1. Some sets of laws

[1] Defining laws for groups: A group is an algebra ...satisfying the laws ...

[2] Defining laws for lattices: A lattice is an algebra ...satisfying the laws ...

[3] Defining laws for relation algebras: A relation algebra is an algebra $\langle R; \vee, \wedge, 0,1, ', \circ, \ ^ {\cup}, \Delta \rangle$ such that

(i) $\langle R; \vee, \wedge, 0, 1 \rangle$ is a Boolean algebra;

(ii) $\circ$ is associative;

(iii) $\Delta \circ x = x \circ \Delta = x$;

(iv) ${\ ^ {\cup}}$ is a Boolean automorphism, $x \ ^ {\cup}\ ^ {\cup} = x$, and $(x \circ y) \ ^ {\cup}= y \ ^ {\cup}\circ x \ ^ {\cup}$;

(v) $(x \circ y ) \wedge z \leq x \circ (y \wedge ( x \ ^ {\cup} \circ z ))$.

[4] Defining laws for Heyting algebras: A Heyting algebra is an algebra $\langle H; \vee, \wedge, \rightarrow , 0 \rangle$ such that

(i) $\langle H; \vee, \wedge, 0 \rangle$ is a lattice with 0;

(ii) $x \wedge (x \rightarrow y) = x \wedge y$;

(iii) $x \wedge (y \rightarrow z ) = x \wedge ((x \wedge y) \rightarrow (x \wedge z))$;

(iv) $z \wedge ((x \wedge y) \rightarrow x) = z$.

[5] Defining laws for implication algebras: An implication algebra is an algebra $\langle A; \rightarrow \rangle$ such that

(i) $(x\rightarrow y)\rightarrow x = x$;

(ii) $(x \rightarrow y) \rightarrow y = (y \rightarrow x ) \rightarrow x$;

(iii) $x \rightarrow (y \rightarrow z ) = y \rightarrow (x \rightarrow z)$.

[6] Tarski's ``high-school identity problem'': Do these laws imply all laws of $\langle \omega; x+y, xy, x ^ y, 1 \rangle$? This was solved; the answer is negative.

$x+y = y+x$	$xy=yx$	$x\!+\!(y\!+\!z)=(x\!+\!y)\!+\!z$	$x(yz)=(xy)z$
$x(y+z)=xy+xz$	$x ^ {y+z} = x ^ y x ^ z$	$(xy)^ z = x ^ z y ^ z$	$(x ^ {y}) ^ z = x ^ {(yz)}$
$x \cdot 1 = x$	$x ^ 1 = x$	$1 ^ x = 1$

[7] Robbins' Problem: Do these laws define Boolean algebras? The answer is ``yes''; the proof was found by computer in 1996.

(i) $\vee$ is commutative;

(ii) $\vee$ is associative;

(iii) $((x \vee y)' \vee (x \vee y')')' = x$.