Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
Survey 1979: Equational Logic - Department of Mathematics ...
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16 EQUATIONAL LOGIC<br />
(Martin [286, page 56]). (Here co= { 0,1,2,...} and xY denotes ordinary<br />
exponentiation with 00 = 1.) The pro<strong>of</strong> is surprisingly long. Cf. 9.20 below.<br />
8.11. The algebra (co,xy,x y) is generic for the laws<br />
(Martin [286, page 78] ).<br />
xy = yx<br />
(xy)z = x(yz)<br />
(xy)Z = xZy z<br />
(xY)Z = xYZ<br />
8.12. The algebra (FZ,+) is generic for the laws<br />
(x+y) + z = x + (y+z)<br />
x+y+x+y=y+x+x+y<br />
x+y+z+x+y=y+x+z+x+y<br />
x+y+x+z+y=y+x+x+z+y<br />
x+y+z+x+w+y=y+x+z+x+w+y<br />
(J. Karn<strong>of</strong>sky [unpublished] - see [286, page 31]). Here + denotes addition on the<br />
class FZ <strong>of</strong> all ordinals - the algebra (FZ,+) could be replaced by a countable one.<br />
8.13. The algebras (co;An)n>3 and (co;0n)n>4 are each generic for the variety <strong>of</strong><br />
all algebras <strong>of</strong> type (2,2,2,..) (i.e., for Z0 = 0). (Martin [286, page 131, page 134] .)<br />
Here A n (n > 3)are the Ackermann operations beyond exponentiation, and the O n are<br />
some related operations invented by Doner and Tarski [110], who conjectured a<br />
somewhat stronger statement.<br />
algebras?<br />
PROBLEM 1. Do these three equations form an axiom base for Boolean<br />
xvy=yvx<br />
xv(yvz)=(xvy)vz<br />
((x v y)'v (x v y')')'= x.<br />
(See [354] for a history <strong>of</strong> this problem.) All finite models <strong>of</strong> these equations are<br />
Boolean algebras.<br />
PROBLEM 2. Do the following eleven equations form an axiom base for<br />
(co,1 ,x+y,xy,xY)?<br />
1. x+y=y+x