Epimorphisms in varieties of square-increasing residuated structures - - PowerPoint PPT Presentation

Epimorphisms in varieties of square-increasing residuated structures

• T. Moraschini1, J.G. Raftery2, and J.J. Wannenburg2,3

1Academy of Sciences of the Czech Republic, Czech Republic 2University of Pretoria, South Africa 3DST-NRF Centre of Excellence in Mathematical and Statistical Sciences

(CoE-MaSS)

LATD, August 2018

Opinions expressed and conclusions arrived at are those of the author and are not necessarily to be attributed to the CoE-MaSS.

ES Property

◮ Let K be a variety of algebras. Let ❆, ❇ ∈ K and f : ❆ → ❇

a homomorphism.

◮ f is a K-epimorphism if, whenever ❈ ∈ K and g, h : ❇ → ❈

are homomorphisms such that g ◦ f = h ◦ f , then f = g.

◮ Note that if f is surjective, then f is a K-epimorphism. ◮ If all K-epimorphisms are surjective, then K has the

epimorphism surjectivity (ES) property.

◮ ❆ is called a K-epic subalgebra of ❇ if the injection map

i : ❆ → ❇ is a K-epimorphism.

◮ K has surjective epimorphisms iff there is no ❇ ∈ K with a

proper K-epic subalgebra. ❆

✒✑ ✓✏

f✲ ❇

✒✑ ✓✏

g

✲

h

✲

❈

✒✑ ✓✏

Beth property

If a variety K algebraizes a logic ⊢, then K has the ES property iff ⊢ has the infinite (deductive) Beth (definability) property, i.e., whenever a set Z of variables is defined implicitly in terms of a disjoint set X of variables by means of some set Γ of formal assertions about X ∪ Z, then Γ also defines Z explicitly in terms of X.

Residuated structures

A square-increasing commutative residuated lattice (SRL) ❆ = A; ∧, ∨, ·, →, e comprises

◮ a lattice A; ∧, ∨, ◮ a commutative monoid A; ·, e ◮ that is square-increasing, i.e., x ≤ x · x := x2, ◮ and a binary operation → satisfying the law of residuation

x · y ≤ z iff y ≤ x → z. We may enrich the language of SRLs with a constant symbol f and define an involution ¬x := x → f that satisfies x = ¬¬x and x → ¬y = y → ¬x, thus obtaining SIRLs. It follows that ¬e = f .

Residuated structures

◮ In any SRL if x, y ≤ e then x · y = x ∧ y. ◮ Integral (x ≤ e) SRLs are called Brouwerian algebras in which

case · coincides with ∧. Given an S[I]RL ❆, its negative cone A− := {a ∈ A : a ≤ e} can be turned into a Brouwerian algebra ❆− = A−; ∧, ∨, e, →−, by restricting the operations ∧, ∨ to A− and defining a →− b = (a → b) ∧ e, for a, b ∈ A−.

◮ Heyting algebras are Brouwerian algebras with a distinguished

bottom element ⊥.

◮ Distributive SRLs are called Dunn monoids. ◮ Distributive SIRLs are called De Morgan monoids. ◮ Idempotent (x = x · x) De Morgan monoids are called

Sugihara monoids. Their structure is very well understood.

Logic

The varieties of algebras on the left algebraize the logics on the right. SRLs LRt+ SIRLs LRt Dunn monoids Rt+ De Morgan monoids Rt Sugihara monoids RMt Brouwerian Algebras positive intuitionistic logic Heyting Algebras intuitionistic logic

Depth

For any Brouwerian algebra ❇, let Pr(❇) be the set of non-empty prime lattice filters F of ❇, including B itself. We define the depth of F in Pr(❇) to be the greatest n ∈ ω (if it exists) such that there is a chain in Pr(❇) of the the form F = F0 F1 · · · Fn = B. If no greatest such n exists, we say F has depth ∞ in Pr(❇). We define depth(❇) = sup{depth(F) : F ∈ Pr(❇)}. If ❆ is an S[I]RL and K is a variety of S[I]RLs, we define depth(❆) = depth(❆−) and depth(K) = sup{depth(❆) : ❆ ∈ K}. If a variety of S[I]RLs is finitely generated, then it has finite depth (but not conversely).

Main Theorem

Thm. Let K be a variety of SRLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

◮ KFSI denotes the class of finitely subdirectly irreducible

members of K. Sg(−) denotes subalgebra generation.

◮ The argument also works for expansions of SRLs, where the

additional operations on each algebra in K are compatible with the congruences of its SRL reduct.

◮ In particular, the theorem holds if we add involution or

distinguished least elements to the type.

◮ Therefore, this theorem generalises the recent result of

Bezhanishvili, Moraschini and Raftery (2017) that every variety of Brouwerian or Heyting algebras of finite depth has the ES property.

(1) can’t be dropped

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

Urquhart (1999) showed that the crystal lattice has a proper epic subalgebra in the variety of De Morgan monoids. Its involution-less reduct also has a proper epic subalgebra in the variety of Dunn monoids.

s s

• s

❅ ❅ s

• s

❅ ❅ s (f )2 f a = ¬a b = ¬b e ¬(f 2) crystal:

(2) can’t be dropped for SRLs

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

Bezhanishvili, Moraschini and Raftery (2017) exhibit a Brouwerian algebra of infinite depth that generates a variety without the ES property.

