Congruence permutable Fregean varieties Katarzyna Somczyska - - PowerPoint PPT Presentation

Outline Introduction The structure of finite algebras Free algebras

Congruence permutable Fregean varieties

Katarzyna Słomczyńska

Pedagogical University in Kraków

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras

1

Introduction Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

2

The structure of finite algebras From algebras to frames From frames to algebras Representation theorem

3

Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Definition (Blok, K¨

• hler & Pigozzi 1984, Idziak, KS & Wroński 2009)

An algebra A with a distinguished constant 1 is called Fregean if it fulfills the following two conditions: () ΘA (1, a) = ΘA (1, b) implies a = b for a, b ∈ A, where ΘA (1, c) is the smallest congruence containing the pair (1, c) (i.e., it is congruence

• rderable),

(R) 1/α = 1/β implies α = β for α, β ∈ Con (A) (i.e., it is point regular or 1−regular). Orderability The condition () allows us to introduce a natural order on A by putting a b iff ΘA (1, b) ⊆ ΘA (1, a) . Clearly, 1 is the largest element in (A, ).

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Definition (Blok, K¨

• hler & Pigozzi 1984, Idziak, KS & Wroński 2009)

An algebra A with a distinguished constant 1 is called Fregean if it fulfills the following two conditions: () ΘA (1, a) = ΘA (1, b) implies a = b for a, b ∈ A, where ΘA (1, c) is the smallest congruence containing the pair (1, c) (i.e., it is congruence

• rderable),

(R) 1/α = 1/β implies α = β for α, β ∈ Con (A) (i.e., it is point regular or 1−regular). Orderability The condition () allows us to introduce a natural order on A by putting a b iff ΘA (1, b) ⊆ ΘA (1, a) . Clearly, 1 is the largest element in (A, ).

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Definition (Blok, K¨

• hler & Pigozzi 1984, Idziak, KS & Wroński 2009)

An algebra A with a distinguished constant 1 is called Fregean if it fulfills the following two conditions: () ΘA (1, a) = ΘA (1, b) implies a = b for a, b ∈ A, where ΘA (1, c) is the smallest congruence containing the pair (1, c) (i.e., it is congruence

• rderable),

(R) 1/α = 1/β implies α = β for α, β ∈ Con (A) (i.e., it is point regular or 1−regular). 1-regularity From the condition (R) it follows that the congruences in A are uniquely determined by their 1–cosets traditionally called filters: Con (A) ← → Φ (A) . In particular, for a ∈ A, one can replace ΘA (1, a) by the filter generated by a: ΘA (1, a) ← → 1/ΘA (1, a) = [a) := {b ∈ A : b a} .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Fregean varieties A class of algebras is called Fregean if all algebras in this class are Fregean with respect to the common constant term 1. Examples Typical well-known examples of Fregean varieties (which are algebraic counterparts of intuitionistic, intermediate and classical logics or their fragments): Boolean algebras (CPC) (Boole 1854) Heyting algebras (IPC) (Heyting 1930) Hilbert algebras (IPC, →) (Henkin 1950, Monteiro 1955) Brouwerian semilattices (IPC, →, ∧) (Monteiro 1955) equivalential algebras (IPC, ↔) (Wroński & Kabziński 1975) Boolean groups (CPC, ↔) (xx = 1, Bernstein 1939) By an equivalential algebra we mean a grupoid A = (A, ↔) that is a subreduct

• f a Heyting algebra with the operation ↔ given by

x ↔ y = (x → y) ∧ (y → x) .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Fregean varieties A class of algebras is called Fregean if all algebras in this class are Fregean with respect to the common constant term 1. Examples Typical well-known examples of Fregean varieties (which are algebraic counterparts of intuitionistic, intermediate and classical logics or their fragments): Boolean algebras (CPC) (Boole 1854) Heyting algebras (IPC) (Heyting 1930) Hilbert algebras (IPC, →) (Henkin 1950, Monteiro 1955) Brouwerian semilattices (IPC, →, ∧) (Monteiro 1955) equivalential algebras (IPC, ↔) (Wroński & Kabziński 1975) Boolean groups (CPC, ↔) (xx = 1, Bernstein 1939) By an equivalential algebra we mean a grupoid A = (A, ↔) that is a subreduct

