Equivalence of varieties of MTL-algebras
built from prelinear semihoops
Tommaso Flaminio
Dipartimento di Scienze Teoriche e Applicate, Università dell’Insubria.
Italy
[email protected]
sites.google.com/site/tomflaminio
(with Stefano Aguzzoli and Brunella Gerla)
T. Flaminio (DiSTA-Varese)
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Introduction
An MTL-algebra is a structure A = (A, , →, ∧, ∨, ⊥, >) where:
- (A, ∧, ∨, ⊥, >) is a bounded distributive lattice,
- (A, , >) is a commutative monoid,
- x y ≤ z ⇔ z ≤ x → y holds for every x, y , z ∈ A,
- (x → y ) ∨ (y → x) = > holds for every x, y ∈ A.
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Introduction
An MTL-algebra is a structure A = (A, , →, ∧, ∨, ⊥, >) where:
- (A, ∧, ∨, ⊥, >) is a bounded distributive lattice,
- (A, , >) is a commutative monoid,
- x y ≤ z ⇔ z ≤ x → y holds for every x, y , z ∈ A,
- (x → y ) ∨ (y → x) = > holds for every x, y ∈ A.
In every MTL-algebra we can define further operations and abbreviations:
¬x = x → ⊥, x ⊕ y = ¬x → y , x 2 = x x, 2x = x ⊕ x.
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Introduction
An MTL-algebra is a structure A = (A, , →, ∧, ∨, ⊥, >) where:
- (A, ∧, ∨, ⊥, >) is a bounded distributive lattice,
- (A, , >) is a commutative monoid,
- x y ≤ z ⇔ z ≤ x → y holds for every x, y , z ∈ A,
- (x → y ) ∨ (y → x) = > holds for every x, y ∈ A.
In every MTL-algebra we can define further operations and abbreviations:
¬x = x → ⊥, x ⊕ y = ¬x → y , x 2 = x x, 2x = x ⊕ x.
MTL-algebras form the variety MTL.
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Introduction
A strongly perfect MTL-algebra (SBP0 -algebra), is any MTL-algebra satisfying
(N) ¬(x)2 → (¬¬x → x) = 1,
(DL) (2x)2 = 2(x 2 ).
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Introduction
A strongly perfect MTL-algebra (SBP0 -algebra), is any MTL-algebra satisfying
(N) ¬(x)2 → (¬¬x → x) = 1,
(DL) (2x)2 = 2(x 2 ).
The class of SBP0 forms a variety denoted by SBP0 . In what follows we will
denote BP0 = MTL + (DL) (i.e., perfect MTL-algebras).
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Introduction
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Introduction
T. Flaminio (DiSTA-Varese)
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Introduction
SMTL is MTL + {x ∧ ¬x = ⊥}
SBL is MTL + {x ∧ y = x (x → y )}
G is MTL + {x 2 = x},
P is SBL + {¬x ∨ ((x → (x y )) → y ) = >}
IBP0 is BP0 + {¬¬x = x}
DLMV is IBP0 + {x ∧ y = x (x → y )}
NM− is IBP0 + {¬(x 2 ) ∨ (x → x 2 ) = >}
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Aim
Provide categorical equivalences between SMTL and IBP0 by “defining” any
algebra in SMTL from any other algebra in IBP0 and vice versa
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From IBP0 -algebras to SMTL-algebras
Let σ be the term
σ(x) = 2x 2 ∧ x = ¬(¬(x x) ¬(x x)) ∧ x
For every IBP0 -algebra A = (A, , →, ∧, ⊥), let S(A) = (S(A), ⊗, →⊗ , ∧, ⊥)
where:
S(A) = σ[A] = {σ(a) | a ∈ A},
a ⊗ b = a b, for every a, b ∈ S(A)
a →⊗ b = σ(a → b), for every a, b ∈ S(A).
Lemma
If A is a IBP0 -algebra, S(A) is an SMTL-algebra.
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Explaining the term σ
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Explaining the term σ
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From SMTL-algebras to IBP0 -algebras
Let now ι be the binary term
ι(x, y ) = (x ∧ y ) ∨ (¬x ∧ ¬y )
For every SMTL-algebra A = (A, ⊗, →⊗ , ∧, ⊥), let
I(A) = (I (A), , →, ∧, (>, ⊥)) where:
I (A) = {(a− , a+ ) | a− , a+ ∈ A, ι(a− , a+ ) = ⊥} = {(a− , a+ ) | a− =
⊥ and a+ 6= ⊥ or a− 6= ⊥ and a+ = ⊥},
(a− , a+ ) (b − , b + ) = (b + →⊗ a− ∧ a+ →⊗ b − , a+ ⊗ b + ),
(a− , a+ ) → (b − , b + ) = (a+ ⊗ b − , a+ →⊗ b + ∧ b − →⊗ a− ),
(a− , a+ ) ∧ (b − , b + ) = (a− ∨ b − , a+ ∧ b + ).