❅ s

• s

❅ ❅ s

• s

❅ ❅ s

• s

❅ ❅ s

• s

❅ ❅ s q q q e b0 c0 b1 c1 ❆1 :

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s s s s

e a D Dunn monoid

Reflection

s

⊥

s s

e′ a′ D′

s s

e a D

s

⊤ R(D)

Reflection

◮ There is a unique way of turning the structure into a De

Morgan monoid R(❉) = D ∪ D′ ∪ {⊥, ⊤}; ∧, ∨, ·, ¬, e of which ❉ is a subreduct and where ¬ extends ′.

◮ If K is a class of Dunn monoids we define

R(K) := {R(❉) : ❉ ∈ K}. Thm. A Dunn monoid ❉ is generated by its negative cone iff R(❉) is. Thm. Let K be a variety of Dunn monoids. Then K has surjective epimorphisms iff V(R(K)) has. In particular, ❉ has a proper K-epic subalgebra iff R(❉) has a proper V(R(K))-epic subalgebra.

(2) can’t be dropped for SIRLs

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

One can show that R(❆1) has infinite depth and has a proper V(R(❆1))-epic subalgebra.

❅ s

• s

❅ ❅ s

• s

❅ ❅ s q q q ❅ s

• s

❅ ❅ s

• s

❅ ❅ s q q q e b0 c0 e′ c′ b′ s⊥ s⊤ R(❆1) :

Applications

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

There are many interesting varieties of S[I]RLs whose FSI members are generated by their negative cones. For example Sugihara monoids, their involution-less subreducts, and Brouwerian algebras. Our main theorem therefore implies (the known results) that each of their finitely generated subvarieties have surjective epimorphisms.

New applications

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

◮ ❈4 is the only 0-generated algebra onto which

finitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms.

◮ There is a largest variety U of De Morgan

monoids consisting of homomorphic preimages

• f ❈4 (and trivial algebras).

s s s s ¬(f 2) e f f 2 ❈4:

DMM

U trivial V(❈4)

New applications

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

V(❈4) has just ten covers in U, each of which satisfies (1) and is generated by a finite algebra, see (Moraschini, Raftery and Wannenburg, 2017). Each of these varieties therefore has surjective epimorphisms. DM

U trivial V(❈4)

Semilinear SRLs

An SRL ❆ is called semilinear if it is a subdirect product of totally

• rdered SRLs. These form a variety.

A semilinear S[I]RL is FSI iff it is totally ordered. Certain semilinear varieties of S[I]RLs enjoy strong positive ES results, see (Bezhanishvili, Moraschini and Raftery, 2017):

◮ Every variety of semilinear Brouwerian or Heyting algebras

(resp. relative Stone and G¨

• del algebras) has surjective

epimorphisms.

◮ Every variety of Sugihara monoids is semilinear and has

surjective epimorphisms,

◮ the same applies to the involution-less subreducts of Sugihara

monoids.

(1) can’t be dropped

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

Thm. Sg❆2(4) = {0, 1, 4, 8} is a proper epic subalgebra of ❆2 in the variety of semilinear S[I]RLs.

Proof.

Let ❈ be a totally ordered S[I]RL and g, h : ❆2 → ❈ such that g(4) = h(4). Suppose w.l.o.g. g(2) ≤ h(2). g(2) · h(2) ≤ h(2) · h(2) = h(4) = g(4) h(2) ≤ g(2) → g(4) = g(2) So, g(2) = h(2), but then g = h.

s s s s s 8 4 2 1 = e ❆2 :

· is integer multiplication truncated at 8. Note that 2 = 2 → 4.

Semilinear SRLs generated by their negative cones

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

Thm. If a totally ordered SRL is generated by idempotent elements, then it is idempotent. Therefore, if K is a variety of semilinear SRLs satisfying (1), then K is a subvariety of GSM, the variety of generalized Sugihara monoids (idempotent semilinear SRLs that satisfy ((x ∨ e) → e) → e = x ∨ e). Galatos and Raftery (2015) provides a transparent description of totally ordered generalized Sugihara monoids, based on results in (Raftery, 2007).

Semilinear SRLs generated by their negative cones

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

Determining which subvarieties of GSM has the ES property is still an open problem. Thm. GSM has surjective epimorphisms. The proof of the theorem above exploits a recent result by Campercholi (2016), which states that a variety K with EDPM has the ES property iff no algebra in KFSI has a proper K-epic subalgebra.

Semilinear SIRLs generated by their negative cones

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

Thm. Let K be a variety of semilinear SIRLs satisfying x ≤ f 2. Then K satisfies (1) iff K = V(R(K′)) for some variety K′ of generalised Sugihara monoids. In particular, C :=

• ❆ :

❆ is a non-trivial totally ordered SIRL satisfying (1) and x ≤ f 2

• = R(GSMFSI).

Thm. It follows that V(C) = V(R(GSM)) has surjective epimorphisms.

Semilinear SIRLs generated by their negative cones

Thm. Let K be a variety of S[I]RLs such that 1) ❆ = Sg(A−) for every ❆ ∈ KFSI and 2) K has finite depth. Then K has surjective epimorphisms.

C ∗ :=

• ❆ : ❆ is a totally ordered

SIRL satisfying (1)

• .

Exploiting a description of finitely subdirectly irreducible De Morgan monoids by Moraschini, Raftery and Wannenburg (2017): Thm. ❆ ∈ C ∗ iff

• 1. ❆ is an FSI Sugihara monoid, or
• 2. A = (⊥] ∪ R(❉) ∪ [⊤) where (⊥] and [⊤) are chains of

idempotents, and ❉ ∈ GSMFSI. V(C ∗) is the largest variety of semilinear SIRLs satisfying (1). Thm. V(C ∗) is locally finite and has surjective epimorphisms.

(⊥] D D′ [⊤)

Definitions Main Theorem Examples Reflection Applications Semilinear SRLs