• f a Heyting algebra with the operation ↔ given by

x ↔ y = (x → y) ∧ (y → x) .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Properties of the congruence lattice in Fregean varieties Fregean varieties are: congruence modular. Fregean varieties need not: be congruence distributive (CD), be congruence permutable (CP), have definable principal congruences (DPC), have the congruence extension property (CEP). CP and CD: Boolean algebras, Heyting algebras, Brouwerian semilattices; CD and not CP: Hilbert algebras; CP and not CD, not DPC, not CEP: equivalential algebras.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Theorem () A variety V is congruence orderable with respect to a constant term 1 if and

• nly if for every subdirectly irreducible algebra A ∈ V with the monolith µ we

have |1/µ| = 2 and all the other µ-cosets have one element. Theorem Let A be an algebra in a Fregean variety V. Then: A is subdirectly irreducible iff there is the largest non-unit element in A; A is simple iff |A| = 2.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Theorem (SC1) Let A be a subdirectly irreducible algebra in a Fregean variety V with the monolith µ. Then the centralizer (0 : µ) does not exceed µ. V fulfills the condition (SC1 - strong C1, Idziak & KS 2001); (SC1) ⇒ (C1); (C1, Freese & McKenzie 1987) [α, β] = (α ∧ [β, β]) ∨ (β ∧ [α, α]) . for A ∈ V, α, β ∈ Con(A). Theorem (Criterion for Fregeanity) The variety V (A) generated by a finite algebra A is Fregean if it is 1-regular and all algebras in HS (A) are congruence orderable.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

Theorem (SC1) Let A be a subdirectly irreducible algebra in a Fregean variety V with the monolith µ. Then the centralizer (0 : µ) does not exceed µ. V fulfills the condition (SC1 - strong C1, Idziak & KS 2001); (SC1) ⇒ (C1); (C1, Freese & McKenzie 1987) [α, β] = (α ∧ [β, β]) ∨ (β ∧ [α, α]) . for A ∈ V, α, β ∈ Con(A). Theorem (Criterion for Fregeanity) The variety V (A) generated by a finite algebra A is Fregean if it is 1-regular and all algebras in HS (A) are congruence orderable.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties

The subject of this talk Congruence permutable Fregean varieties Theorem (on the existence of the equivalential term) Every congruence permutable Fregean variety V has a binary term · that turns each of its algebras into an equivalential algebra. Moreover: ΘA (a, b) = ΘA (1, a · b) for a, b ∈ A, A ∈ V, and so the equivalence operation serves here as the principal congruence term. (Idziak, KS & Wroński 2009)

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Theorem A - a finite algebra from a congruence permutable Fregean variety. Then: f : As → A is a polynomial of A if and only if f preserves congruences of A and its commutator operation (Idziak & KS 2001); The clone of polynomials of A is uniquely determined by the congruence lattice of A expanded by the commutator operation, i.e., by the structure Concom (A) := (Con(A); ∧, ∨, [·, ·]). Recovering the algebra From the information contained in Concom (A) one can recover successively: the universe of algebra, and the equivalential term, a bunch of idempotent unary polynomials, a bunch of binary polynomials, that, after adding all constants, generate the clone of polynomials A.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Theorem A - a finite algebra from a congruence permutable Fregean variety. Then: f : As → A is a polynomial of A if and only if f preserves congruences of A and its commutator operation (Idziak & KS 2001); The clone of polynomials of A is uniquely determined by the congruence lattice of A expanded by the commutator operation, i.e., by the structure Concom (A) := (Con(A); ∧, ∨, [·, ·]). Recovering the algebra From the information contained in Concom (A) one can recover successively: the universe of algebra, and the equivalential term, a bunch of idempotent unary polynomials, a bunch of binary polynomials, that, after adding all constants, generate the clone of polynomials A.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

A - a finite algebra from congruence permutable Fregean variety. Irreducible elements J(A) - ∨-irreducible in Con(A); I(A) := {a ∈ A : ΘA (1, a) ∈ J(A)} - irreducible elements in A; I(A) ∋ a → δA (a) := ΘA (1, a) ∈ J(A) - a natural bijection; δ−

A (a) - the unique subcover of δA(a) in Con(A).

Equivalence relation a, b ∈ I(A) a ∼ b iff I δ−

A (a) , δA(a)

, I δ−

A (b) , δA(b)

are projective in Con(A); Definition In a modular lattice two intervals of the form I[a ∧ b, a] and I[b, a ∨ b] are called transposes. We say that I[a, b] and I[c, d] are projective iff there is a finite sequence of intervals I[a, b] = I1, I2, . . . , In = I[c, d] such that Ik and Ik+1 are transposes for 1 k n − 1.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

A - a finite algebra from congruence permutable Fregean variety. Irreducible elements J(A) - ∨-irreducible in Con(A); I(A) := {a ∈ A : ΘA (1, a) ∈ J(A)} - irreducible elements in A; I(A) ∋ a → δA (a) := ΘA (1, a) ∈ J(A) - a natural bijection; δ−

A (a) - the unique subcover of δA(a) in Con(A).

