CS 486: Spring 2001 Cornell University Prof. Robert Constable

April 17 Lecture Notes - Formalization

Semigroups

Last Thursday we discussed the formalization of semigroups.

Semigroup = Set S, operation ∘, such that ∀x, y, z ∈ S, x ∘ (y ∘ z) = (x ∘ y) ∘ z \underline{Assoc}

We formalized that as first-order language with predicates Eq for equality, =; Op for ∘; and axioms

E_{1} − E_{3} for equality
Op_{1}  for total functionality of Op : ∀xy∃! z Op (x, y, z)
assoc  Op (y, z, u) ∧ Op (x, u, t) ∧ Op (x, y, v) ∧ Op (v, z, s) ⊃ Eq (t, s)

Monoids, Groups, Rings and Fields

Today we will extend that to Monoids, Groups, Rings, and Fields. We will first review what these concepts are and then formalize them.

a) Monoids are semigroups together with an identity for ∘.

axiom is \underline{Id}: ∃! i. ∀x∃S. x ∘ i = x ∧ i ∘ x = x
or formalized: ∃! i. ∀x Op (x, i, x) ∧ Op (i, x, x)

for (ℤ, +) the identity is 0; for (ℤ, ∗) the identity is 1

b) Groups are monoids with inverse elements.

\underline{Inv} ∀x ∈ S. ∃x̄ ∈ S. x ∘ x̄ = i ∧ x̄ ∘ x = i.

Formalized ∀x∃x̄. Op (x, x̄, i) ∧ Op (x̄, x, i).

for (ℤ, +) the inverse of x is −x

for (ℤ, ∗) there are no inverses; it is not a group

for (ℤ_{3}, ∗) the inverse of 1 is 1, and for 2 it is 2; but 0 doesn’t have an inverse

c) Rings. There is more to ℤ than being a group – this comes from the interplay of two operations, + and ∗. (ℤ, +) is a group, (ℤ, ∗) is a monoid, both are commutative (a + b = b + a; a ∗ b = b ∗ a).

If we put that together, we know that (ℤ, +, ∗) satisfies the following axioms:

+ ∗
Assoc A_{2} M_{2}
Comm A_{1} M_{1}
Id  A_{3} M_{3}
Inv A_{4} —

(The names A_{1} − A_{4}, M_{1} − M_{4} refer to the handout.)

And there is one more axiom about the combination of + and ∗:

∀xyz. x ∗ (y + z) = x ∗ y + x ∗ z Distrib (or “D” in the handout)

If we put these together we get what is called a \underline{ring}. Actually for being a ring, it suffices to have A_{1} − A_{4}, M_{2} and D.

Because of: M_{1} (ℤ, +, ∗) is a commutative ring
M_{3} it is a ring with unity

And even more, we know that there is \underline{no} divisor of zero. If x ∗ y = 0 then either x = 0 or y = 0. A ring that has this property,

∀xy. x ∗ y = i_{+} ⊃ x = i_{+} ∨ y = i_{+} (Z)

and M_{1} and M_{3}, is called an integral domain. However, since (ℤ, +, ∗) does not satisfy M_{4} (Inv for ∗) it is not a \underline{field}.

d) Fields.

M_{4} : ∀x. x ≠ i_{+} ⊃ ∃ x̄ ∈ S. x ∗ x̄ = i_{∗} ∧ x̄ ∗ x = i_{∗}

An example for a domain with two operations that is a field is:

(ℚ, +, ∗) with x̄ ≡ \frac{1}{x}

Notes from Professor Constable’s half of lecture:

Connection between logic and boolean rings (relate t, f, &, ∨, ⊃ to ring operations). \underline{Boolean ring}: ring + idempotence x · x = x \underline{note:} this is not a boolean algebra Formalization of basic math is first-order logic

a) from algebraic and ordered structures to natural numbers
b) compare to Peano Arithmetic in Stewart & Tall (assignment next week)

Which axioms do we have to add to uniquely describe ℤ (up to isomorphism)? I.e., what characteristizes ℤ, beyond being an integral domain?

• Ordering (discrete linear order!) connected to + and ∗

• Define predicate \underline{positive} P os(Z), then a < b is P (b − a)

\begin{aligned}P_{1} & ∀xy. P os(x) ∧ P os(y) & ⊃ & P os(x + y) \\ P_{2} & ∀xy. P os(x) ∧ P os(y) & ⊃ & P os(x ∗ y) \\ P_{3} & ∀x. P os(x) ∨ P os(−x) & ∨ & x = 0 \end{aligned}

• Prove the basic laws of <, +, ∗ (monotonicity, strict order, etc.)

e.g. ∀xyz. x < y ∧ y < z > x < z.

\underline{Proof}: we know P os(y − x) ∧ P os(z − y) show P os(z − x) using P_{1} and the ring axioms.

• \underline{But} we need more than just ordered integral domains to characterize ℤ (ℤ_{4} satisfies all these axioms), i.e \underline{induction principle}:

Q(0) ∧ (∀x > 0. Q > Q(x + 1)) ∧ (∀x. x < 0. Q(x + 1) > Q(x)) > ∀x. Qx

ℤ is \underline{the} ordered integral domain with induction. Define Nat(x) ≡ P os(x) ∨ x = 0.