Varieties of De Morgan Monoids T. Moraschini 1 , J.G. Raftery 2 , - - PowerPoint PPT Presentation

Varieties of De Morgan Monoids

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

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

• t ≤ γ1 & . . . & t ≤ γn
• ⇒ 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.

s s

trivial DM

s s

Rt inconsistent

❩❩❩❩❩ ❩ ⑦ ✚✚✚✚✚ ✚ ❃

Subvarieties

• f DM

Axiomatic extensions

• f Rt

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.

s s s ❅ ❅ s

• s

s

trivial

V(2) V(❈4) V(❙3) V(❉4)

RA Subvarieties of RA (´ Swirydowicz 1995)

s ❏ ❏ ❏ s ❜❜❜❜ s ✡ ✡ ✡ s ✧ ✧ ✧ ✧ s s

trivial

V(❈4) V(❙3) V(❉4) V(2)

DM Subvarieties of DM

Important Algebras

❈4 ❉4 2 ❙3

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 s s a t = f ¬a

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

• ne 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.

s s

• s

❅ ❅ s ❅ ❅

• s

s ¬(f 2) t b ¬b f f 2 ❇:

with b · b = f 2, ¬b · ¬b = f 2 and b · ¬b = f .

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, ◮ ((f → x) ∨ (x → t)) → 0 = 0

[0 := ¬(f 2)]. DM

M Odd SM trivial V(2) V(❈4) V(❉4) V(❙3)

Exploring M

Every subdirectly irreducible algebra in M arises by a construction

• f 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

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s s s

t a A Dunn monoid

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s

t′ a′ A′

s s

t a A

Skew Reflection

s s s

t′ a′ A′

s s

t a A

s

1

Skew Reflection

s s s

t′ a′ A′

s s ✑✑✑ ✑

t a A

s

1 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, a · b < c′ iff a < (b · c)′. Then there is a unique way of turning the struc- ture into a De Morgan monoid S<(❆) = A ∪ A′ ∪ {0, 1}; ∨, ∧, ·, ¬, t ∈ M,

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

In particular if we specify that a < b′ for all a, b ∈ A, then we get the reflection construc- tion, which is an older idea, see Meyer (1973) 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)

Thm. Let K be a cover of V(❈4) within M. Then K = V(❆) 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 s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s a t ¬a ❙3:

Constructing R(❙3)

s s s s s s s s a t ¬a ¬a′ t′ a′ 1

Constructing R(❙3)

s s s s s s s s a t ¬a ¬a′ t′ a′ 1 R(❙3):

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ❙3:

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s a t ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ t a ¬a ¬a′ t′ a′

Constructing S<(❙3)

s s s s s s 0 s 1 s ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ t a ¬a ¬a′ t′ a′ S<(❙3):

Covers of V(❈4)

R(2) R(❙3) S<(❙3) S<(❈4)

s s s s t′ f ′ 1 f t s s s s s s s a t ¬a ¬a′ t′ a′ 1 s s s s

• s

❅ ❅ ❅ s

• s
• s

a t ¬a ¬a′ t′ a′ 1 s ❅ ❅ ❅ s s s

• s

❅ ❅ ❅ ❅ ❅ s

• s
• s
• s

¬(f 2) t f f 2 (f 2)′ f ′ t′ ¬(f 2)′ 1 s ❅ ❅ ❅ ❅ ❅ s s s

Covers of V(❈4)

S<(❚5) S<(❚6)

s s s s

• s

❅ ❅ ❅ s

• s
• ❅

❅ s 1 t c → t c t′ (c → t)′ c′ s ❅ ❅ ❅ s

• s

s ❅ ❅ s

• s

s s

• s

❅ ❅ ❅ ❅ ❅ s

• s
• s
• ❅

❅ s t c t′ 1 c → t (c → t)′ c′ s ❅ ❅ ❅ ❅ ❅ s s

• s

❅ ❅ s

• s

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

Structurally Complete Subvarieties of M

Thm. The covers of V(❈4) within M are just V(R(2)), V(R(❙3)), V(S<(❙3)), V(S<(❈4)), V(S<(❚5)) and V(S<(❚6)). Thm. All the covers of V(❈4) within M are structurally complete.

Covers of V(❈4) in M

Here ❆ denotes V(❆), and S is really S<. DM

M R(2) R(❙3) S(❙3) S(❈4) S(❚4) S(❚6) Odd SM trivial 2 ❈4 ❉4 ❙3

Definitions Structural Completeness Constructions Skew Reflection Covers of V(❈4) Reflection of ❙3 Skew Reflection of ❙3 Conclusion