Equivalence relation a, b ∈ I(A) a ∼ b iff I δ−

A (a) , δA(a)

, I δ−

A (b) , δA(b)

are projective in Con(A); Definition In a modular lattice two intervals of the form I[a ∧ b, a] and I[b, a ∨ b] are called transposes. We say that I[a, b] and I[c, d] are projective iff there is a finite sequence of intervals I[a, b] = I1, I2, . . . , In = I[c, d] such that Ik and Ik+1 are transposes for 1 k n − 1.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

A - a finite algebra from congruence permutable Fregean variety. Irreducible elements J(A) - ∨-irreducible in Con(A); I(A) := {a ∈ A : ΘA (1, a) ∈ J(A)} - irreducible elements in A; I(A) ∋ a → δA (a) := ΘA (1, a) ∈ J(A) - a natural bijection; δ−

A (a) - the unique subcover of δA(a) in Con(A).

Equivalence relation a, b ∈ I(A) a ∼ b iff I δ−

A (a) , δA(a)

, I δ−

A (b) , δA(b)

are projective in Con(A); cosets of ∼ have a natural structure of Boolean groups: if U := a/∼, U := U ∪ {1}, then U is closed under the equivalence operation and (U, ·) is a Boolean group; the operation · on U be defined exclusively by means of the lattice structure of Con(A), since for a, b ∈ U, a = b there is exactly one c ∈ U, such that δA (a) ∨ δA (c) = δA (b) ∨ δA (c), namely c = a · b; moreover, U is a δ−

A (a) , δA(a)

• minimal set in A.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

A - a finite algebra from congruence permutable Fregean variety. Irreducible elements J(A) - ∨-irreducible in Con(A); I(A) := {a ∈ A : ΘA (1, a) ∈ J(A)} - irreducible elements in A; I(A) ∋ a → δA (a) := ΘA (1, a) ∈ J(A) - a natural bijection; δ−

A (a) - the unique subcover of δA(a) in Con(A).

Equivalence relation a, b ∈ I(A) a ∼ b iff I δ−

A (a) , δA(a)

, I δ−

A (b) , δA(b)

are projective in Con(A); cosets of ∼ have a natural structure of Boolean groups: if U := a/∼, U := U ∪ {1}, then U is closed under the equivalence operation and (U, ·) is a Boolean group; the operation · on U be defined exclusively by means of the lattice structure of Con(A), since for a, b ∈ U, a = b there is exactly one c ∈ U, such that δA (a) ∨ δA (c) = δA (b) ∨ δA (c), namely c = a · b; moreover, U is a δ−

A (a) , δA(a)

• minimal set in A.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Theorem (Recovering the universe) For every x ∈ A there is exactly one set R (x) ⊂ I(A), called the representation of x, such that:

1

R (x) is an -antichain;

2

R (x) consists of pairwise ∼-nonequivalent elements. Moreover, ΘA (1, x) =

a∈R(x) δA(a).

Representation A ∋ x → R (x) ∈ {M ⊂ I(A) : M fulfills (1),(2)} is a bijection. Recovering the equivalence Operation · on two such antichains is the result of performing first the Boolean group operation on each equivalence class separately, and then choosing minimal elements only, remembering that the result has to be an antichain.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Theorem (Recovering the universe) For every x ∈ A there is exactly one set R (x) ⊂ I(A), called the representation of x, such that:

1

R (x) is an -antichain;

2

R (x) consists of pairwise ∼-nonequivalent elements. Moreover, ΘA (1, x) =

a∈R(x) δA(a).

Representation A ∋ x → R (x) ∈ {M ⊂ I(A) : M fulfills (1),(2)} is a bijection. Recovering the equivalence Operation · on two such antichains is the result of performing first the Boolean group operation on each equivalence class separately, and then choosing minimal elements only, remembering that the result has to be an antichain.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Theorem (Recovering the universe) For every x ∈ A there is exactly one set R (x) ⊂ I(A), called the representation of x, such that:

1

R (x) is an -antichain;

2

R (x) consists of pairwise ∼-nonequivalent elements. Moreover, ΘA (1, x) =

a∈R(x) δA(a).

