Assignment 7 & 8
Due Thursday, November 15, 2012
Recall that a lter of a lattice L = (L, , ) is a nonempty subset F L which is upward-closed and
closed under . If T is a set, let LP(T ) denote the (distributive) lattice (P(T ), , ) of
Due Thursday, November 1, 2012
Denition. A Heyting algebra is an algebra H = H, , , , 0, 1 of type 2, 2, 2, 0, 0 such that
H, , , 0, 1 is a bounded distributive lattice;
H satises the identities
(x y) y y
x (x y) x y
Due Thursday, October 25, 2012
Denition. A semilattice is an algebra A = A, of type 2 whose binary operation is commutative,
associative and idempotent.
Denition. A Boolean algebra is an algebra B = B, , , , 0, 1 of type 2
Due Thursday, October 4, 2012
1. Let G = G, , 1 , e be a group.
(a) If N
G, dene N = cfw_(a, b) G2 : b1 a N . Prove that N Con G.
(b) If Con G, dene N = e/. Prove that N
(c) Prove that the maps N N and N are mutually in
ASSIGNMENT 5: SOLUTIONS TO HS = SH
Solution #1 (David). Let A = (A, +A , f A ) where A = cfw_0, 1, 2, 3, e, f A : A A
is given by f A (x) = x + 1 (mod 4) for x cfw_0, 1, 2, 3 and f A (e) = e, and +A is the
binary operation given by
0 if (x, y) = (e, 0)
Due Thursday, October 18, 2012
Denition. If A is an algebra and 1 , 2 Con A, we say that 1 , 2 permute if 1 2 = 2 1 . We say
that A is congruence permutable if every pair of congruences of A permutes.
The rst problem is wo
Due Thursday, Sept 27, 2012
In this assigment you will need to use
Zorns Lemma. Let P = (P, ) be a poset. Suppose that every chain (i.e., linearly ordered subset) in
P has an upper bound in P. Then P has at least one maxim
Due Thursday, Sept 20, 2012
1. Show that the idempotent laws (id ), (id ) follow from the other six lattice axioms.
2. If P and Q are posets, let QP be the poset of order-preserving maps from P to Q, where for f, g QP
Due Thursday, November 29, 2012
1. In this problem you will prove that every algebra can be embedded in an ultraproduct of its nitely
generated subalgebras. Fix an algebra A. Let T be the set of all nonempty nite subsets o