Lemma
If A is an SMTL-algebra, I(A) is IBP0 .
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Explaining the term ι
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Theorem
For any SMTL-algebra A, A ∼
= S(I(A)) and for every IBP0 -algebra B,
B∼
I(S(B)).
=
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Morphisms
Let f : A1 → A2 be a morphism of IBP0 -algebras, for every x ∈ A1 ,
f (σ(x)) = σ(f (x)).
Thus we define
S(f ) : x ∈ S(A1 ) 7→ f (x) ∈ S(A2 ).
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Morphisms
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Morphisms
If g : B1 → B2 is a morphism of SMTL-algebras, for every (a− , a+ ) ∈ I(B1 ),
ι(g (a− ), g (a+ )) = g (ι(a− , a+ )).
We hence define
I(g ) : (a− , a+ ) ∈ I(B1 ) 7→ (g (a− ), g (a+ )).
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Morphisms
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Categorical equivalence between IBP0 and
SMTL
Theorem
IBP0 and SMTL are equivalent categories.
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Categorical equivalence between IBP0 and
SMTL
Theorem
IBP0 and SMTL are equivalent categories.
A Corollary of McKenzie’s result
R. McKenzie, An algebraic version of categorical equivalence for varieties and more general algebraic categories, in: A. Ursini, P.
Aglianò (Eds.), Logic and Algebra, in: Lecture Notes in Pure and Applied Mathematics, 180, Marcel Dekker, New York, 1996,
pp. 211–243.
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Axiomatization of subvarieties of SMTL and
IBP0 -algebras
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Axiomatization of subvarieties of SMTL and
IBP0 -algebras
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Axiomatization of subvarieties of SMTL and
IBP0 -algebras
Definition
An algebra H = (H, ∗, →, ∧, ∨, >) of type (2, 2, 2, 2, 0) is a prelinear semihoop if
it is an integral commutative residuated lattice satisfying prelinearity
(x → y ) ∨ (y → x) = >.
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Axiomatization of subvarieties of SMTL and
IBP0 -algebras
Proposition
The categories PSH, SMTLd.i. , and IBP0d.i. are equivalent.
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Axiomatization of subvarieties of SMTL and
IBP0 -algebras
Let ϕ(x) be a formula in the language of semihoops in the variables
x = x1 , . . . , xn . We define the following formulas in the language of MTL:
1
s(ϕ(x)) stands for ¬x1 ∨ . . . ∨ ¬xn ∨ ϕ(x),
2
i(ϕ(x)) stands for ¬(x1 )2 ∨ . . . ∨ ¬(xn )2 ∨ ϕ(x).
If H is any subvariety of PSH and let {ψx = > | x ∈ X } the (possibly infinite)
equations which axiomatize H within PSH. Then, with a small abuse, we will
henceforth adopt the following notation:
s(H) = {s(ψx ) = > | x ∈ X },
i(H) = {i(ψx ) = > | x ∈ X }.
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Axiomatization of subvarieties of SMTL and
IBP0 -algebras
Finally, for every variety V let us denote Λ(V) be the lattice of subvarieties of V
and let
S : Λ(PSH) → Λ(SMTL), I : Λ(PSH) → Λ(IBP0 )
be defined as follows: for every H ∈ Λ(PSH),
S(H) = {A ∈ SMTL | A |= s(H)};
I (H) = {A ∈ IBP0 | A |= i(H)};
Since each Z (H), for Z ∈ {S, I } is an equational class, S(H), and I (H) are
subvarieties of SMTL, and IBP0 , respectively.
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Isomorphism between lattices of subvarieties
via axiomatization
Proposition
The maps S and I are well defined lattice isomorphisms.
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Isomorphism between lattices of subvarieties
via axiomatization
Proposition
The maps S and I are well defined lattice isomorphisms.
/ Λ(IBP0 )
9
Λ(SMTL) o
f
S
T. Flaminio (DiSTA-Varese)
&
y
Λ(PSH)
I
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Isomorphism between lattices of subvarieties
via axiomatization
Proposition
The maps S and I are well defined lattice isomorphisms.
/ Λ(IBP0 )
9
Λ(SMTL) o
f
S
&
y
Λ(PSH)
I
Corollary
For every subvariety C of X, there exists a subvariety H of PSH such that C is
axiomatized, within X, by i(H). Moreover, if H is finitely axiomatizable in PSH,
then so it C within X.
(X is either SMTL or IBP0 )
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Thank you.
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