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Static Program Analysis 前置知识: Lattice Theory Contents: Data Flow Analysis
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Let G be a group. H, K is normal subgroup of G and H is subset of K. Then K/H isomorphic to G/H
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A group G is cyclic if G=(a), a is in G 1. If G has infinite order, then G isomorphic to Z 2. If G has finite order, say n, then G isomorphic to Z/nZ
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Let R be a commutative ring. Units of R are elements who are invertible. Let a be a nonzero nonunit element, if a can be written as bc, where b and c are not unit, a is called reducible. Otherwise, it is called irreducible.
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A real Lie Group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. For example, rotation group SO(2)
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Let R be a ring, then R x R\{0} is a set and define an relation on it to be [a, b] related to [c, d] iff ad=bc, it is an equivalent relation. Prove that the quotient set is a field
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Let R be a commutative ring. Then ideal of R is trivial if and only if R is a field. Prove that Z[sqrt(2), sqrt[3]] = Z[sqrt(2)][sqrt(3)]
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Let R be a set with two operations, denoted by + and *, such that (R, +) is abelian group, (R, *) is semi group, and two operations satisfy distributed rules. R is called a ring. In addition, if * is commutative, R is called commutative ring.
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Let G be a nonempty set. A binary operation is a function that takes two elements of G and produces an element in G. Denoted by * : G x G -> G, and *(x, y) denoted by x*y or xy. 1. For a in G, if for any element g in G, g*a = a*g = g, a is called identity of G. Denoted by 1
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R is a commutative ring. I is an ideal of R, then I is a maximal ideal if and only if R/I is a field
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