1991 MTC Qual Questions

Vaughan Pratt

1. (i) Show that a function f : B_{1} → B_{2} between two Boolean algebras which preserves binary meet, binary join, 0, and 1 (f(x ∧ y) = f(x) ∧ f(y), f(x ∨ y) = f(x) ∨ f(y), f(0) = 0, and f(1) = 1), also preserves complement. (Hint: consider f(x) ∧ f(¬x).)

(ii) Show that for any two monoids L, M the set of monoid homomorphisms f : L → M forms a monoid under the operation f ∘ g defined by (f ∘ g)(x) = f(x) ∘ g(x).

(iii) Show that for any lattices L, M the set of lattice homomorphisms f : L → M forms a lattice when ordered so that f ≤ g iff ∀x ∈ L[f(x) ≤ g(x)].

2. (i) Let C be any category. Explain the difference between a binary operation f : ob(C)^{2} → ob(C) and a functor F : C^{2} → C.

(ii) Define coproduct + : ob(C)^{2} → ob(C) as a binary operation mapping pairs of objects of a category C to objects of C. What is this operation when C is the category Set of sets and their functions?

(iii) Describe the extension of the coproduct operation of (ii) to a functor + : C^{2} → C. What is this extension when C is the category Set of sets and their functions?

(iv) A pointed set is an algebra with one constant c and no other operations. In elementary terms (i.e. using the language of set theory rather than category theory), define a binary operation X + Y on pointed sets X and Y such that X + Y is a product of X and Y. Do the same for coproduct.