APPENDIX B Algebraic Systems
B.1Introduction
This Appendix investigates some of the major algebraic systems in mathematics: semigroups, groups, rings,
and fields. We also define the notion of a homomorphism and the notion of a quotient structure. We begin with
the formal definition of an operation, and discuss various types of operations.
B.2Operations
The reader is familiar with the operations of addition and multiplication of numbers, union and intersection
of sets, and the composition of functions. These operations are denoted as follows:
a+b=c, a·b=c, A∪B=C, A∩B=C, gof=h.In each situation, an element (c,C,orh) is assigned to an original pair of elements. We make this notion precise.
Definition B.1: LetSbe a nonempty set. AnoperationonSis a function∗fromS×SintoS. In such a case,
instead of∗(a,b), we usually write
a∗b or sometimes abThe setSand an operation∗onSis denoted by (S,∗) or simplySwhen the operation is
understood.Remark:An operation∗fromS×SintoSis sometimes called abinary operation.Aunaryoperation is a
function fromSintoS. For example, the absolute value|n|of an integernis a unary operation onZ, and the
complementACof a setAis a unary operation on the power setP(X)of a setX.Aternary(3-ary) operation is a
function fromS×S×SintoS. More generally, ann-ary operation is a function fromS×S×···×S(nfactors)
intoS. Unless otherwise stated, the word operation shall mean binary operation. We will also assume that our
underlying setSis nonempty.
SupposeSis a finite set. Then an operation∗onScan be presented by its operation (multiplication) table
where the entry in the row labeledaand the column labeledbisa∗b.
SupposeSis a set with an operation∗, and supposeAis a subset ofS. ThenAis said to beclosed under∗
ifa∗bbelongs toAfor any elementsaandbinA.
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