Varieties, Quasivarieties
and Prevarieties:
Completing the Picture
Wataru Hino & Ichiro Hasuo
The University of Tokyo
Wataru Hino
1
Outline
• Review 1: variety and quasivariety
• Review 2: orthogonality and prevariety
• New notion: sort-of-variety
New
characterization
logical
orthogonality
Variety
equations
∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴
closure property
H, S, P
Sort-of-variety Quasivariety
Prevariety
implications
preequations
∀∃!-formulas
∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴
𝐴↠𝐵
𝐴⟶𝐵
?
Wataru Hino
S, P, FC
𝐴-pure S, P, FC
2
Review: variety and quasivariety
characterization
logical
orthogonality
Variety
equations
∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴
closure property
H, S, P
Sort-of-variety Quasivariety
Prevariety
implications
preequations
∀∃!-formulas
∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴
𝐴↠𝐵
𝐴⟶𝐵
?
Wataru Hino
S, P, FC
𝐴-pure S, P, FC
3
Variety: definition
Def. (variety)
𝑉 ⊂ AlgΣ is a variety
∀𝑥 𝑠 = 𝑡
∃
⟺ 𝐸: a set of equations
signature-relevant
s.t. 𝑉 = 𝐴 ∈ AlgΣ 𝐴 ⊨ 𝐸}
definition
Example
• The class of groups is a variety for ∙ , 1, −1 , but not for ∙
• Rings, lattices (for appropriate signatures)
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4
Variety: HSP theorem
Thm. (Birkhoff)
𝑉 ⊂ AlgΣ is a variety
⟺ 𝑉 is closed under
• (H) homomorphic images
• (S) subalgebras
• (P) products
Variety can be characterized by closure property
Cor. The class of torsion-free abelian groups isn’t a variety for +, 0, −
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5
Quasivariety
Def. (quasivariety)
𝑉 ⊂ AlgΣ is a quasivariety
∀𝑥 ∧ 𝑠𝑖 = 𝑡𝑖 → 𝑠 = 𝑡
∃
⟺ 𝐸: a set of implications
s.t. 𝑉 = 𝐴 ∈ AlgΣ 𝐴 ⊨ 𝐸}
Example Torsion-free abelian groups for +, 0, −
• group-equations & ∀𝑥 𝑛 ⋅ 𝑥 = 0 → 𝑥 = 0 for each 𝑛
Thm. 𝑉 ⊂ AlgΣ is a quasivariety
⟺ 𝑉 is closed under
(S) subalgebras, (P) products and (FC) filtered colimits
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6
Review: orthogonality
characterization
logical
orthogonality
Variety
equations
∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴
closure property
H, S, P
Sort-of-variety Quasivariety
Prevariety
implications
preequations
∀∃!-formulas
∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴
𝐴↠𝐵
𝐴⟶𝐵
?
Wataru Hino
S, P, FC
𝐴-pure S, P, FC
7
Orthogonality [Freyd, Kelly 1972]
Def. (orthogonality)
𝑓: 𝐴 → 𝐵 is orthogonal to 𝐶 ⟺
(𝑓 ⊥ 𝐶)
Example
In the category of groups,
𝐵
𝑓
∃! 𝑔
𝐶
∀𝑔
𝐴
𝜋: ℤ ↠ ℤ 𝑛ℤ ⊥ 𝐺 ⟺ 𝐺 ⊨ ∀𝑥 𝑥 𝑛 = 𝑒
𝜄: 2ℤ ↪ ℤ ⊥ 𝐺 ⟺ 𝐺 ⊨ ∀𝑥∃!𝑦 (𝑦 2 = 𝑥)
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Orthogonality
Observation 𝐹𝑋, 𝐴, 𝐵: finitely presentable (FP)
• 𝜑: equation ⟷ 𝑓: 𝐹𝑋 ↠ 𝐴 (𝐹𝑋: free)
• 𝜑: implication ⟷ 𝑓: 𝐴 ↠ 𝐵
Variety and quasivariety are characterized
by orthogonality
→ What if we drop these conditions on morphisms:
surjectivenss and free-domain?
