By Stanley Burris
"As a graduate textbook, the paintings is a convinced winner. With its transparent, leisurely exposition and beneficiant collection of routines, the e-book attains its pedagogical ambitions stylishly. additionally, the paintings will serve good as a learn tool…[offering] a wealthy collection of important new effects that have been formerly scattered through the technical literature. often, the proofs within the booklet are tidier than the unique arguments." — Mathematical Reviews of the yank Mathematical Society.
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Extra info for A Course in Universal Algebra
8. Every algebraic lattice is isomorphic to the lattice of closed subsets of some set A with an algebraic closure operator C. P ROOF. Let L be an algebraic lattice, and let A be the subset of compact elements. For X ❸ A define ➟ ✭ ✥ C ♣X q ✏ a A : a ↕ X . C is a closure operator, and from the definition of compact elements it follows that C is algebraic. The map a ÞÑ tb A : b ↕ a✉ gives the desired isomorphism as L is compactly generated. ❧ R EFERENCES 1. M. Cohn  2. A. Tarski  §5 Closure Operators 21 E XERCISES §5 1.
An q, f A ♣b1 , . . , bn q f A ♣a1 , . . , an q B θ . ❉ θ, B θ . Then §6 Homomorphisms and the Homomorphism and Isomorphism Theorems 49 Thus B θ is a subuniverse of A. Next, to verify that θ æB is a congruence on B is straightforward. 18 (Third Isomorphism Theorem). If B is a subalgebra of A and θ Con A, then ♣see Figure 15q B④θ æB ✕ Bθ ④θ æB θ . α B/θ| B Bθ/θ| Bθ Figure 15 P ROOF. We leave it to the reader to verify that the map α which is defined by α♣b④θ æB q ✏ b④θ æB θ gives the desired isomorphism.
P ROOF. From the definition of E (see §3) and the fact that α is a homomorphism we have αE ♣Y q ✏ E ♣αY q for all Y ❸ A. Thus, by induction on n, αE n ♣X q ✏ E n ♣αX q for n ➙ 1; hence α Sg♣X q ✏ α X ✟ ❨ E ♣X q ❨ E 2 ♣X q ❨ ☎ ☎ ☎ ✏ αX ❨ αE ♣X q ❨ αE 2♣X q ❨ ☎ ☎ ☎ ✏ αX ❨ E ♣αX q ❨ E 2♣αX q ❨ ☎ ☎ ☎ ✏ Sg♣αX q. 7. Let α : A Ñ B be a homomorphism. Then the kernel of α, written ker♣αq, and sometimes just ker α, is defined by ker♣αq ✏ t①a, b② A2 : α♣aq ✏ α♣bq✉. 8. Let α : A Ñ B be a homomorphism. Then ker♣αq is a congruence on A.