By Bass

Lawsuits Of The convention Held on the Seattle learn heart Of Battelle Memorial Institute, August 28 - September eight, 1972

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A semigroup of functions from X to itself may include the identity function, which is a model for the general definition of an identity element. 2 Let S be a semigroup. An element e 2 S is called an identity element if we have, for any a 2 S , ae D ea D a: A semigroup with an identity element is called a monoid. It is easy to prove that there can be at most one identity element; in most cases, we simply call it 1. If M1 and M2 are monoids, we say that a function f W M1 ! 1/ D 1. The general rule is that a homomorphism must preserve all the available structure: for monoids, products and identity elements.

The conjugate of h by g is hg D g 1 hg. If H < G is a subgroup, the conjugate of H by g is the subgroup H g consisting of the conjugates of all elements h 2 H . g 1 yg/. In fact, mapping g to the automorphism “conjugation by g 1 ” gives a homomorphism ˇ W G ! G/. ) Since conjugation by g is an automorphism, for any subgroup H < G it gives an isomorphism from H to H g . This means that conjugation also defines a right action of G on the set of all its subgroups. Being an automorphism, it preserves cardinality, so we can also see it as a right action of G on the set of all subsets of G with a given cardinality.

Other extensions of the real numbers. ” Algebras are extremely important in applications, and so deserve special attention. We limit our definitions to algebras over commutative rings. Most of our algebras are associative. Let R be a commutative ring. An associative algebra over R is an Rmodule A which has a ring structure compatible with the R-module structure. r b/: It turns out that this can be expressed in fewer words. Since A is a ring, it has a multiplicative identity 1. , elements of R commute with all elements of A.