Katarzyna Slomczynska

1
Free algebras for the ?,fragment of IPC and KC

• Katarzyna Slomczynska
• (Pedagogical University, Kraków)
• e-mail kslomcz_at_ap.krakow.pl

2
Equivalential algebras

• equivalential algebras (E ) ? equivalential
  fragment of intuitionistic propositional logic
  (IPC? )

3
Equivalential algebras with 0

• E 0 - equivalential algebras with (constant) 0 ?
  ?, fragment of intuitionistic propositional
  logic (IPC?, ), x x 0, 0 1

• E ? ?, fragment of weak excluded middle
  logic (KC?, )
• KC IPC (p v (p) 1)

• E - linear equivalential algebras with 0 ?
  ?, fragment of Gödel-Dummett logic

4
How to construct effectively free finitely
generated algebras in E 0?

• Such constructions are known both for
• intuitionistic propositional logic (free Heyting
  algebras -
• Urquhart73, Bellissima86, Grigolia87,
  Ghilardi92, Butz98, Darnière Junker06,
  Bezhanishvili07, OConnor07)
• as well as for some of its fragments
• implication (?) (free Hilbert, or positive
  implication, algebras - Urquhart74, de
  Bruijn75)
• implication-conjunction (?,?) (free Brouwerian,
  or implicative, semilattices - de Bruijn75,
  Köhler81)
• equivalence (?) (free equivalential algebras
  Wronski93, Slomczynska05, Slomczynska08)
• and many others - de Jongh, Hendriks Renardel
  de Lavalette91, Hendriks96.

5
Fregean varieties and frames

• Fregean variety ? 1-regular congruence
  orderable.
• The equivalential algebras form a paradigm of
  congruence permutable Fregean varieties, in the
  sense that every such variety has a binary term
  that turns every of its members into an
  equivalential algebra (Idziak, Slomczynska
  Wronski).
• Fregean frame ? (Cm(A), , , ?), where A an
  algebra from a Fregean variety.
• equivalence relation µ ? if and only if I
  µ,µ and I ?,? are projective, where µ,? ?
  Cm(A) and f - unique cover of f.
• Boolean group operation Let µ ? (Cm(A). Define
  f ? ? (f ? ) n µ for f,? ? µ/
  ? µ. Then (µ/ ? µ,?) ? E 2 .
• Two extreme cases (µ,? ? Cm(A))
• congruence distributive varieties µ ? if and
  only if µ ?
• equivalential algebras µ ? if and only if µ
  ? .

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Fregean varieties and frames

• Representation theorem the elements of a finite
  algebra A from a congruence permutable Fregean
  variety can be represented as certain upwards
  closed subsets in the frame (Cm(A), , , ?)
  called hereditary subsets.
• To describe a finitely generated free algebra in
  a locally finite congruence permutable Fregean
  variety it suffices to find its Fregean frame.
  This was done for some congruence distributive
  varieties, as well as for E in (Slomczynska08).
  Here we show how it works for E 0 and some of its
  subvarieties.
• F 0(n) the n-generated free equivalential
  algebra with 0.

7
Construction the frame

• We start from the construction of the frame of F
  0(n)
• the frame is a poset divided into n1 layers
  (levels) Ek(n) (k 1,...,n1) whose
  elements are labelled by proper subsets of the
  set 0,1,...,n
• if two elements of the poset are comparable, then
  the larger lies in an upper layer
• each layer is divided into equivalence classes
  consisting of elements with the same successors
  (covers)
• the top layer E1(n) consists of only one
  equivalence class labelled by all proper subsets
  of the set 0,1,...,n and hence it has 2n1-1
  elements
• elements of any other class are labelled by all
  proper subsets of a subset K of 1,...,n
• each class supplemented by K is endowed with the
  natural Boolean group operation (the complement
  of the symmetric difference with respect to K)
  and K is the unit of this group
• the layers are constructed inductively.

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Construction the frame (top layer)

• The top layer E1(n) consists of only one
  equivalence class labelled by all proper subsets
  of the set 0,1,...,n and hence it has 2n1-1
  elements.

