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Boolean Semilattices Boolean Semilattices Clifford Bergman Iowa State University June 2015 Motivating Construction Let G = hG; ·i be a groupoid (i.e., 1 binary operation) Form the complex algebra G+ = hSb(G); \; [; ∼; ·; ;; Gi X · Y = f x · y : x 2 X; y 2 Y g “complex operation” G+ is an expansion of a Boolean algebra X ·; = ;· X = ; (normality) X · (Y [ Z) = (X · Y ) [ (X · Z ) and (Y [ Z ) · X = (Y · X) [ (Z · X) (additivity) In fact, complete and atomic, and completely additive Boolean Groupoids Definition A Boolean groupoid is an algebra B = hB; ^; _; 0; ·; 0; 1i such that hB; ^; _; 0; 0; 1i is a Boolean algebra x · 0 ≈ 0 · x ≈ 0 x · (y _ z) ≈ (x · y) _ (x · z) (y _ z) · x ≈ (y · x) _ (z · x): BG = the variety of Boolean groupoids Crash Course in Universal Algebra Algebra Structure with no relational parameters Variety Class of algebras closed under homomorphic image, subalgebra, and product Birkhoff’s Theorem K is a variety iff it is axiomatized by a set of equations Equation Universal, atomic sentence Variety Generated by K V(K) is the smallest variety containing K. V(K) = Mod(Eq(K)) Boolean Groupoids Definition A Boolean groupoid is an algebra B = hB; ^; _; 0; ·; 0; 1i such that hB; ^; _; 0; 0; 1i is a Boolean algebra x · 0 ≈ 0 · x ≈ 0 x · (y _ z) ≈ (x · y) _ (x · z) (y _ z) · x ≈ (y · x) _ (z · x): BG = the variety of Boolean groupoids G a groupoid =) G+ 2 BG + Converse? Nope. jG j 6= @0 Is BG generated by complex algebras? Yes! Can retrieve G from G+: a · b = c in G () fag · fbg = fcg in G+ Note that fag, fbg, fcg are atoms of G+ Not all Boolean groupoids have atoms The product of two atoms might not be an atom We need a more general approach Definition Let B be a Boolean groupoid. B+ = hA; θi in which A is the set of atoms of B and θ = (x; y; z) 2 A3 : z ≤ x · y (a ternary relational structure) For a groupoid G, let θ = f (x; y; x · y) g. + Then hG; θi = (G )+ Definition Let H = hH; i be a ternary relational structure H+ = hSb(H); \; [; ∼; ·; ;; Hi in which X · Y = f z 2 H :(9x 2 X)(9y 2 Y )(x; y; z) 2 g. + ∼ H a ternary relational structure =) (H )+ = H B a complete and atomic Boolean algebra + ∼ =) (B+) = B There is a duality between the categories of ternary relational structures and complete, atomic Boolean algebras Canonical Extensions Theorem (Jonsson-Tarski,´ 1951) Let B be a Boolean groupoid. There is a complete, atomic Boolean groupoid Bσ extending B such that ^ p · q = f a · b : a; b 2 B; a ≥ p; b ≥ q g for atoms p; q of Bσ. σ Idea: (B )+ can serve as an approximation to a groupoid induced by B. Lemma (Jipsen) Every Boolean groupoid can be embedded into P+ for some partial groupoid P. σ (B )+ P G B Bσ P+ G+ Theorem Every Boolean groupoid lies in SH(G+) for some groupoid G. Corollary BG = V f G+ : G a groupoid g V f G+ : G a groupoid g is axiomatized by the identities that “have to be true” Will this hold for other classes of groupoids? Properties Reflected in the Complex Algebra normality additivity ) monotonicity x ≤ y ! x · z ≤ y · z p ≈ q is linear if each variable appears exactly once in p and in q + G p ≈ q () G p ≈ q x · (y · z) ≈ (x · y) · z, x · y ≈ y · x, x · (x · y) ≈ y p ≈ q is semilinear if p has no repeated variables and each variable of q occurs in p + G p ≈ q () G p ≤ q x ≈ x 2, x · y ≈ x Some Generic Questions Let V be a variety of Boolean groupoids, K a class of groupoids or TRS. K+ = f G+ : G 2 K g Is V = V(K+)? Is V = SP(K+)? (“V is representable by K”) Is V(K+) finitely axiomatizable? Some Nice Axiomatizations V f G+ : G a commutative groupoid g axiomatized by x · y ≈ y · x V f G+ : G an idempotent groupoid g: x ≤ x · x V f G+ : G a commutative, idempotent groupoid g: x · y ≈ y · x, x ≤ x · x Together with axioms for Boolean algebras, normality, and additivity. (Jipsen) So all are finitely based. Are these varieties represented by a class of groupoids? On the Other Hand. What about the associative law? Let Sg (CSg) be the variety of (commutative) semigroups. Theorem (Jipsen) Neither V(Sg+) nor V(CSg+) is finitely based. On the Other, Other Hand. Lz = left-zero