5. Problems

Problem U-1. For each of these algebras K, find (i) a 1-variable law of the algebra that does not hold in all algebras of the same type, and (ii) (if you can) a law in 2 or more variables that is not an obvious consequence of a 1-variable law of the algebra. No proofs are required.

(a) Perkins' semigroup;

(b) Murskii's 1-binary algebra;

(c) Shallon's graph algebra [note: the operation is idempotent];

(d) the permutation group $S _ 3$.

(e) the tournament (8).

(A tournament is a directed graph in which every two vertices are joined by a single edge oriented one way or the other. It can be envisioned as a record of who won each match in a ``round-robin'' tournament, where each player has played every other player once--the arrow points towards the player who won. A tournament can be made into an algebra by letting $x \vee y$ be the winner and $x \wedge y$ the loser of the game between $x$ and $y$.)

Problem U-2. For the 1-unary algebra $\langle A; f \rangle$ of Example (7), find its equational theory (the set of all laws that hold). You'll need to consider the possibilities $f ^ n (x) = f ^ m (y)$ and $f ^ n (x) = f ^ m (x)$ ( $m \geq n \geq 0$). Sketch your reasoning.

Problem U-3. For the two-element group $C _ 2 = \{e,a\}$, invent a procedure for telling whether a given group law holds in $C _ 2$. (For example, $((xy)z ^ {-1}) ^ {-1} = x ^ {-1} (zy)$?)

Problem U-4. For each of the algebras of examples (4)(for R), (6), (7), (8), (10), (12), (15), (16), (17) in §, comment on its subalgebras. If there are just a couple, say what they are; if there are many, either describe them all or describe a typical one. No proofs are required.

Problem U-5. Of the binary operations involved in the examples from §, list those that are not commutative.