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Review: prevariety
characterization
logical
orthogonality
Variety
equations
∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴
closure property
H, S, P
Sort-of-variety Quasivariety
Prevariety
implications
preequations
∀∃!-formulas
∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴
𝐴↠𝐵
𝐴⟶𝐵
?
Wataru Hino
S, P, FC
𝐴-pure S, P, FC
10
Prevariety [Adámek, Sousa 2004]
Def. (prevariety a.k.a. 𝜔-orthogonality class) correspond to pre-equation
𝑉 ⊂ AlgΣ is a prevariety
∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
∃
⟺ 𝑀: a set of morphisms (between FP algebra)
s.t. 𝑉 = 𝐴 ∈ AlgΣ 𝑀 ⊥ 𝐴}
• It also can be characterized by closure property
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11
2 axes for variety-like notions
Surjective
Free-domain
Variety: 𝐹𝑋 ↠ 𝐴
Arbitrary
Quasi-variety: 𝐴 ↠ 𝐵
Arbitrary
Sort-of-variety: 𝐹𝑋 → 𝐴
Prevariety: 𝐴 → 𝐵
characterization
logical
orthogonality
Variety
equations
∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴
closure property
H, S, P
Sort-of-variety Quasivariety
Prevariety
implications
preequations
∀∃!-formulas
∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴
𝐴↠𝐵
𝐴⟶𝐵
?
Wataru Hino
S, P, FC
𝐴-pure S, P, FC
12
New notion: sort-of-variety
characterization
logical
orthogonality
Variety
equations
∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴
closure property
H, S, P
Sort-of-variety Quasivariety
Prevariety
implications
preequations
∀∃!-formulas
∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴
𝐴↠𝐵
𝐴⟶𝐵
?
Wataru Hino
S, P, FC
𝐴-pure S, P, FC
13
Sort-of-variety: definition
orthogonality to
𝑓: 𝐹𝑋 → 𝐴
Def. (sort-of-variety)
∀𝑥 ∃! 𝑦 𝐸
𝑉 ⊂ AlgΣ is a sort-of-variety
∃
⟺ 𝐸: a set of ∀∃!-formulas
s.t. 𝑉 = 𝐴 ∈ AlgΣ 𝐴 ⊨ 𝐸}
Remark
• ∀∃!-formula defines “extra” operations on algebras
• Σ-morphisms must preserve them
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Sort-of-variety: example
Example
The class of groups is a sort-of-variety for ∙
• neither a variety nor a quasivariety
• associativity, ∀𝑥𝑦∃! 𝑧 𝑥 ⋅ 𝑧 = 𝑦 , ∀𝑥𝑦∃! 𝑧 𝑧 ⋅ 𝑥 = 𝑦 and ∃! 𝑒 𝑒2 = 𝑒
• They define 3 extra operations ∖, / , 𝑒
• which satisfy 𝑥 ⋅ 𝑥 ∖ 𝑦 = 𝑦 etc.
• 𝑒 is unit and 𝑥 −1 = 𝑥 ∖ 𝑒
• ∙ -morphisms preserve unit and multiplication
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Relation between other variety-like notions
Thm.
A sort-of-variety for Σ is isomorphic to a quasivariety for an
extended signature Σ’ (by the forgetful functor).
Surjective
Free-domain
Variety
Arbitrary
Quasi-variety
Arbitrary
Sort-of-variety
Prevariety
Expanding 𝚺
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Converse?
Observation
• The quasivariety of torsion-free abelian groups and that of
positive monoids are a sort-of-variety.
• The class of left-cancellative monoids is quasivariety, but
doesn’t seem to be a sort-of-variety.
Surjective
Free-domain
Variety
Arbitrary
Quasi-variety
Arbitrary
Sort-of-variety
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??
Prevariety
17
Summary
• Reviewed variety, quasivariety and prevariety
• Introduced a new variety-like notion: sort-of-variety
Future work
• Find the closure property for sort-of-variety
• Exploit its relation between other variety-like notions
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