Survey 1979: Equational Logic - Department of Mathematics ...

WALTER TAYLOR 11
One might also consider Albert's deduction [5] of full power-associativity of
"algebras" (multiplicative vector spaces) over fields of characteristic 4:2,3,5 from the
laws xy = yx and (x2x)x = x2x 2.
7. Equivalent varieties. We mention two of the many possible ways of
axiomatizing group theory equationally (not to mention non-equational forms such as
"for all x there exists y (xy = e)").
Fl: x(yz) = (xy)z
u.(xx-1) = (y-l.y).u = u
?2:(xY)z=x(Yz) ex=xe=x
x/x=e x/y=x(e/y)
u(e/u) = (e/u)u = e,
(where / denotes "division"). Clearly Pl and P2 do not define the same variety, for
they are of different types - (2,1) and (2,2,0). But examination of the models of P l
and the models of P2 will convince one that there is no essential difference between a
non-empty Pl-group and a non-empty 1`2-group. To make this sameness precise we
introduce equations which will serve as definitions:
Now one may check that
Al:x/y=x-y -1
e = x.x-1
A2: x -1 = e/x.
(*) F1, z51 I- I' 2 and P2,z52 I- P 1.
One more point is important. If we take one of the A 1 definitions of an operation F,
i.e. F = c, and substitute into c all the A 2 definitions, we get F = c[A2]; then one
should have
(**) I'21- F = c[A2] and likewise with the roles of Fi,P2; A 1,A 2 reversed.
(E.g. A 1 says x/y = x-y -1 . Upon substituting the A 2 definitions, we get x/y = x'(e/y),
and this is indeed provable from F2. ) Now generally, equational theories Pl,P2 are
said to be equivalent iff there exist sets of definitions A1,A 2 such that (*) and (**)
hold.
(There is one intrinsic difference: F 1 has an empty model, but I' 2 does not.
Nonetheless, 1'1 and 1'2 are generally regarded as equivalent. To this extent, empty