Primer of Quasivariety Lattices

Transcription

1 A Kira Adaricheva, Jennifer Hyndman, J. B. Nation, Joy Nishida Hofstra University, UNBC, University of Hawai i Hofstra, October 2017

2 Backus International Airport

3 Two problems of Birkhoff and Mal cev Describe lattices of subvarieties L v (V) Describe lattices of subquasivarieties L q (Q)

4 Outline 1 The fundamental representation theorems 2 Equaclosure operators revisited: new restrictions 3 A construction project 4 Problems 5 Advertisement

5 Two lectures on equality Mrs. Smith Solve x + 4 = 7 Bjarni Jónsson The lattice L v (V) of subvarieties of a group variety V is dually isomorphic to the lattice of fully invariant subgroups of F V (ω).

6 Endomorphism action For an algebra A and ε End A and ϕ Con A, define (a, b) ε (ϕ) iff (εa, εb) ϕ. E = {ε : ε End A} acts on Con A preserving, arbitrary meets and nonempty directed joins. Aside: When Q is a quasivariety and A Q, E acts on Con Q A. A congruence ϕ is fully invariant if ε (ϕ) ϕ for all ε E. The fully invariant congruences form a complete sublattice of Con A or Con Q A.

7 Representation theorem for varieties For a variety V, the lattice L v (V) is dually isomorphic to the lattice of fully invariant congruences of F V (ω). Let F = F V (ω). Subvarieties correspond to filters ϕ of Con F with ϕ fully invariant. ϕ is fully invariant if and only if F/ϕ is relatively free.

8 Natural equaclosure operator on L q (K) Dziobiak observed that there is a natural closure operator η on the lattice L q (K). For Q K let η(q) = K HSP(Q) = K H(Q). Moreover, there is a least subquasivariety L = λ(q) with η(l) = η(q), which is generated by F Q (ω). The map η is the equaclosure operator on L q (K).

9 Example of equaclosure operator Figure: L q (L)

10 Lattices of algebraic subsets Let S be an algebraic lattice. A subset X S is an algebraic subset if it contains 1 S and is closed under arbitrary meets and nonempty directed joins. An operator h : S S is continuous if it preserves 1 S, arbitrary meets and nonempty directed joins. If H is a monoid of continuous operators on S, then S p (S, H) denotes the lattice of all H-closed algebraic subsets of S, ordered by inclusion.

11 Natural closure operator on S p (S, H) Closure operator γ on S p (S, H) Each X in S p (S, H) contains a least element x 0. Define γ(x) = x 0 which is H-closed. The least Y with γ(y ) = γ(x) is the H-closed algebraic subset generated by x 0, which is DMO(x 0 ).

12 Representation theorem for quasivarieties Hoehnke, AN, HNN For a quasivariety K, the lattice L q (K) is isomorphic to the lattice S p (S, H) where S = Con K F K (ω) H = E. Moreover, if f denotes the isomorphism, then f (η(q)) = γ(f (Q)). Plan Use the representation to find properties of the equaclosure operator.

13 Generalization With appropriate definitions, the representation theorem applies verbatim to quasivarieties of algebras quasivarieties of structures with operations and/or relations implicational classes of structures in a language that may not contain equality. Partial converse If S = S p (S, H) then S = L q (K) for an implicational class of structures in a language without equality. With equality, additional restrictions apply.

14 Question Given a pair (L, γ) with L a dually algebraic lattice and γ a closure operator satisfying the known properties of natural equaclosure operators, when can we represent L as L q (K) for a quasivariety of algebras L q (K) for a quasivariety of structures S p (S, H) with γ corresponding to the natural equaclosure operator?

15 Classical properties of equaclosure operators (DAG) Let L be a dually algebraic lattice. (I1) x γ(x). (I2) x y implies γ(x) γ(y). (I3) γ 2 (x) = γ(x). (I4) γ(0) = 0. (I5) γ(x) = u for all x X implies γ( X) = u. (I6) γ(x) (y z) = (γ(x) y) (γ(x) z). (I7) γ(l) is the complete meet subsemilattice of L generated by γ(l) K, the semilattice of dually compact elements. (I8) There is a dually compact element w L such that γ(w) = w and the interval [0, w] is isomorphic to S p (S) for some algebraic lattice S.

16 Asides (I5) defines τ(x) abstractly τ(a b) τ(a) τ(b)

17 Condition (K9) γ[x τ(x z)] x τ(z)

18 Condition (K19) τb τd & γc γd γ(a c) & c γ(b) γa γb γa.

19 Construction project Given a pair (L, γ) with L a finite lower bounded lattice and γ satisfying the known properties of natural equaclosure operators, can we represent L as S p (S, H) L q (K) for a quasivariety of structures with γ corresponding to the natural equaclosure operator?

20 Hint for where to start Let L = S p (S, H) and let γ be the natural closure operator. Define f : γ d (L) S via f [X] = x 0. Then f is a complete lattice embedding.

21 Six-step program - Step 1: L = Sub(S,, ˆ0, h) ẑ â â & ˆb ẑ ˆ0 â ˆb ẑ h ˆ0 â ˆ0 â

22 Steps 2 5 (routine): L = L q (K 0 ) Convert Sub(S,, ˆ0, h) to L q (K 0 ) in a language without equality. K 0 has the operations e, µ and predicates O, A, B with laws P(e), P(µe) for P = O, A, B P(µ 2 x) P(µx) for P = O, A, B O(x) A(x) O(x) B(x) B(µx) O(µx) A(µx)

23 Step 6: L = L q (K 1 ) Convert K 0 to a quasivariety K 1 with equality (if possible) by interpreting O(x) as x e. K 1 has the operations e, µ and a predicate A with laws A(e) µe e µ 2 x µx A(µx) A(x) & µx e x e A(x) µx x

24 Step 6 simplified: L = L q (K 2 ) With the interpretation A(x) µx x we obtain an equivalent quasivariety K 2 with operations e, µ and the laws µe e µ 2 x µx

25 L q (K 2 ) 0: x e a: µx x b: µx e z: µx e x e 1: µe e, µ 2 x µx

26 Problems Find more restrictions on pairs (L, γ) to be representable. Finish the construction project. Decide the test cases: Can you represent Fin(X) + 1 as S p (S, H), where X is an infinite set? Can you represent the leaf 1 + Co(4) as L q (K) in a language with equality? Can you represent the lattice in the next slide?

27 Leaf lattice

28 Advertisement HN The Lattice of Subquasivarieties of a Locally Finite Quasivariety of Finite Type MAHALO!