math question

Name:

Homework 1 for MTH421B

due on Friday, February 8th

1. Let S be a set with a binary operation ∗.

(a) We say that x ∈ S is a left identity if x∗s = s for all s ∈ S. We say that y ∈ S is a right identity if s ∗ y = s for all s ∈ S. Show that, if x ∈ S is a left identity and y ∈ S is a right identity, then x = y. [Thus, under the given assumptions, x is an identity in the sense defined in class, and there is no other left or right identity in S.]

Give an example of a set S with binary operation ∗ having more than one left identity. Give an example of a set S with binary operation ∗ having more than one right identity. (Your best bet here is to describe (S,∗) using a multiplication table. I used such a table to define an operation ? in class. You should use similar tables to generate the examples that are required in the other parts of this problem.)

(b) We say that x ∈ S is a left zero if x∗s = x for all s ∈ S. We say that y ∈ S is a right zero if s ∗ y = y for all s ∈ S. An element z ∈ S is said to be a zero (or two-sided zero) if it is simultaneously a left and right zero; that is, if z ∗s = z = s∗z, for all s ∈ S.

Show that, if x ∈ S is a left zero and y ∈ S is a right zero, then x = y, so x is in fact a (two-sided) zero. [And there will be no left, right, or two-sided zero other than x.]

Give an example of a set S with binary operation ∗ having more than one left zero. Give an example of a set S with binary operation ∗ having more than one right zero. (Of course, these two examples have to be different; the same goes for the examples from (a).)

(c) Assume that S has an identity, which we will denote 1 (or 1S). Let x ∈ S. A left inverse for x is an element s ∈ S such that s ∗ x = 1; a right inverse for x is an element t ∈ S such that x ∗ t = 1. Assuming that ∗ is an associative operation, show that s = t (s and t are as above).

Give an example of a set S with binary operation ∗ such that S has an identity, and there is an element with exactly two left inverses and one right inverse.

2. Let S be a commutative monoid with a 0. (Thus, 0 has the formal properties of a zero as described in the previous question. It need not be the number 0.) Assume that 0 6= 1, where 1 is the identity of S.

An element s ∈ S is said to be a zero divisor if there is a non-zero t ∈ S such that st = 0. We say that s ∈ S is a unit if there is some w ∈ S such that sw = 1.

Show that it is impossible for an element s to be both a zero divisor and a unit.

3. Let S be a semigroup. We say that s ∈ S is an idempotent if s2 = s. We say that S is a regular semigroup if, for each x ∈ S, there is y ∈ S such that xyx = x. We say that S is an inverse semigroup if S is a regular semigroup and any two idempotents in S commute. [That is, if e and f are idempotents, then ef = fe.]

(a) Let S be an inverse semigroup; let x ∈ S. An inverse for x is an element y such that xyx = x and yxy = y. Show that each x ∈ S has an inverse. [Big hint: since S is regular, we know that there is y ∈ S such that xyx = x. Try to argue that yxy is the inverse for x.]

(b) The present definition of “inverse” (from (a)) is very different from the definition of inverse that we saw in class. To see this, consider the example S = {−1, 0, 1}, where the operation is multiplication of real numbers. Show that S is an inverse semigroup (actually, an inverse monoid, since it has a 1). Identify the idempotents in S, and find the inverse of each element (in the sense of (a)). Show that not all elements have inverses in the sense of group theory (i.e., in the sense defined in class).

(c) We say that (S,∗) has the left cancellation property if, whenever a, b, c ∈ S and ab = ac, b = c. Show, using the example from (b), that inverse semigroups need not have the left cancellation property. [In contrast, every group does have this prop- erty.] Similarly, inverse semigroups need not have the right cancellation property (although every group does).

(d) Let S be an inverse semigroup; let x ∈ S. Show that the inverse of x is unique, as follows. Suppose that y and z are both inverses for x:

xyx = x; yxy = y; xzx = x; zxz = x.

i. Show that yx, xy, zx, and xz are idempotents.

ii. Use the fact that idempotents commute to show that yx = zx and xy = xz. [Note: this is not quite good enough yet, because of the failure of the cancellation properties.]

iii. Use the previous part to argue that z = y. (Some experimentation may be necessary here.)