semigroups (x · y ≈ x) V(Lz+) is finitely based and represented by Lz. Rb = rectangular bands (x · (y · z) ≈ (x · y) · z, x ≈ x 2, x · y · z ≈ x · z) V(Rb+) is finitely based and represented by Rb. Boolean Semilattices Let Sl denote the variety of semilattices, i.e. groupoids satisfying x · (y · z) ≈ (x · y) · z x · y ≈ y · x x · x ≈ x: These are the identities of associativity, commutativity, and idempotence Definition A Boolean semilattice is a Boolean groupoid satisfying x · (y · z) ≈ (x · y) · z x · y ≈ y · x x ≤ x · x The variety is denoted BSl. Easy to check that V(Sl+) ⊆ BSl. One wishes that these are equal. They are not. Example Let H = fa; bg. Define TRS on H with θ = (a; a; a); (a; b; b); (b; a; b); (b; b; a); (b; b; b) . H+ 2 BSl + + but Sl x ^ (y · 1) ≤ x · y while H fails this identity Open Questions + 1 Is V(Sl ) finitely based? Is the equational theory decidable? + 2 Is either BSl or V(Sl ) generated by its finite members? 3 Is there a finitely axiomatizable class K of TRS such that BSl = V(K+)? Algebraic Theory of BSl Let B 2 BSl, x 2 B. #x = x · 1 Theorem ‘#’ yields a closure operator on B: x ≤ #x = ## x and x ≤ y =)#x ≤ #y x is closed if x = #x For a semilattice S and X 2 S+ #X = X · S = f y 2 S :(9x 2 X) y ≤ x g the downset generated by X. Congruence Ideals Let θ 2 Con(B). I = 0/θ = f x 2 B :(0; x) 2 θ g is an ideal of the Boolean algebra x 2 I =) x θ 0 =)#x θ #0 = 0 =)#x 2 I Conversely I an ideal closed under # implies I = 0/θ for some θ 2 Con(B) “Congruence ideals” Consequences Let B be a Boolean semilattice 1 Let a 2 B. The smallest congruence ideal containing a is (#a]. 2 BSl has equationally definable principal congruences (EDPC) 3 B is subdirectly irreducible iff it has a smallest nonzero closed element 4 B is simple iff x > 0 =)#x = 1 Thus, for S 2 Sl, S+ is SI iff S has a least element S+ is simple iff S is trivial 5 B subdirectly irreducible implies Bσ subdirectly irreducible B simple implies Bσ simple Discriminator Algebras Let A be a set. The discriminator on A is ( z if x = y dA(x; y; z) = x if x 6= y. An algebra A is a discriminator algebra if dA is a term of A. Discriminator algebras are simple every subalgebra is a discriminator alg generate arithmetical varieties Theorem Every simple BSl is a discriminator algebra, with term d(x; y; z) = x ^ #(x ⊕ y) _ z ^ #(x ⊕ y)0 Theorem The subvariety of BSl generated by all simple algebras is axiomatized, relative to BSl, by #(#x)0 ≈ (#x)0 This is the largest discriminator subvariety of BSl. Problem: Is this variety representable by a nice family of TRS? An Interesting Subvariety Tot = hH; H3i : H any set . Every member of Tot+ is a discriminator algebra. + V(Tot ) is axiomatized (relative to BSlD) by x · y · 1 ≈ x · y. Linear Semilattices Recall BSl x ≤ x · x. What about IBSl = f B 2 BSl : B x ≈ x · x g? + Let S 2 Sl. S x ≈ x · x iff S is linearly ordered Proof: X · X ⊆ X =) X is a subsemilattice So S has the property that every subset is a subsemilattice. Thus S is linearly ordered Theorem Let LS denote the class of linearly ordered semilattices. Then IBSl = SP(LS+). In IBSl, x · y = (x ^ #y) _ (y ^ #x) Thus IBSl is term-equivalent to its closure-reduct This variety of modal algebras is S4;3 Consequences of EDPC For a class K, H!(K) = f B/θ : B 2 K; θ a compact congruence g If K ⊆ BSl, then B/θ = B=(c] for a closed element c. A is splitting (relative to V) if subdirectly irreducible and there is a largest subvariety excluding A. Denoted V=A, defined by a single equation. Theorem (Blok-Pigozzi) If V has EDPC then every finite subdirectly irreducible algebra is splitting. In this case V=A = f B 2 V : A 2= SH!(B) g : 2@0 subvarieties For n 2 !, a1 a2 a3 an Yn = + Each Yn is subdirectly irreducible, hence splitting. Easy to check: + + n 6= m =) Ym 2= SH(Yn ) + + Since Yn is finite, H(Yn ) = H!(Yn ) + Let T ⊆ ! and VT = V f Yn : n 2 T g. Then + + m 2= T =) Ym 2= SH Yn : n 2 T =) + + Yn : n 2 T ⊆ BSl=Ym + Splitting, so BSl=Ym is a variety. Therefore + + VT = V Yn : n 2 T ⊆ BSl=Ym + so Ym 2= VT . + m 2= T =) Ym 2= VT Thus S 6= T =)VS 6= VT .
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