Representation A ∋ x → R (x) ∈ {M ⊂ I(A) : M fulfills (1),(2)} is a bijection. Recovering the equivalence Operation · on two such antichains is the result of performing first the Boolean group operation on each equivalence class separately, and then choosing minimal elements only, remembering that the result has to be an antichain.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Unary polynomials For every U ∈ I(A)/∼ the mapping given by pU (x) :=

• 1,

if R (x) ∩ U = ∅; xU, if R (x) ∩ U = {xU}, for x ∈ A, is an idempotent unary polynomial on A, whose range is U. Binary polynomials In(A) := {a ∈ I(A) : [δA (a) , δA (a)] = δA (a)} If a ∈ In(A), then V := a/∼ = {a} and the mapping given by mV (x, y) :=

• 1,

if pV (x) = pV (y) = 1 a,

• therwise

, for x, y ∈ A, is a binary polynomial of A, whose range is V . Moreover,

• V , mV |V ×V
• is a semilattice.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Unary polynomials For every U ∈ I(A)/∼ the mapping given by pU (x) :=

• 1,

if R (x) ∩ U = ∅; xU, if R (x) ∩ U = {xU}, for x ∈ A, is an idempotent unary polynomial on A, whose range is U. Binary polynomials In(A) := {a ∈ I(A) : [δA (a) , δA (a)] = δA (a)} If a ∈ In(A), then V := a/∼ = {a} and the mapping given by mV (x, y) :=

• 1,

if pV (x) = pV (y) = 1 a,

• therwise

, for x, y ∈ A, is a binary polynomial of A, whose range is V . Moreover,

• V , mV |V ×V
• is a semilattice.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

An algebra polynomially equivalent to original we define a new algebra: A∗ := A; ·, {pU}U∈I(A)/∼ , {mV }V ∈In(A)/∼

• ;

it is polynomially equivalent to A; the universe and operations of this algebra can be described in terms of a relational-algebraic structure generated by Concom (A): A → relational-algebraic structure A∗ := (I (A) , , ∼, ·, In (A) , 1)

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

Axiomatic definition of Fregean frames By a Fregean ∨-frame we mean a structure of the form F = (F; , ∼, e, Fn, 1), where: (F; , 1) is a finite poset with the largest element 1; ∼ is an equivalence relation on F + := F\ {1}; e is a partial binary operation on F, such that (U, e) is a Boolean group for all U ∈ F +/∼, where U := U ∪ {1}; Fn ⊂ F and x/∼ = {x} for every x ∈ Fn, satisfying the following two compatibility conditions: if x, y, z ∈ F +, x < y and y ∼ z then x < z; if x, y, z ∈ F +, y ∼ z and e (y, z) < x then y < x or z < x.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

A∗ := (I (A) , , ∼, ·, In (A) , 1) → A∗ := A; ·, {pU}U∈I(A)/∼ , {mV }V ∈In(A)/∼

• Recovering an algebra A (A∗) from its frame A∗

1

(Universe) Every element of A can be uniquely decomposed into a finite antichain of elements coming from different equivalence classes. We can identify the universe with the collection of such antichains.

2

(Equivalence) The operation · on two such antichains is given as the result of performing first the Boolean group operation on each equivalence class separately, and then choosing minimal elements only.

3

(Family of unary operations P) It consists of ‘projections’ of antichains

• n given equivalence classes (supplemented by 1).

4

(Family of binary operations M) It consists of meets of ‘projections’ of two antichains on the equivalence classes (supplemented by 1) of elements from In (A).

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

F = (F; , ∼, e, Fn, 1) → F∗ = F; E, {PU}U∈F +/∼ , {MV }V ∈Fn/∼

• Building the algebra F∗ from a frame F

1

(Universe) The universe F of the algebra F∗ is the collection of antichains

• f elements coming from different equivalence classes.

2

(Equivalence) The operation · on two such antichains is given as the result of performing first the Boolean group operation on each equivalence class separately, and then choosing minimal elements; (F,·) is an equivalential algebra.

3

(Family of unary operations P) It consists of ‘projections’ of antichains

• n given equivalence classes (supplemented by 1).

