Translating open formulas
Miguel Campercholi
FaMAF
Universidad Nacional de Córdoba
Argentina
Pucón, December 2012
Introduction
Disjunctions to equations
In the class
R = frings without zero divisorsg
we have
x = y _ z = w $ (x
y ) (z
w) = 0
Introduction
Disjunctions to equations
In the class
R = frings without zero divisorsg
we have
x = y _ z = w $ (x
y ) (z
w) = 0
Thus, every positive open formula is equivalent to a conjunction of
equations over R.
Introduction
Everything to equations
On a boolean algebra B de…ne the unary operation
(
1 if x 6= 0,
9x =
0 if x = 0.
Then, for the class
M = fhB, 9i j B boolean algebrag = fsimple monadic algebrasg
Introduction
Everything to equations
On a boolean algebra B de…ne the unary operation
(
1 if x 6= 0,
9x =
0 if x = 0.
Then, for the class
M = fhB, 9i j B boolean algebrag = fsimple monadic algebrasg
we have
x 6 = y $ 9 (x 4 y ) = 1
x = y _ z = w $ 9 (x 4 y ) ^ 9 (z 4 w ) = 0
Introduction
Everything to equations
On a boolean algebra B de…ne the unary operation
(
1 if x 6= 0,
9x =
0 if x = 0.
Then, for the class
M = fhB, 9i j B boolean algebrag = fsimple monadic algebrasg
we have
x 6 = y $ 9 (x 4 y ) = 1
x = y _ z = w $ 9 (x 4 y ) ^ 9 (z 4 w ) = 0
Hence, every open formula is equivalent to a conjunction of
equations over M.
Introduction
The discriminator
The (quaternary) discriminator on a set A is the function
(
z if x = y ,
A
d (x, y , z, w ) =
w if x 6= y .
Introduction
The discriminator
The (quaternary) discriminator on a set A is the function
(
z if x = y ,
A
d (x, y , z, w ) =
w if x 6= y .
Fact
Let K be a class of algebras with a term t (x, y , z, w ) satisfying
t A = d A , for all A 2 K. Then:
Introduction
The discriminator
The (quaternary) discriminator on a set A is the function
(
z if x = y ,
A
d (x, y , z, w ) =
w if x 6= y .
Fact
Let K be a class of algebras with a term t (x, y , z, w ) satisfying
t A = d A , for all A 2 K. Then:
every trivial-satis…able open formula is equivalent to a
conjunction of equations over K.
Introduction
The discriminator
The (quaternary) discriminator on a set A is the function
(
z if x = y ,
A
d (x, y , z, w ) =
w if x 6= y .
Fact
Let K be a class of algebras with a term t (x, y , z, w ) satisfying
t A = d A , for all A 2 K. Then:
every trivial-satis…able open formula is equivalent to a
conjunction of equations over K.
If there are constants 0, 1 such that K 0 = 1 ! x = y ,
then the translation works for all open formulas over K .
Introduction
Questions
A formula is equational over a class K if it is equivalent to a
conjunction of equations over K.
Introduction
Questions
A formula is equational over a class K if it is equivalent to a
conjunction of equations over K.
Problem
When is every positive open formula equational over K?
Introduction
Questions
A formula is equational over a class K if it is equivalent to a
conjunction of equations over K.
Problem
When is every positive open formula equational over K?
An answer to this question is provided in [Cze&Dzi1990].
Introduction
Questions
A formula is equational over a class K if it is equivalent to a
conjunction of equations over K.
Problem
When is every positive open formula equational over K?
An answer to this question is provided in [Cze&Dzi1990].
Problem
When is every trivial-satis…able open formula equational over K?
