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 Introduction 1 Fregean varieties Properties of Fregean varieties Congruence permutable Fregean varieties The structure of finite algebras 2 From algebras to frames From frames to algebras Representation theorem Free algebras 3 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 Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras Definition (Blok, K¨ ohler & Pigozzi 1984, Idziak, KS & Wroński 2009) An algebra A with a distinguished constant 1 is called Fregean if it fulfills the following two conditions: ( � ) Θ A ( 1 , a ) = Θ A ( 1 , b ) implies a = b for a , b ∈ A , where Θ A ( 1 , c ) is the smallest congruence containing the pair ( 1 , c ) (i.e., it is congruence orderable ), ( 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 Θ A ( 1 , b ) ⊆ Θ A ( 1 , a ) . Clearly, 1 is the largest element in ( A , � ) . Katarzyna Słomczyńska Congruence permutable Fregean varieties
Outline Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras Definition (Blok, K¨ ohler & Pigozzi 1984, Idziak, KS & Wroński 2009) An algebra A with a distinguished constant 1 is called Fregean if it fulfills the following two conditions: ( � ) Θ A ( 1 , a ) = Θ A ( 1 , b ) implies a = b for a , b ∈ A , where Θ A ( 1 , c ) is the smallest congruence containing the pair ( 1 , c ) (i.e., it is congruence orderable ), ( 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 Θ A ( 1 , b ) ⊆ Θ A ( 1 , a ) . Clearly, 1 is the largest element in ( A , � ) . Katarzyna Słomczyńska Congruence permutable Fregean varieties
Outline Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras Definition (Blok, K¨ ohler & Pigozzi 1984, Idziak, KS & Wroński 2009) An algebra A with a distinguished constant 1 is called Fregean if it fulfills the following two conditions: ( � ) Θ A ( 1 , a ) = Θ A ( 1 , b ) implies a = b for a , b ∈ A , where Θ A ( 1 , c ) is the smallest congruence containing the pair ( 1 , c ) (i.e., it is congruence orderable ), ( 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 Θ A ( 1 , a ) by the filter generated by a : Θ A ( 1 , a ) ← → 1 / Θ A ( 1 , a ) = [ a ) := { b ∈ A : b � a } . Katarzyna Słomczyńska Congruence permutable Fregean varieties
Outline Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras 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 of a Heyting algebra with the operation ↔ given by x ↔ y = ( x → y ) ∧ ( y → x ) . Katarzyna Słomczyńska Congruence permutable Fregean varieties
Outline Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras 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 of a Heyting algebra with the operation ↔ given by x ↔ y = ( x → y ) ∧ ( y → x ) . Katarzyna Słomczyńska Congruence permutable Fregean varieties
Outline Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras 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 Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras Theorem ( � ) A variety V is congruence orderable with respect to a constant term 1 if and only 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 Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras 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 Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras 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 Fregean varieties Introduction Properties of Fregean varieties The structure of finite algebras Congruence permutable Fregean varieties Free algebras 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: Θ A ( a , b ) = Θ A ( 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
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