By Stanley Burris

"As a graduate textbook, the paintings is a certain winner. With its transparent, leisurely exposition and beneficiant number of workouts, the e-book attains its pedagogical targets stylishly. furthermore, the paintings will serve good as a study tool…[offering] a wealthy collection of vital new effects that have been formerly scattered during the technical literature. mostly, the proofs within the e-book are tidier than the unique arguments." —

*Mathematical Reviews*of the yank Mathematical Society.

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Anq ✏ f B♣αa1, . . , αanq ✟ ✏ f B β ♣a1④θq, . . , β ♣an④θq . 12 we see that an algebra is a homomorphic image of an algebra A iff it is isomorphic to a quotient of the algebra A. Thus the “external” problem of finding all homomorphic images of A reduces to the “internal” problem of finding all congruences on A. The homomorphism theorem is also called “the first isomorphism theorem”. §6 Homomorphisms and the Homomorphism and Isomorphism Theorems 47 Con A with θ ❸ φ. Then let ✥ ✭ φ④θ ✏ ①a④θ, b④θ② ♣A④θq2 : ①a, b② φ .

3. Suppose C is a closure operator on S. A minimal generating set of S is called an irredundant basis. Let IrB♣C q ✏ tn ➔ ω : S has an irredundant basis of n elements✉. The next result shows that the length of the finite gaps in IrB♣C q is bounded by n ✁ 2 if C is an n-ary closure operator. 4 (Tarski). If C is an n-ary closure operator on S with n ➙ 2, and if i ➔ j with i, j IrB♣C q such that ti 1, . . , j ✁ 1✉ ❳ IrB♣C q ✏ Ø, ( ✝) then j ✁ i ↕ n ✁ 1. , a sequence of consecutive numbers. P ROOF.

3. If A is a unary algebra show that ✞IrB♣Sgq✞ ↕ 1. 4. Give an example of an algebra A such that IrB♣Sgq is not convex. §5 Congruences and Quotient Algebras §5 35 Congruences and Quotient Algebras The concepts of congruence, quotient algebra, and homomorphism are all closely related. These will be the subjects of this and the next section. Normal subgroups, which were introduced by Galois at the beginning of the century, play a fundamental role in defining quotient groups and in the socalled homomorphism and isomorphism theorems which are so basic to the general development of group theory.

### A Course in Universal Algebra by Stanley Burris

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