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
A is RFSI if ∆ is meet irreducible in ConQ (A)
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
A is RFSI if ∆ is meet irreducible in ConQ (A)
A is Relatively Simple (RS) if jConQ (A)j
2
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
A is RFSI if ∆ is meet irreducible in ConQ (A)
A is Relatively Simple (RS) if jConQ (A)j
2
Q is congruence distributive if every ConQ (A) is distributive
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
A is RFSI if ∆ is meet irreducible in ConQ (A)
A is Relatively Simple (RS) if jConQ (A)j
2
Q is congruence distributive if every ConQ (A) is distributive
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
A is RFSI if ∆ is meet irreducible in ConQ (A)
A is Relatively Simple (RS) if jConQ (A)j
2
Q is congruence distributive if every ConQ (A) is distributive
Theorem ([Cze&Dzi1990])
The following are equivalent for a class K:
Every positive open formula is equational over K.
Preliminaries
Quasivarieties and the positive case
Let Q be a quasivariety, i.e., Q = SPPu (Q), and let A 2 Q.
ConQ (A) := fθ 2 Con(A) j A/θ 2 Qg
A is RFSI if ∆ is meet irreducible in ConQ (A)
A is Relatively Simple (RS) if jConQ (A)j
2
Q is congruence distributive if every ConQ (A) is distributive
Theorem ([Cze&Dzi1990])
The following are equivalent for a class K:
Every positive open formula is equational over K.
Q (K) is congruence distributive and
K Q (K)RFSI = SPu (K).
The general case
Gathering facts
Suppose every trivial-satis…able open formula is equational over K.
Then:
Translations work for SPu (K).
The general case
Gathering facts
Suppose every trivial-satis…able open formula is equational over K.
Then:
Translations work for SPu (K).
Since
x =y !z =w
is equational, it must be preserved by homomorphisms
between algebras in SPu (K). Thus,
SPu (K) = Q (K)RS .
The general case
Gathering facts
Suppose every trivial-satis…able open formula is equational over K.
Then:
Translations work for SPu (K).
Since
x =y !z =w
is equational, it must be preserved by homomorphisms
between algebras in SPu (K). Thus,
SPu (K) = Q (K)RS .
By the [Cze&Dzi1990] Theorem we have that Q (K) is
congruence distributive.
The general case
Gathering facts
Suppose every trivial-satis…able open formula is equational over K.
Then:
Translations work for SPu (K).
Since
x =y !z =w
is equational, it must be preserved by homomorphisms
between algebras in SPu (K). Thus,
SPu (K) = Q (K)RS .
By the [Cze&Dzi1990] Theorem we have that Q (K) is
congruence distributive.
The quaternary discriminator is equationally de…nable in K.
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
Every trivial-satis…able open formula is equational over K.
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
2
Every trivial-satis…able open formula is equational over K.
The formula
x =y !z =w
is equational over K.
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
2
Every trivial-satis…able open formula is equational over K.
The formula
x =y !z =w
3
is equational over K.
The quaternary discriminator is equationally de…nable in K.
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
2
Every trivial-satis…able open formula is equational over K.
The formula
x =y !z =w
3
4
is equational over K.
The quaternary discriminator is equationally de…nable in K.
Q (K) is congruence distributive and
K Q (K)RS = SPu (K)
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
2
Every trivial-satis…able open formula is equational over K.
The formula
x =y !z =w
3
4
5
is equational over K.
The quaternary discriminator is equationally de…nable in K.
Q (K) is congruence distributive and
K Q (K)RS = SPu (K)
K
QRS for some quasivariety Q with EDRPC1 .
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
2
Every trivial-satis…able open formula is equational over K.
The formula
x =y !z =w
3
4
5
is equational over K.
The quaternary discriminator is equationally de…nable in K.
Q (K) is congruence distributive and
K Q (K)RS = SPu (K)
K
QRS for some quasivariety Q with EDRPC1 .
Main Theorem
Theorem
Let K be class of algebras. T.f.a.e.:
1
2
Every trivial-satis…able open formula is equational over K.
The formula
x =y !z =w
3
4
5
is equational over K.
