊤ c → t c′ s ❅ ❅ ❅ ❅ ❅ s s

• s

❅ ❅ s

• s

❚5 is idempotent and ❚6 is idempotent except for t′ ∧ (c → t).

Summary

Thm. Every cover of V(❈4) within M has no proper nontrivial subquasivariety other than V(❈4). DM

U M trivial V(2) V(❈4) V(❉4) V(❙3) V(❙5)

Definitions Atoms Covers of V(2) and V(❙3) Covers of V(❉4) Covers of V(❈4) Skew Reflection Covers of V(❈4) within M