E1(2)
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Construction the frame (inductive step)

• Let k 1,...,n. Assume that we have already
  defined the poset E1(n) ? ? ? Ek(n).
• We consider all subsets S of the set E1(n) ? ? ?
  Ek(n) such that
• S is an upwards closed set
• the intersection of S with each equivalence
  class (supplemented by the unit) is a Boolean
  subgroup of this class (also supplemented by the
  unit)
• S intersects non-empty the lowest layer Ek(n)
  defined so far
• the labellings of all elements of S intersect
  non-empty and the intersection does not contain
  0.
• To each such set S we associate exactly one
  equivalence class in the layer Ek1(n) labelled
  by all proper subsets of the intersection of the
  labellings of all elements of S. The elements of
  this class are smaller than all elements of the
  set S.

E1(2)
10
Construction the frame (inductive step)

• Let k 1,...,n. Assume that we have already
  defined the poset E1(n) ? ? ? Ek(n).
• We consider all subsets S of the set E1(n) ? ? ?
  Ek(n) such that
• S is an upwards closed set
• the intersection of S with each equivalence
  class (supplemented by the unit) is a Boolean
  subgroup of this class (also supplemented by the
  unit)
• S intersects non-empty the lowest layer Ek(n)
  defined so far
• the labellings of all elements of S intersect
  non-empty and the intersection does not contain
  0.
• To each such set S we associate exactly one
  equivalence class in the layer Ek1(n) labelled
  by all proper subsets of the intersection of the
  labellings of all elements of S. The elements of
  this class are smaller than all elements of the
  set S.

E1(2) ? E2(2)
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Construction the frame (inductive step)

• Let k 1,...,n. Assume that we have already
  defined the poset E1(n) ? ? ? Ek(n).
• We consider all subsets S of the set E1(n) ? ?
  ? Ek(n) such that
• S is an upwards closed set
• the intersection of S with each equivalence
  class (supplemented by the unit) is a Boolean
  subgroup of this class (also supplemented by the
  unit)
• S intersects non-empty the lowest layer Ek(n)
  defined so far
• the labellings of all elements of S intersect
  non-empty and the intersection does not contain
  0.
• To each such set S we associate exactly one
  equivalence class in the layer Ek1(n) labelled
  by all proper subsets of the intersection of the
  labellings of all elements of S. The elements of
  this class are smaller than all elements of the
  set S.

E1(2) ? E2(2) ? E3(2)
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Construction the free algebra

• Having the frame defined, we describe the
  universe of the free n-generated equivalential
  algebra as the collection of all hereditary
  sets, i.e., the subsets Z of the frame such that
• 1. Z is an upwards closed set
• 2. if all the elements larger than the elements
  of a given equivalence class are contained in Z,
  then the intersection of Z (supplemented by the
  unit) with this class is either a maximal Boolean
  subgroup of this class (also supplemented by the
  unit) or is equal to this class.
• The family of hereditary sets is endowed with the
  natural equivalence operation of the dual
  pseudo-difference (i.e., for hereditary sets S
  and T we define S ? T ((S T)?).
• The free equivalential algebra F 0(n)
  (hereditary sets, ?, 0), where 0 is the
  hereditary set consisting of all elements
  labelled by subsets containing 0.

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Construction the free generators

• The sets Zk (k 0,1,...,n) consisted of all
  elements having k in their labelling are 0 and
  free generators of the free algebra F0(n).
• For n 0 we have E1(1) 1, and
  2 hereditary sets.
• For n 1 we have E1(1) 3, E2(1) 1, and
  6 hereditary sets.
• For n 2 we have E1(2) 7, E2(2) 7,
  E3(2) 4, and 538
  hereditary sets.

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How to construct effectively free finitely
generated algebras for subvarieties of E0?

• E - frame every equivalence class in the
  layer Ek1(n) has exactly one larger element in
  the top layer E1(n)

• E - frame every equivalence class in the
  layer Ek1(n) has exactly one larger element in
  the layers E1(n), , Ek(n)

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Free spectra for E and E
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Frame for E (n 3)
Frame 70 elements Free algebra 28 170 470 730
elements
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Frame for E (n 3)
Frame 58 elements Free algebra 40 706 210
elements
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Free spectra (and the cardinalities of the frames)