4

(Family of binary operations M) It consists of the meets of ‘projections’

• f two antichains on the equivalence classes (supplemented by 1) of

elements from Fn. After adding the constants that correspond to the elements of F the algebra F∗ generates a congruence permutable Fregean variety.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

frames ∋ F → F∗ ∈ algebras algebras ∋ A → A∗ ∈ frames Theorem (on the correspondence between algebras and frames)

1

If A is a finite algebra from a congruence permutable Fregean variety then the algebra (A∗)∗ is isomorphic to the algebra A∗ polynomially equivalent to A;

2

If F is a finite Fregean frame then the frame (F∗)∗ is isomorphic to F. In (A) = ∅ → A is solvable In (A) = I (A) → the frame is fully described by (I (A) , ) → A is polynomially equivalent to a Brouwerian semilattice

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras From algebras to frames From frames to algebras Representation theorem

frames ∋ F → F∗ ∈ algebras algebras ∋ A → A∗ ∈ frames Theorem (on the correspondence between algebras and frames)

1

If A is a finite algebra from a congruence permutable Fregean variety then the algebra (A∗)∗ is isomorphic to the algebra A∗ polynomially equivalent to A;

2

If F is a finite Fregean frame then the frame (F∗)∗ is isomorphic to F. In (A) = ∅ → A is solvable In (A) = I (A) → the frame is fully described by (I (A) , ) → A is polynomially equivalent to a Brouwerian semilattice

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

V - Fregean variety, FV (n) - n-generated free algebra in V two problems:

construction, computing (or estimating) of free spectra: (|FV (1)| , |FV (2)| , . . .);

they are studied e.g. for Hilbert algebras and Brouwerian semilattices. Equivalential algebras |F (n)| < ∞ |F (1)| = 2 |F (2)| = 9 (Wroński 1975) |F (3)| = 4 415 434 (Wroński 1993, KS 1994) |F (4)| = ? construction (KS 2008)

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

V - Fregean variety, FV (n) - n-generated free algebra in V two problems:

construction, computing (or estimating) of free spectra: (|FV (1)| , |FV (2)| , . . .);

they are studied e.g. for Hilbert algebras and Brouwerian semilattices. Equivalential algebras |F (n)| < ∞ |F (1)| = 2 |F (2)| = 9 (Wroński 1975) |F (3)| = 4 415 434 (Wroński 1993, KS 1994) |F (4)| = ? construction (KS 2008)

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

∧−irreducible congruences and hereditary sets A - a finite equivalential algebra (M (A) , ) := poset of meet-irreducible congruences of A an equivalence relation ∼ in M (A) : µ ∼ ν ⇔ µ+ = ν+ every equivalence class U = µ/∼ (supplemented with 1U = µ+) is endowed with Boolean group operation: (U, •), µ • ν := (µ ÷ ν)′ ∩ µ+ we call a set Z ⊂ M (A) hereditary, iff:

Z = Z ↑; for all U ∈ M (A) /∼ , if U ↑\ U ⊂ Z, then either Z ∩ U = U or Z ∩ U is a maximal subgroup in (U, •)

Theorem (on ∧−representation, KS 2005) The map A ∋ a → {ϕ ∈ M (A) : (1, a) ∈ ϕ} ∈ H (A) := {Z ⊂ M (A) : Z-hereditary} establishes the isomorphism between the algebras A and (H (A) , ↔), where Z1 ↔ Z2 := ((Z1 ÷ Z2)↓)′ for Z1, Z2 ∈ H (A) .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

∧−irreducible congruences and hereditary sets A - a finite equivalential algebra (M (A) , ) := poset of meet-irreducible congruences of A an equivalence relation ∼ in M (A) : µ ∼ ν ⇔ µ+ = ν+ every equivalence class U = µ/∼ (supplemented with 1U = µ+) is endowed with Boolean group operation: (U, •), µ • ν := (µ ÷ ν)′ ∩ µ+ we call a set Z ⊂ M (A) hereditary, iff:

Z = Z ↑; for all U ∈ M (A) /∼ , if U ↑\ U ⊂ Z, then either Z ∩ U = U or Z ∩ U is a maximal subgroup in (U, •)

Theorem (on ∧−representation, KS 2005) The map A ∋ a → {ϕ ∈ M (A) : (1, a) ∈ ϕ} ∈ H (A) := {Z ⊂ M (A) : Z-hereditary} establishes the isomorphism between the algebras A and (H (A) , ↔), where Z1 ↔ Z2 := ((Z1 ÷ Z2)↓)′ for Z1, Z2 ∈ H (A) .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