The quaternary discriminator is equationally de…nable in K.
Q (K) is congruence distributive and
K Q (K)RS = SPu (K)
K
1 There
QRS for some quasivariety Q with EDRPC1 .
is a conj. of eq. ε(x, y , z, w ) such that
θA
Q (a, b ) = fhc, d i j A
ε (a, b, c, d )g, 8A 2 Q.
Bonus
Translating all open formulas
Lemma
Let K be an elementary class. T.f.a.e.:
No member of K properly contains a trivial subalgebra.
Bonus
Translating all open formulas
Lemma
Let K be an elementary class. T.f.a.e.:
No member of K properly contains a trivial subalgebra.
There are unary terms 11 (w ), . . . , 1k (w ), 01 (w ), . . . , 0k (w )
such that
K ~1 (w ) = ~0 (w ) ! x = y .
Bonus
Translating all open formulas
Lemma
Let K be an elementary class. T.f.a.e.:
No member of K properly contains a trivial subalgebra.
There are unary terms 11 (w ), . . . , 1k (w ), 01 (w ), . . . , 0k (w )
such that
K ~1 (w ) = ~0 (w ) ! x = y .
Bonus
Translating all open formulas
Lemma
Let K be an elementary class. T.f.a.e.:
No member of K properly contains a trivial subalgebra.
There are unary terms 11 (w ), . . . , 1k (w ), 01 (w ), . . . , 0k (w )
such that
K ~1 (w ) = ~0 (w ) ! x = y .
Corollary
Let K be a class of algebras. T.f.a.e.:
Every open formula is equational over K .
Bonus
Translating all open formulas
Lemma
Let K be an elementary class. T.f.a.e.:
No member of K properly contains a trivial subalgebra.
There are unary terms 11 (w ), . . . , 1k (w ), 01 (w ), . . . , 0k (w )
such that
K ~1 (w ) = ~0 (w ) ! x = y .
Corollary
Let K be a class of algebras. T.f.a.e.:
Every open formula is equational over K .
Every trivial-satis…able open formula is equational over K, and
no member of K properly contains a trivial subalgebra.
Example
S.I. Heyting algebras with constant coatom
Recall that a Heyting algebra H is S.I. i¤ H r f1H g has a greatest
element c H .
Example
S.I. Heyting algebras with constant coatom
Recall that a Heyting algebra H is S.I. i¤ H r f1H g has a greatest
element c H .
Let L = f0, 1, ^, _, !g [ fc g, and let
D
E
K = f H, c H j H 2 HSI g.
Example
S.I. Heyting algebras with constant coatom
Recall that a Heyting algebra H is S.I. i¤ H r f1H g has a greatest
element c H .
Let L = f0, 1, ^, _, !g [ fc g, and let
D
E
K = f H, c H j H 2 HSI g.
The following hold in K
x 6 = y , (x $ y ) _ c = c
x = y or z = w , (x $ y ) ^ (z $ w ) = 1
Example
S.I. Heyting algebras with constant coatom
Recall that a Heyting algebra H is S.I. i¤ H r f1H g has a greatest
element c H .
Let L = f0, 1, ^, _, !g [ fc g, and let
D
E
K = f H, c H j H 2 HSI g.
The following hold in K
x 6 = y , (x $ y ) _ c = c
x = y or z = w , (x $ y ) ^ (z $ w ) = 1
Thus, K satis…es all equivalent conditions of the theorem. In
particular Q (K) has EDRPC and is semisimple.
References
Czelakowski, J., Dziobiak, W.: Congruence distributive
quasivarieties whose …nitely subdirectly irreducible members
form a universal class, Algebra Universalis 27, 128–149 (1990)
References
Czelakowski, J., Dziobiak, W.: Congruence distributive
quasivarieties whose …nitely subdirectly irreducible members
form a universal class, Algebra Universalis 27, 128–149 (1990)
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