Varieties, Quasivarieties
and Prevarieties:
Completing the Picture
Wataru Hino & Ichiro Hasuo
The University of Tokyo

Wataru Hino

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Outline• Review 1: variety and quasivariety• Review 2: orthogonality and prevariety• New notion: sort-of-variety
Newcharacterization
logicalorthogonality

Varietyequations∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴closure property

H, S, P

Sort-of-variety Quasivariety
Prevariety
implications
preequations∀∃!-formulas∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴𝐴↠𝐵𝐴⟶𝐵?
Wataru Hino

S, P, FC

𝐴-pure S, P, FC2

Review: variety and quasivariety
characterization
logicalorthogonality

Varietyequations∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴closure 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

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Variety: definition
Def. (variety)𝑉 ⊂ AlgΣ is a variety∀𝑥 𝑠 = 𝑡∃⟺ 𝐸: a set of equations
signature-relevants.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)
Wataru Hino

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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, −Wataru Hino

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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
Wataru Hino

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Review: orthogonality
characterization
logicalorthogonality

Varietyequations∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴closure 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↠𝐵𝐴⟶𝐵?

Wataru Hino

S, P, FC

𝐴-pure S, P, FC

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Orthogonality [Freyd, Kelly 1972]
Def. (orthogonality)
𝑓: 𝐴 → 𝐵 is orthogonal to 𝐶 ⟺(𝑓 ⊥ 𝐶)
ExampleIn the category of groups,𝐵𝑓∃! 𝑔𝐶∀𝑔𝐴

𝜋: ℤ ↠ ℤ 𝑛ℤ ⊥ 𝐺 ⟺ 𝐺 ⊨ ∀𝑥 𝑥 𝑛 = 𝑒

𝜄: 2ℤ ↪ ℤ ⊥ 𝐺 ⟺ 𝐺 ⊨ ∀𝑥∃!𝑦 (𝑦 2 = 𝑥)
Wataru Hino

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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?
Wataru Hino

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Review: prevariety
characterization
logicalorthogonality

Varietyequations∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴closure 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↠𝐵𝐴⟶𝐵?

Wataru Hino

S, P, FC

𝐴-pure S, P, FC

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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

Wataru Hino11

2 axes for variety-like notions
SurjectiveFree-domain
Variety: 𝐹𝑋 ↠ 𝐴

Arbitrary
Quasi-variety: 𝐴 ↠ 𝐵

Arbitrary

Sort-of-variety: 𝐹𝑋 → 𝐴

Prevariety: 𝐴 → 𝐵characterization
logicalorthogonality

Varietyequations∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴closure 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↠𝐵𝐴⟶𝐵?
Wataru Hino

S, P, FC

𝐴-pure S, P, FC
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New notion: sort-of-variety
characterization
logicalorthogonality

Varietyequations∀𝑥 (𝑠 = 𝑡)
𝐹𝑋 ↠ 𝐴closure property

H, S, P

Sort-of-variety Quasivariety
Prevariety
implications
preequations∀∃!-formulas∀𝑥 ∃! 𝑦 𝐸
∀𝑥 (𝐸 → 𝑠 = 𝑡) ∀𝑥 (𝐸 → ∃! 𝑦 𝐸′)
𝐹𝑋 ⟶ 𝐴𝐴↠𝐵𝐴⟶𝐵?

Wataru Hino

S, P, FC

𝐴-pure S, P, FC

13

Sort-of-variety: definition
orthogonality to𝑓: 𝐹𝑋 → 𝐴Def. (sort-of-variety)∀𝑥 ∃! 𝑦 𝐸𝑉 ⊂ AlgΣ is a sort-of-variety∃⟺ 𝐸: a set of ∀∃!-formulass.t. 𝑉 = 𝐴 ∈ AlgΣ 𝐴 ⊨ 𝐸}
Remark• ∀∃!-formula defines “extra” operations on algebras• Σ-morphisms must preserve them

Wataru Hino

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Sort-of-variety: example
ExampleThe 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
Wataru Hino

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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 𝚺Wataru Hino

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Converse?
Observation• The quasivariety of torsion-free abelian groups and that ofpositive monoids are a sort-of-variety.
• The class of left-cancellative monoids is quasivariety, butdoesn’t seem to be a sort-of-variety.
SurjectiveFree-domain
Variety

Arbitrary
Quasi-variety

Arbitrary

Sort-of-variety
Wataru Hino??

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

Wataru Hino

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Wataru Hino

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