690
Modeling Arithmetic, Computation, and Languages
Although the requirements for a structure to be a semigroup are relatively
modest, not every arithmetic structure qualifies.
PrACtiCe 4 Prove that 3Z, − 4 is not a semigroup, where − denotes integer subtraction.
■
PrACtiCe 5 Let S be the set of noninteger rational numbers, and let # denote multiplication. Is 3S, # 4
a semigroup?
■
example 4
Each of the following is an instance of a commutative semigroup. You should be
able to verify closure, associativity, and commutativity for each:
3N, + 4, 3N, # 4, 3Q, # 4, 3R
+
, + 4, 3R, + 4
example 5
For any Boolean algebra 3B, +, # , ′, 0, 14, 3B, + 4 and 3B, # 4 are commutative
semigroups. Therefore for any set S, 3`(S ), c 4 and 3`(S ), d 4 are commutative
semigroups.
Because the requirements that must be satisfied in going from semigroup to
monoid to group keep getting stiffer, we expect some examples to drop out, but
those remaining should have richer and more interesting personalities.
Next we look at a selection of other examples of semigroups, monoids, and
groups where the elements are not just simple numbers or where the operations
are less familiar.
PrACtiCe 6 Which of the following semigroups are monoids? Name the identities.
3N, + 4, 3N, # 4, 3Q, # 4, 3R
+
, + 4, 3R, + 4, 3`(S ), c 4, 3`(S ), d 4
■
PrACtiCe 7 Which of the monoids from the list in Practice 6 are groups?
■
example 6
An expression of the form
a n x
n
+ a n−1 x
n−1
+ c + a 0
where a i [ R, i = 0, 1, … , n, and n [ N is a polynomial in x with real-number
coefficients (or a polynomial in x over R.) For each i, a i is the coefficient of x
i
.
Modeling Arithmetic, Computation, and Languages
Although the requirements for a structure to be a semigroup are relatively
modest, not every arithmetic structure qualifies.
PrACtiCe 4 Prove that 3Z, − 4 is not a semigroup, where − denotes integer subtraction.
■
PrACtiCe 5 Let S be the set of noninteger rational numbers, and let # denote multiplication. Is 3S, # 4
a semigroup?
■
example 4
Each of the following is an instance of a commutative semigroup. You should be
able to verify closure, associativity, and commutativity for each:
3N, + 4, 3N, # 4, 3Q, # 4, 3R
+
, + 4, 3R, + 4
example 5
For any Boolean algebra 3B, +, # , ′, 0, 14, 3B, + 4 and 3B, # 4 are commutative
semigroups. Therefore for any set S, 3`(S ), c 4 and 3`(S ), d 4 are commutative
semigroups.
Because the requirements that must be satisfied in going from semigroup to
monoid to group keep getting stiffer, we expect some examples to drop out, but
those remaining should have richer and more interesting personalities.
Next we look at a selection of other examples of semigroups, monoids, and
groups where the elements are not just simple numbers or where the operations
are less familiar.
PrACtiCe 6 Which of the following semigroups are monoids? Name the identities.
3N, + 4, 3N, # 4, 3Q, # 4, 3R
+
, + 4, 3R, + 4, 3`(S ), c 4, 3`(S ), d 4
■
PrACtiCe 7 Which of the monoids from the list in Practice 6 are groups?
■
example 6
An expression of the form
a n x
n
+ a n−1 x
n−1
+ c + a 0
where a i [ R, i = 0, 1, … , n, and n [ N is a polynomial in x with real-number
coefficients (or a polynomial in x over R.) For each i, a i is the coefficient of x
i
.