Download Álgebra y Análisis de Funciones Elementales by M. Potápov - V. Alexándrov - P. Pasichenko PDF

By M. Potápov - V. Alexándrov - P. Pasichenko

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Extra info for Álgebra y Análisis de Funciones Elementales

Example text

Proof. 10, the map re is a {∧, 0, 1}-homomorphism. Let Θ and Φ be congruences of L; we have to prove that Θ K ∨ Φ K = (Θ ∨ Φ) K. Since ≤ is trivial, we prove ≥. So let a, b ∈ K, a ≡ b ((Θ ∨ Φ) K); we want to prove that a ≡ b (Θ K ∨ Φ K). 2, there is a sequence z 0 = a ∧ b ≤ z1 ≤ · · · ≤ z n = a ∨ b such that, for each j with 0 ≤ j < n, either zj ≡ zj+1 (Θ) or zj ≡ zj+1 (Φ) holds in L. Since a, b ∈ K and K is an ideal, it follows that z0 , z1 , . . , zn ∈ K, so for each j with 0 ≤ j < n, either zj ≡ zj+1 (Θ K) or zj ≡ zj+1 (Φ K) holds, proving that a ≡ b (Θ K ∨ Φ K).

To state it, we need one more concept: Let ϕ : L → L1 be a homomorphism of the lattice L into the lattice L1 , and define the binary relation Θ on L by x Θ y iff xϕ = yϕ; the relation Θ is a congruence relation of L, called the kernel of ϕ, in notation, ker(ϕ) = Θ. 4 (Homomorphism Theorem). Let L be a lattice. Any homomorphic image of L is isomorphic to a suitable quotient lattice of L. 8 ) is given by ψ : x/Θ → xϕ, for x ∈ L. 8: The Homomorphism Theorem. 5 (Second Isomorphism Theorem). Let L be a lattice and let Θ be a congruence relation of L.

Down(J(D)) and P ∼ Let D and E be nontrivial finite distributive lattices, and let ϕ : D → E be a {0, 1}-homomorphism. Then with every x ∈ J(E), we can associate the smallest y ∈ D with yϕ ≥ x. It turns out that y ∈ J(D), so we obtain an isotone map J(ϕ) : J(E) → J(D). 17. Let D and E be nontrivial finite distributive lattices, and let ϕ : D → E be a {0, 1}-homomorphism. Let ϕD and ϕE be the isomorphisms between D and Down(J(D)) and between E and Down(J(E)), respectively. Then the diagram ψD D −−−−→ Down(J(D)) ⏐ ⏐ ⏐ ⏐Down(J(ϕ)) ϕ ψE E −−−−→ Down(J(E)) commutes, that is, ψD Down(J(ϕ)) = ϕψE .

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