∧−irreducible congruences and hereditary sets A - a finite algebra from a congruence permutable Fregean variety (M (A) , ) := poset of meet-irreducible congruences of A an equivalence relation ∼ in M (A) : µ ∼ ν ⇔ (I[µ, µ+], I[ν, ν+] projective in Con(A)) every equivalence class U = µ/∼ (supplemented with 1U = µ+) is endowed with Boolean group operation: (U, •), µ • ν := (µ ÷ ν)′ ∩ µ+ we call a set Z ⊂ M (A) hereditary, iff:

Z = Z ↑; for all U ∈ M (A) /∼ , if U ↑\ U ⊂ Z, then either Z ∩ U = U or Z ∩ U is a maximal subgroup in (U, •)

Theorem (on ∧−representation, KS 2005) The map A ∋ a → {ϕ ∈ M (A) : (1, a) ∈ ϕ} ∈ H (A) := {Z ⊂ M (A) : Z-hereditary} establishes the one-to-one correspondence between A and H (A). ·

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

General idea To construct the n-generated equivalential algebra F(n) it suffices to describe its ∧−frame: (M (F(n)) , , ∼, •) , and then to find all hereditary sets H (F(n)). The ∧−frame of F(n) is constructed recursively from the highest layer, and then moving down layer-by-layer to the lowest, where each equivalence class is contained in one layer. The members of the k-th layer (k = 1, . . . , n) are precisely µ ∈ M (F(n)) such that k is the maximal cardinality of a chain in M (F(n)) with the least element µ. The number of layers is equal to the number of free generators.

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Free equivalential algebras for ‘small’ n (KS 2008) n layer 1 layer 2 layer 3 layer 4 |M (F (n))| |F (n)| 1 1 − − − 1 2 2 3 2 − − 5 9 3 7 15 15 − 37 4 415 434 4 15 88 45 126 ? ? ?

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Free equivalential algebras for ‘small’ n (KS 2008) n layer 1 layer 2 layer 3 layer 4 |M (F (n))| |F (n)| 1 1 − − − 1 2 2 3 2 − − 5 9 3 7 15 15 − 37 4 415 434 4 15 88 45 126 ? ? ?

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Linear equivalential algebras (KS 2005, KS 2007) Linear equivalential algebras form a subvariety of the variety of equivalential algebras. This subvariety is generated by the algebra ω := (N1, ·) with the binary

• peration: i · j := max (i, j) for i = j, and i · j := 1 for i = j.

They give the algebraic semantics of the equivalential fragment of one of the most important intermediate logics - G¨

• del-Dummett logic (G¨
• del

1932, Dummett 1959).

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Theorem (on cardinalities of free linear equivalential algebras, KS 2005)

1

(the formula for the number of elements of the ∧-frame, n 1) |M (F (n))| = 2 b (n) − 1 , where b (n) are the ordered Bell numbers.

2

(the recurrence formula for the free spectrum, n 2) |F (n)| = Wn (|F (1)| , . . . , |F (n − 1)|) , where a polynomial Wn is given by Wn (s1, . . . , sn−1) := n

k=0

n

k

n−1

m=1s

• p(

n−k n−m−2p)( k 2p)

m

.

3

(an estimate and an approximate formula for the free spectrum) 2nb(n−1) |F (n)| 22b(n)−1 ln ln |F (n)| ∼ n ln n .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Free linear equivalential algebras for ‘small’ n (KS 2005) n |M (F (n))| |F (n)| 1 1 1 2 2 5 9 3 25 6380 4 149 14 088 274 367 211 230 781 681 5 1081 ≈ 6. 919 438 668 729 81 × 10159 The formula for the free spectrum for linear equivalential algebras of height 3 |FE3 (n)| = 2n(2n−1−1) + n · 2(n+1)2n−2−n + 2n2n−2 − (n + 1) 2n(2n−2−1) .

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

Thank you for attention

Katarzyna Słomczyńska Congruence permutable Fregean varieties

Outline Introduction The structure of finite algebras Free algebras Free algebras in Fregean varieties Representation theorem for equivalential algebras Construction of free equivalential algebras Free equivalential algebras for ‘small’ n Free linear equivalential algebras

KS, Free spectra of linear equivalential algebras, Journal of Symbolic Logic 70 (2005), 1341-1358. KS, Free equivalential algebras, Annals of Pure and Applied Logic 155 (2008), 86-96. Idziak P., KS & Wroński A., Fregean Varieties, International Journal of Algebra and Computation 19 (2009), 595-645.

Katarzyna Słomczyńska Congruence permutable Fregean varieties